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    <title>web.youwen.dev</title>
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    <link href="https://web.youwen.dev" />
    <id>https://web.youwen.dev/atom.xml</id>
    <author>
        <name>Youwen Wu</name>
        
        <email>youwen@functor.systems</email>
        
    </author>
    <updated>2026-08-03T00:00:00Z</updated>
    <entry>
    <title>CogSec</title>
    <link href="https://web.youwen.dev/misc/cogsec" />
    <id>https://web.youwen.dev/misc/cogsec</id>
    <published>2026-08-03</published>
    <updated>2026-08-03T00:00:00Z</updated>
    <summary type="html"><![CDATA[    <p>With the advent of models with high cyber capabilities like Mythos, the tech sphere has been increasingly concerned with CyberSec and OpSec.</p>
    <p>One other threat vector that such models greatly amplify, which is perhaps more insidious, is towards <em>CogSec</em>—cognitive security.</p>
    <p>Information hazards, or <em>infohazards</em>, have been discussed at length in <a class="text-link external-link" href="https://scp-db.fandom.com/wiki/Infohazard" target="_blank">sci-fi</a>. However, they increasingly exist on the real-life internet. In sci-fi, this is mere <em>information</em>, e.g. memes, images, text, etc. which affect the consumer in ways ranging from amnesia to instant brain death. In real life, infohazards manifest as low-entropy slop disguised as high-entropy information worth consuming. Such slop is emitted either by LLMs, or <em>proids</em><a id="loc-1" href="#loc-3" role="doc-noteref"><sup>1</sup></a>.</p>
    <p>Examples:</p>
    <ul>
      <li>Every LinkedIn post</li>
      <li>90% of posts on AI Twitter</li>
      <li>YouTube comments</li>
      <li>Probably a lot of YouTube videos</li>
      <li>Fox News</li>
      <li>(Most) short-form video</li>
      <li>Many LLM outputs, even from frontier models, <em>even</em> if they are factually correct, due to slop writing and slop framing. Also, even when models aren’t slop, slop users direct them to produce slop which they plaster over the internet</li>
      <li>etc.</li>
    </ul>
    <p>Infohazards gunk up valuable information processing channels. I postulate the following:</p>
    <div class="py-2 px-4 text-[0.8em] rounded-md border-1 border-slate-200 dark:border-zinc-700 bg-slate-50 dark:bg-overlay leading-[1.5em]">
      <div><span style="font-variant-caps: all-small-caps">Conjecture</span></div>
      <div class="!mb-0 mt-2 prose-p:my-1 prose-p:mt-1">Regularly consuming low-entropy information degrades cognitive functions over time, in a roughly power law relationship.</div>
    </div>
    <p>Infohazards of this sort have probably existed since the printing press, but the dual proliferation of the internet and LLMs means infohazards probably now outnumber <em>infogems</em><a id="loc-2" href="#loc-4" role="doc-noteref"><sup>2</sup></a> in the average person’s information channels.</p>
    <p>For the sake of your CogSec, you must shut out as many infohazards as possible. I truly believe that unless the sci-fi scenarios of AI doom or utopia play out in the next decade, the most important thing one can do is prevent their brain from melting into soup under a torrent of infohazardous slop. Those retaining higher-order brain functionality will form the true overclass.</p>
    <p>AGI is here, and you still don’t have a CogSec threat model?</p>
    <section role="doc-endnotes">
      <ol style="list-style-type: none">
        <li id="loc-3"><a href="#loc-1" role="doc-backlink"><sup>1</sup></a>A human sufficiently low-entropy to be indistinguishable from a prompt.</li>
        <li id="loc-4"><a href="#loc-2" role="doc-backlink"><sup>2</sup></a><a class="text-link external-link" href="https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTFZk9D52CNswdvU-W5EnD8Kvg6DvTT7JMPljV3hPDMEg&amp;s=10" target="_blank">Gem alert.</a></li>
      </ol>
    </section>
]]></summary>
</entry>
<entry>
    <title>Transferring</title>
    <link href="https://web.youwen.dev/misc/transferring" />
    <id>https://web.youwen.dev/misc/transferring</id>
    <published>2026-05-15</published>
    <updated>2026-05-15T00:00:00Z</updated>
    <summary type="html"><![CDATA[    <p>As some readers of this blog may know, I’m transferring from <a class="text-link external-link" href="https://ucsb.edu/" target="_blank">UC Santa Barbara</a> to <a class="text-link external-link" href="https://www.berkeley.edu/" target="_blank">UC Berkeley</a>.</p>
    <p>I’ve been asked about this way too many times, by other transfer students, friends, and (unsurprisingly) parents unsatisfied with their kids’ college results, with questions along the lines of “why?” and “how?” This undoubtedly stems from the fact that 95% of Berkeley transfer admits come from community colleges so there doesn’t exist much information on gaining transfer admission from another University of California (UC) campus.</p>
    <p>I’ve texted/emailed people an array of the same scattered resources and thoughts. As such, I’m going to collect everything I know here in one place, so I can just send this webpage to people in the future. I’ll briefly outline why I decided to transfer, but if you don’t care, then just skip to the <span style="font-variant-caps: small-caps"><a href="#how"><span class="inline-block smallcaps">How?</span></a></span> section.</p>
    <p>In addition to some general advice at the end, I’ll also discuss my thoughts on whether you should attempt to transfer between UC campuses. If you are actually considering attempting transferring between UCs, you should read there.</p>
    <p>This post will focus primarily on transferring up to UC Berkeley and UCLA, which have substantially higher standards. I was also admitted to UC San Diego, which seems fairly easy to get into as a transfer if you have a high GPA (and does not consider application essays at all, as of the time of writing!) This article will mostly apply to those attempting to transfer to a lesser ranked UC (≤ UCSD) too, but the admission standards will be lower and the likelihood of success substantially greater. (Anecdotally, it seems everyone I know who tried transferring from UCSB to UCSD got in.)</p>
    <p>Lastly, I will note that it seems your source school doesn’t really matter. That is, it doesn’t seem to matter whether you are at UC Merced or UCSB. The only caveat is that it seems to be more difficult to go between UCLA <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 1em; height: 0.683em;" viewBox="0 0 11 7.513000000000001" width="11pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g56DE1EF8E5523E602C0C258B8C8717C0" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g56DE1EF8E5523E602C0C258B8C8717C0" overflow="visible"><path d="M 0 0m 10.241 2.574 c 0.076999664 0.032999992 0.120999336 0.099000216 0.120999336 0.17600012 c 0 0.0769999 -0.043999672 0.14299989 -0.120999336 0.17599988 c -0.5279999 0.17600012 -1.0229998 0.5500002 -1.4850006 1.122 c -0.3079996 0.38500023 -0.51699924 0.83599997 -0.6159992 1.3530002 c -0.032999992 0.14300013 -0.12100029 0.20900011 -0.26400042 0.20900011 c -0.17600012 0 -0.26399994 -0.0880003 -0.26399994 -0.26399994 v -0.032999992 l 0.011000156 -0.022000313 c 0.1869998 -0.9460001 0.6709995 -1.7049999 1.474 -2.277 h -7.1940002 c 0.8030001 0.572 1.2870001 1.3309999 1.4740001 2.277 l 0.010999918 0.022000313 v 0.032999992 c 0 0.17599964 -0.08800006 0.26399994 -0.26399994 0.26399994 c -0.14300013 0 -0.23100019 -0.065999985 -0.26400018 -0.20900011 c -0.09899998 -0.5170002 -0.30799985 -0.96799994 -0.61599994 -1.3530002 c -0.462 -0.5719998 -0.957 -0.94599986 -1.4849999 -1.122 c -0.07700002 -0.032999992 -0.12099999 -0.09899998 -0.12099999 -0.17599988 c 0 -0.0769999 0.04399997 -0.14300013 0.12099999 -0.17600012 c 0.52799994 -0.17599988 1.023 -0.54999995 1.4849999 -1.1219999 c 0.3080001 -0.385 0.51699996 -0.836 0.61599994 -1.353 c 0.032999992 -0.143 0.12100005 -0.20899999 0.26400018 -0.20899999 c 0.17599988 0 0.26399994 0.088 0.26399994 0.264 v 0.033000007 l -0.010999918 0.021999985 c -0.18700004 0.946 -0.671 1.705 -1.4740001 2.277 h 7.1940002 c -0.80300045 -0.572 -1.2870002 -1.3310001 -1.474 -2.277 l -0.011000156 -0.021999985 v -0.033000007 c 0 -0.176 0.08799982 -0.264 0.26399994 -0.264 c 0.14300013 0 0.23100042 0.066 0.26400042 0.20899999 c 0.09899998 0.517 0.3079996 0.96800005 0.6159992 1.353 c 0.46200085 0.5719999 0.95700073 0.946 1.4850006 1.1219999 Z "/></symbol></defs></svg></span> UC Berkeley.</p>
    <p>Please note the table of contents below consists of clickable sections.</p>
    <nav role="doc-toc">
      <h2>Contents</h2>
      <ol style="list-style-type: none">
        <li><a href="#loc-1">Why?</a></li>
        <li>
          <div><a href="#how">How?</a></div>
          <ol style="list-style-type: none">
            <li><a href="#loc-3">My application</a></li>
            <li><a href="#loc-4">General requirements</a></li>
            <li><a href="#loc-5">UC-UC specific requirements and remarks</a></li>
          </ol>
        </li>
        <li>
          <div><a href="#loc-6">Advice</a></div>
          <ol style="list-style-type: none">
            <li><a href="#resources">Useful resources</a></li>
            <li><a href="#loc-7">Should I attempt an intercampus transfer?</a></li>
            <li><a href="#loc-8">Miscellaneous advice</a></li>
          </ol>
        </li>
      </ol>
    </nav>
    <h2 id="loc-1">Why?</h2>
    <p>There were plenty of reasons to transfer, and also anti-reasons. I’m a math and CS double major, so if that is not you, then feel free to cherry-pick and apply only the relevant remarks.</p>
    <p>Firstly, the UCSB math department, wherein I took the vast majority of my classes, is fantastic and will afford you the latitude to basically do whatever you want.</p>
    <p>Likewise, the CS department supports its students well and (at least in my opinion) there is no shortage of interesting research to work on (the <a class="text-link external-link" href="https://www.arch.cs.ucsb.edu/" target="_blank">Arch Lab</a>, makers of PyRTL and formal verification hardware quickly comes to mind). And there are wonderful campus organizations like <a class="text-link external-link" href="https://www.ucsbacm.com/" target="_blank">ACM at UCSB</a> and (my very own) <a class="text-link external-link" href="https://sbhacks.com/" target="_blank">SB Hacks</a>.</p>
    <p>So in summary, I don’t really think I was being held back by UCSB, and I applied to UC Berkeley (and UCLA, and a few other top schools) back in the September 2025 application filing period mostly on a whim, since one of my top school choices <em>was</em> Berkeley back in high school.</p>
    <p>Ultimately, the main reason I transferred was due to the housing.<a id="loc-2" href="#loc-9" role="doc-noteref"><sup class="font-index text-iris text-[0.83em] hover:bg-love/15 transition-colors] ml-0.5 [vertical-align: baseline] relative -top-[0.33em]">1</sup></a></p>
    <p>You see, there’s something of a critical housing crisis in Isla Vista (the unincorporated community that serves as UCSB’s college town), and after various fiascos this year with the lease I signed for an off-campus apartment, I really didn’t want to deal with finding a new roommate and signing a new lease for the next year.</p>
    <p>So I applied for the cheap and affordable university-owned apartments back in January. Splendid!</p>
    <p>Come February, I received the following in my mail:</p>
    <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">Unfortunately, we are unable to offer you a space in UCSB campus housing.</blockquote>
    <p>Such is the woeful tale of many a UCSB underclassman. At this point I was at my wits end and I decided it was probably less annoying to just get into UC Berkeley and get guaranteed housing there than figure out off-campus housing at UCSB. I therefore made my peace with UCSB and patiently awaited my application decision in mid-May. So long and thanks for all the fish!</p>
    <h2 id="how">How?</h2>
    <p>I applied to UC Berkeley and UCLA in the 2026 Fall application cycle, although I actually became ineligible for UCLA due to excess upper division unit restrictions, so Berkeley was my only actual option.</p>
    <div class="py-2 px-4 text-[0.8em] rounded-md border-1 border-slate-200 dark:border-zinc-700 bg-slate-50 dark:bg-overlay leading-[1.5em]">
      <div><span style="font-variant-caps: all-small-caps"><span class="inline-block all-smallcaps">By the way</span></span></div>
      <div class="!mb-0 mt-2 prose-p:my-1 prose-p:mt-1">
        <p>Nomenclature: I use CC to mean Community College, and CCC is California Community College. UC-UC transfer means applying to another UC as a transfer student when you’re already at one of the UCs.</p>
        <p>In case you didn’t already know, UCs give heavy priority to UC transfer applicants from a CCC. Therefore if you apply as an applicant attending any out-of-state community or 4 year college, or an in-state applicant from any non-CCC (e.g. UC, CSU, other 4-year), you are at a heavy disadvantage. In particular, over 95% of transfer admits to Berkeley last year were from CCCs, leaving just 5% of the admit pool for <em>everyone</em> else.</p>
        <p>That means the 25% or so admit rate for Berkeley transfers is <em>only</em> for CCC applicants. As a UC-UC applicant, for instance, you can expect <strong>an admit rate of around 6-8%</strong> (per <a class="text-link external-link" href="https://www.universityofcalifornia.edu/about-us/information-center/transfer-admissions-summary" target="_blank">this website</a>). Yes, this is actually <strong>worse than the 1st year admit rate</strong> from high school.</p>
        <p><strong>Therefore if your goal is to maximize your odds of getting into Berkeley or UCLA it is highly recommended to attempt it from a CCC.</strong></p>
      </div>
    </div>
    <p>With that out of the way, I will outline some initial context: I applied under the regular transfer student application. At the time I applied I was a 2nd year. This is the standard time to apply, although you can apply in your 1st year if you have enough units <em>and</em> you feel you’ve a strong enough application (elaborated on below).</p>
    <p>Note, the transfer application is nearly identical to the first-year one, except for a single PIQ prompt.</p>
    <h3 id="loc-3">My application</h3>
    <p>What made my application strong? Briefly, this is what I conjecture, in order of importance:</p>
    <ul>
      <li>I had a 3.94 GPA. In my opinion, this is not actually that impressive by itself but I had earned this GPA even after taking 24 units of upper division, proof-based math classes, including Abstract Algebra and Real Analysis, the hardest classes in the pure mathematics major (by extension, some of the hardest STEM classes altogether, in my opinion). In terms of transfer admissions, the most important factor seems to be academic preparation, even moreso than high school admissions, so earning as close to a 4.0 as possible while demonstrating readiness for your selected major is key. The fact that I had basically finished 60% of my major before transferring was probably not unhelpful.</li>
      <li>I won various hackathon awards. Again, I don’t think this is very impressive excepting the fact that one of the awards was 1st place at HackMIT, a fairly prestigious annual hackathon at MIT. You can read all about that experience in <a class="text-link internal-link" href="/writing/postscript-on-hackmit/">this blog post I wrote.</a></li>
      <li>I contributed a good deal to organizing SB Hacks, UCSB’s largest hackathon, and I ended up in a high level leadership role. Obviously this demonstrates leadership experience, initiative, etc.</li>
      <li>I had a lot of personal projects which you may find traces of all over this website.</li>
      <li>I earned a couple merit scholarships from UCSB.</li>
    </ul>
    <p>As always, when giving any college application advice, I would be remiss not to recommend <a class="text-link external-link" href="https://mitadmissions.org/blogs/entry/applying_sideways/" target="_blank">Applying Sideways</a> from the director of MIT Admissions. Do <em>not</em> treat the above list as a checklist of things you yourself should do.</p>
    <h3 id="loc-4">General requirements</h3>
    <p>The information that follows was (to the best of my knowledge) accurate for the 2025-2026 application cycle, and I presume will be fairly accurate through the 2026-2027 cycle. If you are reading this in a time beyond that, please check any important info yourself as the UCs update requirements and procedures frequently. I actually wrote the following as a response to someone’s question about transferring UC-UC and then pasted it here near verbatim so apologies in advance for any awkward phrasing.</p>
    <p>Firstly, I will be as clear as possible about what a transfer student is.</p>
    <p>Essentially, you may apply as a transfer as soon as you have been enrolled in some college for more than a term after high school (dual enrollment doesn’t count). However, <strong>not all transfers are equal</strong>.</p>
    <p>Broadly speaking, the UCs classify transfer applicants based on their <strong>class level</strong> and <strong>the institution that they are transferring from</strong>.</p>
    <p>Firstly, class level. There are sophomore-level, junior-level, and senior-level transfers. The nomenclature works like this: your “transfer level” is the class level you will have <em>when you enroll, if admitted</em>. So for instance, if you apply to UCLA’s transfer class of Fall 2026, from Santa Monica Community College (SMCC), the application is actually due end-of-year 2025, and you would <em>start</em> at UCLA in the fall term of 2026, if admitted. Your class level for transfer is not determined by how many units you have at the time of application, but by how many units you <em>expect to have</em> in your last term before Fall 2026, which would be the spring term at Santa Monica Community College in this case. So, if by the end of the 2025-2026 school year, that is, at the end of the Spring 2026 SMCC term, transferring to UCLA, you will have enough credits to be classified as a junior by UCLA, then you are a <em>junior-level</em> transfer, <em>even if you may have been a sophomore level student back in 2025 when you were applying</em>.</p>
    <p>Your class level is not based on how much <em>time</em> you’ve spent in college (that is, how many terms), but rather solely based on how many transferable units you’ll have before matriculating at the new institution. The threshold is typically</p>
    <ul>
      <li>45 quarter units = sophomore-level</li>
      <li>90 quarter units = junior-level</li>
      <li>120 or 135 quarter units = senior-level (depends on institution)</li>
    </ul>
    <p>Note these are <strong>quarter</strong> units. Semester units are typically converted to quarter units at a ×1.5 conversion factor. When in doubt, refer to the counselor guide, which is provided below in the <a href="#resources"><span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">Useful resources</span></span></a> section. Do not rely on the figures above.</p>
    <p>In general, the UCs will <em>only</em> accept junior-level transfers. This means you need between 90–120 units, but not any more or less. Note that typically AP credits are counted <em>towards</em> the unit baseline (thus helping you reach junior-level), but will <em>not</em> be used to count for the unit limit (that is, AP credits won’t push you over the 120 unit senior-level cap that makes you ineligible for admission). This is great for those with lots of AP credits, since they won’t be at risk of becoming ineligible when transferring in their second year of university.</p>
    <p>This class-level calculation is why a 1st year can apply for admission (and many do), given that they have sufficient units from prior dual enrollment or AP exams. When I say “one-year” or “first-year” transfer, I mean someone who applies in their first year of college, and they would be admitted to start in the fall of their 2nd year. A “two-year” or “second-year” transfer is someone who applies in their 2nd year of college, and would start in the fall of their 3rd year.</p>
    <p>Two-year transfers are a lot more typical. However, one-year transfers are certainly possible, although UC-UC one-year transfers seem exceedingly rare. I was a two-year transfer.</p>
    <p>However, if you are a student at a 4 year college, be sure not to obtain <em>too many units</em> and gain senior level standing, which makes you ineligible for transfer admission at most UCs. Therefore most people have at most two opportunities to apply, in their 1st and 2nd year of college. However, CCC applicants cannot have senior standing so they can technically stay in CC and apply as much as they want, although almost no one spends more than two years in practice. Also, one-year transfers will have high school grades taken into account, whereas 2nd year transfers will be evaluated only based on college coursework. So if your high school grades were subpar, a second year transfer will let you be seen from an entirely blank slate.</p>
    <p>As for filing period, the top UCs only accept applications for the standard Fall term start, i.e. you can only apply once per year during the regular filing period starting in August and closing in December (same as the first-year application). Therefore a one-year transfer must apply in the fall of their first year of college, almost immediately after they just started, while a second-year transfer would apply in their second fall term, after a full year of grades have been entered. Since the applications are usually filed before fall grades are finalized, applicants are asked to submit an academic update February listing their fall grades.</p>
    <p>So the bottom line is:</p>
    <ul>
      <li>You file your application similarly to the first-year application</li>
      <li>As a first-year transfer you have only a single fall term of grades (as well as high school and prior dual enrollment grades).</li>
      <li>As a second-year transfer you have an entire year’s worth and a fall term of grades, and dual enrollment grades, but <em>not</em> high school grades. Second-year transfer is more standard and generally leads to more favorable outcomes, especially with a weaker high school transcript.</li>
    </ul>
    <h3 id="loc-5">UC-UC specific requirements and remarks</h3>
    <p>The transfer process is similar in terms of requirements for UC-UC transfer and CCC transfers. You need 90 quarter units (= 60 semester units) to be considered a junior transfer, while you must have no more than 120/135 quarter units (UCLA / Berkeley respectively), under which you would be considered a senior transfer and ineligible. For UCLA, they also require less than 39 upper-division units (at most UCs upper division means courses numbered with 3 digits, e.g. Math 117), which is why I became ineligible for admission. Berkeley has no such restrictions. Most people will not hit the upper division limit but if you’re taking a ton of classes in your major like me, it’s possible.</p>
    <p>There are also a few course requirements, namely, for Berkeley they require the “7-course pattern,” which includes some breadth classes across STEM/humanities, and writing requirements, while UCLA doesn’t impose any additional requirements since they consider all UC-UC applicants to be UC-eligible already based on your prior admission to a UC. It’s generally recommended you complete as much major prep as possible, I would recommend at least roughly trying to match the articulations on ASSIST. But since you are already at a UC I would probably expect you to be able to do a lot more major prep than a CC applicant so for me this wasn’t a concern at all. I finished the lower division major prep expected of CC applicants in my second quarter and I took an additional 9 upper division classes in my major since then..</p>
    <p>The application process is essentially the same for all transfer applicants, and pretty much identical to the first year except for one PIQ prompt. In my case, I just looked at some of the UC-UC transfer resources on this subreddit when I first enrolled and then just tried to take classes, get good grades, and get involved in campus activities or clubs, especially in my major. The accepted wisdom for transfer admissions (and official statements from Berkeley admissions corroborate this) is that grades are given a higher weight relatively speaking compared to HS admissions (they call this “Comprehensive Review,” where academics are most important but other factors are considered, as opposed to “Holistic Review” for HS admissions, where academics, ECs, essays, etc are all equally important). Therefore, it’s very important you obtain as close to a 4.0 as possible, although in my experience the classes at UCs, especially when you get more advanced, are much harder, so I would say hovering around a 3.9 is very strong (you can search around the UC website, there is a page where you can find the average admit GPA for transfer admits from each UC campus. It’s usually a bit lower than the average from CCs, for mine, it was around 3.8 for Berkeley). Please note that UC classes are probably at least a bit harder than your HS AP classes, so getting a high GPA is very much not a guarantee. In my classes, for instance, A/A+ grades are given to only around the top 20-30%, while the bad students who bring down curves are progressively filtered out, so it is actually quite hard to earn a 3.9/4.0. As an intercampus transfer it’s also helpful to come up with a bit of a compelling story for why you want/need to transfer (specific opportunities, geography, career goals, etc) but not <em>strictly</em> necessary.</p>
    <p>Coming from another UC, you’re disadvantaged compared to a CCC in terms of admission odds but you’re still much better off than a non-UC 4 year to UC transfer. All your credits will transfer (any credits earned at any UC will be transferred to any other UC). However, there is nothing like ASSIST which guarantees course articulation, but you can generally be sure that if you take standard classes like Linear Algebra, Calculus, Physics, they will be articulated. Also, there is a thing called UC Reciprocity wherein if you complete the general education requirements at your home UC and then transfer, the new campus will waive their general education requirements. This is essentially the same as or even stronger than IGETC for CCC.</p>
    <p>Moreover, for UCLA at least, I’ve heard from some reputable sources and also read in official material that UCLA in particular gives high priority for UC-UC transfers, so you’re not actually too disadvantaged compared to CCC transfers (acceptance rate is around 15-17% which is 10% lower than the 26% acceptance rate for CCC, and can be partially accounted for by other factors). For Berkeley, non-CCC is at a big disadvantage and the admit rate for non-CCC applicants is 6-8%, although this figure doesn’t distinguish between inter-UC and non-UC 4 years (presumably intercampus is a little higher, I would guess around 10-12%, similar to the 1st year admit rate).</p>
    <p>Being at a UC, you also have access to much more opportunities in terms of extracurriculars and advanced classes, so I would definitely suggest you take advantage of those. I basically took the hardest classes possible every quarter (which is why I’m exceeding upper division unit limits right now) and I’m in a few high level leadership roles in major campus orgs for my major, so even if transfer apps didn’t work out, I’m still in a very strong position for grad school / jobs. Being in this position is not really possible if you go the 2 year CC transfer route since most CCs, even ones like DVC or Santa Monica have much less opportunity and classes. At least in my personal experience, I started taking upper division classes 2nd quarter and by 3rd quarter of my freshman year not a single one of my classes was offered at <em>any</em> CCC in the entire state since they were junior/senior level. Upper division classes, by definition, are not supposed to be offered at any CCC. So I definitely would not have chosen to go to a CC, especially knowing what I know now.</p>
    <h2 id="loc-6">Advice</h2>
    <p>Here I will give some more resources and advice.</p>
    <h3 id="resources">Useful resources</h3>
    <p>There are a few resources that were most helpful when I was planning all of this out.</p>
    <ul>
      <li>The <a class="text-link external-link" href="https://admission.universityofcalifornia.edu/counselors/_files/documents/quick-reference-guide-to-uc-admissions.pdf" target="_blank">UC Admissions counselor’s guide</a>. Seriously, they literally tell you exactly what to do, almost all the non-anecdotal information above can be found in here. I highly recommend you read the transfer portions of this document in their entirety. Note that the link breaks frequently so you should google “quick reference guide to uc admissions” and look for the latest one for your relevant application cycle.</li>
      <li>This <a class="text-link external-link" href="https://www.reddit.com/r/ApplyingToCollege/comments/128zxgo/guide_to_transferring_from_a_uc_to_a_uc/" target="_blank">Reddit post</a>.</li>
      <li>If you like poring over asinine statistics, you can check out admit rates over time <a class="text-link external-link" href="https://www.universityofcalifornia.edu/about-us/information-center/transfer-admissions-summary" target="_blank">here</a>, and you’ll be able to find information about average GPAs by school <a class="text-link external-link" href="https://www.universityofcalifornia.edu/about-us/information-center/admissions-source-school" target="_blank">here.</a> This can help you feel either better or worse about your odds but in the end, remember that statistics are meaningless.</li>
      <li>Berkeley outlines its specific requirements <a class="text-link external-link" href="https://admissions.berkeley.edu/transfer-admissions-information-by-college/" target="_blank">here</a>, and also general requirements <a class="text-link external-link" href="https://admissions.berkeley.edu/apply-to-berkeley/transfer-students/" target="_blank">here.</a></li>
    </ul>
    <h3 id="loc-7">Should I attempt an intercampus transfer?</h3>
    <p>The most common advice counselors and people on the internet will give is not to attend another UC with the express purpose of transferring to a better one. There is a highly nontrivial chance you will end up disappointed with the outcome. They will tell you to attend a CCC instead if your goal is transferring.</p>
    <p><em>However</em>, I can offer a more nuanced perspective. If the financial aspect of attending CCC (i.e. money saved) is not a concern, then attending another UC and planning to transfer <em>may</em> be a good option.</p>
    <p>This is because you will still get a normal 1st year experience, and have access to all the opportunities of a research university. In my opinion there is absolutely no aspect of CC that is better than any UC. However, if you do decide to go this route, you must keep two things in mind.</p>
    <p>First, you should know that it is not very uncommon for disgruntled students to attend a lower ranked UC and vaguely plan on transferring out—but then after the first year will completely discard the idea. This is usually because they either underestimated how difficult UC coursework would be and were no longer competitive for admission, or simply decided it wasn’t worth the hassle to uproot their new college life which they enjoyed. Note that in the second scenario, this is not actually a bad thing since these people ended up happy at their UC! However, the first scenario is an unsatisfying outcome.</p>
    <p>Anecdotally, not a single person I know who planned to transfer was successful, usually because they didn’t even bother applying. Once again the acceptance rate is dismal, hovering at around 8%.</p>
    <p>If you are OK with this first point, then you should ask yourself whether you are actually OK with ending up remaining at the original UC for 4 years and obtaining your degree there, if transfer applications aren’t successful. Honestly, I feel like most people would be fine with this. But if there is any substantive doubt, you should go to CCC.</p>
    <p>If you accept that there is a high likelihood you will not be successful in transferring for myriad reasons, and you are also OK with not transferring out of the UC you attend, then, counterintuitively: I think you have a good case for attending a UC with the goal of UC-UC transfer, and <strong>you should seriously consider it.</strong></p>
    <h3 id="loc-8">Miscellaneous advice</h3>
    <ul>
      <li>Save your syllabi, it will be useful for course articulations.</li>
      <li>
        <p>I think as a motivated student you can take a 3.9+ unweighted GPA for granted in high school. The same does not go for at a UC. On the other hand, a 3.9+ GPA holds a lot more academic weight from a UC than from high school, so it does a lot more heavy lifting for your application.</p>
        <ul>
          <li>I do think CC is quite a bit easier than a UC. Especially if you take upper division classes in STEM at UC. Yet another reason why CCC is a better option if you intend on transferring unless you <em>know</em> you can keep really high grades.</li>
        </ul>
      </li>
      <li>In general clubs and extracurriculars at 4-year colleges are a lot more engaging than high school. A major factor is that most people are not here to leverage the club for college admissions. Therefore you might actually be able to get some interesting things done.</li>
      <li>As for essays, the general wisdom about UC essays applies, the same stuff as high school admissions. Or not, if you are really crazy, you can actually ignore all advice and do whatever you want. When I wrote my essays I basically just figured out what I wanted to talk about first and then matched them to prompts. I think my essays were a bit deranged and somewhat apropos to nothing at times. Have faith in yourself.</li>
    </ul>
    <section role="doc-endnotes">
      <ol style="list-style-type: none">
        <li id="loc-9"><a href="#loc-2" role="doc-backlink"><sup>1</sup></a>I’m getting mild <a class="text-link external-link" href="https://blog.evanchen.cc/transfer/#:~:text=Ultimately%2C%20the%20main%20reason%20I%20transferred%20was%20due%20to%20the%20housing" target="_blank">déjà vu…</a></li>
      </ol>
    </section>
]]></summary>
</entry>
<entry>
    <title>Universal constructions and the universal property of quotients</title>
    <link href="https://web.youwen.dev/math/universal-constructions" />
    <id>https://web.youwen.dev/math/universal-constructions</id>
    <published>2026-01-04</published>
    <updated>2026-01-04T00:00:00Z</updated>
    <summary type="html"><![CDATA[    <p>This is going to be a short introduction to the concept of how we may create a <em>universal construction</em> of a certain concept through defining a <em>universal property</em>. We’ll demonstrate by defining the <em>universal property of quotients</em>, to characterize the concept of taking a quotient, e.g. a quotient group in group theory or quotient space in linear algebra.</p>
    <h2>Notions</h2>
    <p>From <a class="text-link external-link" href="https://en.wikipedia.org/wiki/Universal_property" target="_blank">Wikipedia</a>:</p>
    <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">In mathematics, more specifically in category theory, a <strong>universal property</strong> is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field of their coefficients can all be done in terms of universal properties. In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property.</blockquote>
    <p>The whole idea is that when we want to do a construction in mathematics, the concrete details might be very messy (for example, for quotient groups, we may reach for the notions of <em>coset</em> or <em>fiber</em>), but any construction satisfying a given universal property is going to give an equivalent result. <em>All there is to know about the construction is already contained in the universal property</em>.</p>
    <h2>The universal morphism</h2>
    <p>One way of formulating the universal property is as follows. 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    <p>Then a <em>universal morphism</em> from <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 0.879em; height: 0.683em;" viewBox="0 0 9.669 7.513000000000001" width="9.669pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g7A8DDFE05440C530D4D3F2EDC19C377B" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g7A8DDFE05440C530D4D3F2EDC19C377B" overflow="visible"><path d="M 0 0m 9.174 7.513 c -0.18700027 0 -0.8359995 -0.032999992 -1.0229998 -0.032999992 c -0.22000027 0 -0.99000025 0.032999992 -1.21 0.032999992 c -0.16499996 0 -0.25300026 -0.08799982 -0.25300026 -0.26399994 c 0 -0.09899998 0.055000305 -0.15400028 0.17600012 -0.16499996 c 0.26399994 -0.032999992 0.3959999 -0.12100029 0.3959999 -0.28600025 c 0 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    </figure>
    <p>Here is a concrete example. Let <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 6.209111111111112em; height: 0.683em;" viewBox="0 0 68.30022222222223 7.513000000000001" width="68.30022222222223pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g93F710FCBC47C017329EE0E62637FE62" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 10.249555555555556 7.513000000000001)"><use xlink:href="#g1B387A2F9014F570A058F676AC9B160C" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 21.863111111111113 7.513000000000001)"><use xlink:href="#gA22004C89EE11B3A4F12BE4B6C4A7BFD" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 34.49966666666667 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Now the functor <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 7.200111111111112em; height: 0.683em;" viewBox="0 0 79.20122222222223 7.513000000000001" width="79.20122222222223pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#gAEA65B0F6D0EC66E9A115A307318509C" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 11.602555555555556 7.513000000000001)"><use xlink:href="#g3C74867A0B23EAA8D2A33B2EBDF4296C" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 17.716111111111115 7.513000000000001)"><use xlink:href="#g1BC3DF4E92D302D36DC039D739EF8809" x="0" y="0" fill="#000000" fill-rule="nonzero"/><use xlink:href="#g2E7E58C57982DD32ED1FD5FED19D4941" 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    <p>Now let <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 3.333455555555555em; height: 0.683em;" viewBox="0 0 36.668011111111106 7.513000000000001" width="36.668011111111106pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g7D8824C7D2BC9D6FDC59C635AC9A4FED" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 8.25 3.5200000000000005)"><use xlink:href="#g1D6B2CF355F1F6936C2544278DFAFD9" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 15.055455555555556 7.513000000000001)"><use xlink:href="#g1B387A2F9014F570A058F676AC9B160C" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 26.66901111111111 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0.43120027 Z "/></symbol><symbol id="g1B387A2F9014F570A058F676AC9B160C" overflow="visible"><path d="M 0 0m 7.678 4.037 h -6.798 c -0.176 0 -0.264 -0.0880003 -0.264 -0.25300026 c 0 -0.16499996 0.088 -0.25300002 0.264 -0.25300002 h 6.798 c 0.17600012 0 0.26399994 0.08800006 0.26399994 0.25300002 c 0 0.13199997 -0.12099981 0.25300026 -0.26399994 0.25300026 Z m 0 -2.0680003 h -6.798 c -0.176 0 -0.264 -0.08799994 -0.264 -0.25300002 c 0 -0.16499996 0.088 -0.25300002 0.264 -0.25300002 h 6.798 c 0.17600012 0 0.26399994 0.08800006 0.26399994 0.25300002 c 0 0.143 -0.12099981 0.25300002 -0.26399994 0.25300002 Z "/></symbol><symbol id="g129389D1F686C6700782CC8DE75EA3DE" overflow="visible"><path d="M 0 0m 9.691 7.348 c 0 0.11000013 -0.065999985 0.16499996 -0.19799995 0.16499996 l -1.3970003 -0.032999992 l -1.3969998 0.032999992 c -0.17600012 0 -0.25299978 -0.09899998 -0.25299978 -0.26399994 c 0 -0.07700014 0.032999992 -0.13199997 0.08799982 -0.14300013 c 0.11000013 -0.011000156 0.19799995 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0.26399994 0.088 0.26399994 0.264 c 0 0.110000014 -0.12100029 0.16499999 -0.35200024 0.16499999 c -0.45099974 0 -0.671 0.044 -0.671 0.14300004 c 0 0 0.011000156 0.032999992 0.032999992 0.176 l 1.4740005 5.8739996 c 0.05499935 0.23100042 0.15399933 0.37400007 0.29699993 0.42900038 c 0.065999985 0.021999836 0.2529993 0.032999992 0.57199955 0.032999992 c 0.28600025 0 0.41800022 0 0.41800022 0.26399994 Z "/></symbol></defs></svg></span>, where <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 0.9089999999999999em; height: 0.683em;" viewBox="0 0 9.998999999999999 7.513000000000001" width="9.998999999999999pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g129389D1F686C6700782CC8DE75EA3DE" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g129389D1F686C6700782CC8DE75EA3DE" overflow="visible"><path d="M 0 0m 9.691 7.348 c 0 0.11000013 -0.065999985 0.16499996 -0.19799995 0.16499996 l -1.3970003 -0.032999992 l -1.3969998 0.032999992 c -0.17600012 0 -0.25299978 -0.09899998 -0.25299978 -0.26399994 c 0 -0.07700014 0.032999992 -0.13199997 0.08799982 -0.14300013 c 0.11000013 -0.011000156 0.19799995 -0.021999836 0.25300026 -0.021999836 c 0.3409996 -0.011000156 0.5279999 -0.032999992 0.5829997 -0.04400015 c 0.055000305 -0.011000156 0.0880003 -0.04400015 0.0880003 -0.09899998 c -0.011000156 -0.032999992 -0.022000313 -0.09899998 -0.04400015 -0.19799995 l -0.64900017 -2.618 h -3.234 l 0.63800025 2.4969997 c 0.05499983 0.23100042 0.1539998 0.37400007 0.29699993 0.42900038 c 0.076999664 0.021999836 0.26399994 0.032999992 0.572 0.032999992 c 0.29699993 0 0.41799974 0 0.41799974 0.26399994 c 0 0.11000013 -0.065999985 0.16499996 -0.19799995 0.16499996 l -1.3969998 -0.032999992 l -1.408 0.032999992 c -0.17600012 0 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c -0.032999992 -0.16500002 -0.16499996 -0.27500004 -0.38499975 -0.30800003 c -0.05499983 -0.011000007 -0.2420001 -0.011000007 -0.5609999 -0.011000007 c -0.20900011 0 -0.3080001 -0.088 -0.3080001 -0.264 c 0 -0.10999999 0.065999985 -0.16499999 0.19799995 -0.16499999 l 1.3860002 0.033 l 0.69299984 -0.011 c 0.12099981 0 0.572 -0.022 0.71500015 -0.022 c 0.17599964 0 0.26399994 0.088 0.26399994 0.264 c 0 0.110000014 -0.12100029 0.16499999 -0.35200024 0.16499999 c -0.45099974 0 -0.671 0.044 -0.671 0.14300004 c 0 0 0.011000156 0.032999992 0.032999992 0.176 l 1.4740005 5.8739996 c 0.05499935 0.23100042 0.15399933 0.37400007 0.29699993 0.42900038 c 0.065999985 0.021999836 0.2529993 0.032999992 0.57199955 0.032999992 c 0.28600025 0 0.41800022 0 0.41800022 0.26399994 Z "/></symbol></defs></svg></span> is another group and let <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 4.6641111111111115em; height: 0.683em;" viewBox="0 0 51.30522222222223 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    <p>You may notice that this is a classic result about quotient groups in group theory, and part of the proof of the <a class="text-link external-link" href="https://en.wikipedia.org/wiki/Isomorphism_theorems" target="_blank">first isomorphism theorem</a>! 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The diagram above closely resembles this one which appears in many standard algebra textbooks (e.g. Dummit and Foote).</p>
    <figure class="math block-math">
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    </figure>
    <p>OK, so what? The point is that this universal property characterizes <em>any</em> construction of the quotient group <em>up to isomorphism</em>, which means any construction of a quotient that satisfies the property is essentially the same. In Dummit and Foote, quotients are formulated using <a class="text-link external-link" href="https://en.wikipedia.org/wiki/Fiber_(mathematics)" target="_blank">fibers</a>, while a more standard approach is to use <a class="text-link external-link" href="https://en.wikipedia.org/wiki/Coset" target="_blank">cosets</a>, but with both constructions we can show the universal property above, so both notions of quotient are equivalent, and the quotients they define are isomorphic.</p>
    <p>From another point of view, the universal property is the <em>definition itself</em>. Thus, the diagram above could be seen as the definition of the quotient group.</p>
    <p>The proof that the universal property characterizes constructions up to isomorphism involves high-powered machinery I cannot reproduce here, but it’s essentially due to the <a class="text-link external-link" href="https://ncatlab.org/nlab/show/Yoneda+lemma" target="_blank">Yoneda lemma</a> (perhaps among the most fundamental results in category theory) and how specifying all the maps in and out of an object uniquely determines it.</p>
    <h2>The comma category</h2>
    <p>An alternative, equivalent, and more abstract formulation of the universal property involves defining a <em>comma category</em> with a particular initial object. A comma category essentially generalizes the slice category, as well as other categories involving arrows such as the coslice category (which is the dual of the slice category).</p>
    <p>Firstly, we’ll give a definition of the coslice category (there is a reason why we chose the dual of the slice instead of the slice itself).</p>
    <p>Let <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 0.75em; height: 0.683em;" viewBox="0 0 8.25 7.513000000000001" width="8.25pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g7D8824C7D2BC9D6FDC59C635AC9A4FED" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g7D8824C7D2BC9D6FDC59C635AC9A4FED" overflow="visible"><path d="M 0 0m 1.507 0.033 c 0.19800007 0 0.8800001 -0.033 1.089 -0.033 c 0.16499996 0 0.2420001 0.088 0.2420001 0.253 c 0 0.09900001 -0.065999985 0.16499999 -0.20900011 0.176 c -0.319 0.011000007 -0.47300005 0.12100002 -0.47300005 0.33 c 0 0.09899998 0.09899998 0.31899995 0.3080001 0.649 c 0.29699993 0.495 0.50600004 0.85800004 0.6489999 1.0999999 h 2.6729999 c 0 -0.0769999 0.011000156 -0.23099995 0.04400015 -0.4729998 c 0.07700014 -0.8030001 0.12099981 -1.243 0.12099981 -1.2980001 c 0 -0.20899999 -0.24199963 -0.308 -0.737 -0.308 c -0.20899963 0 -0.3079996 -0.088 -0.3079996 -0.264 c 0 -0.10999999 0.065999985 -0.16499999 0.19799995 -0.16499999 c 0.26399994 0 1.0999999 0.033 1.3639998 0.033 c 0.21999979 0 1.0009999 -0.033 1.2210002 -0.033 c 0.16499996 0 0.24199963 0.088 0.24199963 0.264 c 0 0.110000014 -0.09899998 0.16499999 -0.29699993 0.16499999 c -0.3080001 0 -0.4949999 0.022000015 -0.5499997 0.055000007 c -0.055000305 0.033000022 -0.09899998 0.132 -0.11000013 0.29700002 l -0.65999985 6.798 c -0.032999992 0.20900011 -0.022000313 0.29699993 -0.25300026 0.29699993 c -0.13199997 0 -0.2420001 -0.065999985 -0.31899977 -0.20900011 l -3.7840002 -6.347 c -0.32999992 -0.54999995 -0.77 -0.84699994 -1.309 -0.8909999 c -0.176 -0.011000007 -0.264 -0.09900001 -0.264 -0.264 c 0 -0.10999999 0.066000015 -0.16499999 0.18700004 -0.16499999 c 0.176 0 0.75899994 0.033 0.93499994 0.033 Z m 3.9050002 6.314 l 0.32999992 -3.41 h -2.365 Z "/></symbol></defs></svg></span> be an object in a category <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 0.654em; height: 0.683em;" viewBox="0 0 7.194 7.513000000000001" width="7.194pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g93F710FCBC47C017329EE0E62637FE62" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g93F710FCBC47C017329EE0E62637FE62" overflow="visible"><path d="M 0 0m 2.519 -0.16499999 c 0.90199995 0 1.6500001 0.24199998 2.244 0.72599995 c 0.1869998 0.143 0.36299992 0.352 0.5500002 0.627 l -0.40700006 0.561 l -0.19799995 -0.09899998 c -0.17600012 -0.40699995 -0.38500023 -0.715 -0.6160002 -0.90199995 c -0.385 -0.29700002 -0.78100014 -0.45100003 -1.21 -0.45100003 c -1.012 0 -1.485 0.935 -1.485 2.0240002 c 0 1.188 0.41799998 2.3979998 0.8579999 3.212 c 0.605 1.1329999 1.463 1.6939998 2.563 1.6939998 c 0.71500015 0 1.0669999 -0.28599977 1.0669999 -0.84699965 c 0 -0.44000006 -0.14299965 -0.8470001 -0.4289999 -1.2210002 c -0.3080001 -0.40700006 -0.671 -0.605 -1.0889997 -0.605 c -0.26399994 0 -0.39600015 0.07700014 -0.39600015 0.23099995 c 0 0.09899998 0.18700004 0.572 0.18700004 0.671 c 0 0.18700027 -0.12099981 0.2750001 -0.37400007 0.2750001 c -0.41799998 0 -0.671 -0.31899977 -0.671 -0.803 c 0 -0.5170002 0.39600015 -0.78100014 1.1989999 -0.78100014 c 1.2210002 0 2.1230001 0.8910003 2.376 1.914 c 0.04400015 0.16499996 0.06600046 0.32999992 0.06600046 0.47300005 c 0 0.7589998 -0.5500002 1.1329999 -1.6500001 1.1329999 c -1.6170001 0 -2.8490002 -0.737 -3.7070003 -2.2219996 c -0.45099998 -0.7920003 -0.95699996 -2.1780002 -0.95699996 -3.344 c 0 -1.3640001 0.73700005 -2.266 2.079 -2.266 Z "/></symbol></defs></svg></span>. Then the <strong>coslice</strong> category is denoted <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 3.2375555555555557em; height: 0.683em;" viewBox="0 0 35.61311111111111 7.513000000000001" width="35.61311111111111pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#gC2B624863CDE6828147FA47EC1496FCA" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 4.279 7.513000000000001)"><use xlink:href="#g93F710FCBC47C017329EE0E62637FE62" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 14.528555555555556 7.513000000000001)"><use xlink:href="#g70CD59EFF0AA5AC6D13B80D8AABF5124" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 23.084111111111113 7.513000000000001)"><use xlink:href="#g7D8824C7D2BC9D6FDC59C635AC9A4FED" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 31.334111111111113 7.513000000000001)"><use xlink:href="#g6B8228307ED2E64A8E947B770237E2C7" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="gC2B624863CDE6828147FA47EC1496FCA" overflow="visible"><path d="M 0 0m 3.498 -2.728 c 0.09899998 0 0.15400004 0.05499983 0.15400004 0.15400004 c 0 0.032999992 -0.022000074 0.0769999 -0.055000067 0.12099981 c -0.5719998 0.44000006 -1.0339999 1.1660001 -1.375 2.167 c -0.29699993 0.869 -0.45099986 1.727 -0.45099986 2.5740001 v 0.9239998 c 0 0.8470001 0.15399992 1.7049999 0.45099986 2.574 c 0.34100008 1.0010004 0.8030002 1.7270002 1.375 2.1670003 c 0.032999992 0.032999992 0.055000067 0.076999664 0.055000067 0.12100029 c 0 0.09899998 -0.055000067 0.15399933 -0.15400004 0.15399933 c -0.010999918 0 -0.04399991 -0.01099968 -0.0769999 -0.032999992 c -0.6600001 -0.50599957 -1.21 -1.2539997 -1.661 -2.2549996 c -0.42900002 -0.95700026 -0.64900005 -1.8590002 -0.64900005 -2.7280002 v -0.9239998 c 0 -0.8690001 0.22000003 -1.7710001 0.64900005 -2.7280002 c 0.45099998 -1.0009999 1.0009999 -1.7489998 1.661 -2.2549999 c 0.032999992 -0.022000074 0.065999985 -0.032999992 0.0769999 -0.032999992 Z "/></symbol><symbol id="g93F710FCBC47C017329EE0E62637FE62" overflow="visible"><path d="M 0 0m 2.519 -0.16499999 c 0.90199995 0 1.6500001 0.24199998 2.244 0.72599995 c 0.1869998 0.143 0.36299992 0.352 0.5500002 0.627 l -0.40700006 0.561 l -0.19799995 -0.09899998 c -0.17600012 -0.40699995 -0.38500023 -0.715 -0.6160002 -0.90199995 c -0.385 -0.29700002 -0.78100014 -0.45100003 -1.21 -0.45100003 c -1.012 0 -1.485 0.935 -1.485 2.0240002 c 0 1.188 0.41799998 2.3979998 0.8579999 3.212 c 0.605 1.1329999 1.463 1.6939998 2.563 1.6939998 c 0.71500015 0 1.0669999 -0.28599977 1.0669999 -0.84699965 c 0 -0.44000006 -0.14299965 -0.8470001 -0.4289999 -1.2210002 c -0.3080001 -0.40700006 -0.671 -0.605 -1.0889997 -0.605 c -0.26399994 0 -0.39600015 0.07700014 -0.39600015 0.23099995 c 0 0.09899998 0.18700004 0.572 0.18700004 0.671 c 0 0.18700027 -0.12099981 0.2750001 -0.37400007 0.2750001 c -0.41799998 0 -0.671 -0.31899977 -0.671 -0.803 c 0 -0.5170002 0.39600015 -0.78100014 1.1989999 -0.78100014 c 1.2210002 0 2.1230001 0.8910003 2.376 1.914 c 0.04400015 0.16499996 0.06600046 0.32999992 0.06600046 0.47300005 c 0 0.7589998 -0.5500002 1.1329999 -1.6500001 1.1329999 c -1.6170001 0 -2.8490002 -0.737 -3.7070003 -2.2219996 c -0.45099998 -0.7920003 -0.95699996 -2.1780002 -0.95699996 -3.344 c 0 -1.3640001 0.73700005 -2.266 2.079 -2.266 Z "/></symbol><symbol id="g70CD59EFF0AA5AC6D13B80D8AABF5124" overflow="visible"><path d="M 0 0m 5.093 0.41799998 c 0.12099981 0.033000022 0.1869998 0.12099999 0.1869998 0.25300002 c 0 0.176 -0.08799982 0.264 -0.26399994 0.264 c -0.032999992 0 -0.05499983 -0.010999978 -0.076999664 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-1.7050002 -0.45099998 -2.5740001 c -0.34099996 -1.0009999 -0.803 -1.727 -1.375 -2.167 c -0.032999992 -0.04399991 -0.055000007 -0.08799982 -0.055000007 -0.12099981 c 0 -0.099000216 0.055000007 -0.15400004 0.15400004 -0.15400004 c 0.010999978 0 0.04399997 0.010999918 0.07699996 0.032999992 Z "/></symbol></defs></svg></span> and represents the category of all the arrows originating from <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 0.75em; height: 0.683em;" viewBox="0 0 8.25 7.513000000000001" width="8.25pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g7D8824C7D2BC9D6FDC59C635AC9A4FED" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g7D8824C7D2BC9D6FDC59C635AC9A4FED" overflow="visible"><path d="M 0 0m 1.507 0.033 c 0.19800007 0 0.8800001 -0.033 1.089 -0.033 c 0.16499996 0 0.2420001 0.088 0.2420001 0.253 c 0 0.09900001 -0.065999985 0.16499999 -0.20900011 0.176 c -0.319 0.011000007 -0.47300005 0.12100002 -0.47300005 0.33 c 0 0.09899998 0.09899998 0.31899995 0.3080001 0.649 c 0.29699993 0.495 0.50600004 0.85800004 0.6489999 1.0999999 h 2.6729999 c 0 -0.0769999 0.011000156 -0.23099995 0.04400015 -0.4729998 c 0.07700014 -0.8030001 0.12099981 -1.243 0.12099981 -1.2980001 c 0 -0.20899999 -0.24199963 -0.308 -0.737 -0.308 c -0.20899963 0 -0.3079996 -0.088 -0.3079996 -0.264 c 0 -0.10999999 0.065999985 -0.16499999 0.19799995 -0.16499999 c 0.26399994 0 1.0999999 0.033 1.3639998 0.033 c 0.21999979 0 1.0009999 -0.033 1.2210002 -0.033 c 0.16499996 0 0.24199963 0.088 0.24199963 0.264 c 0 0.110000014 -0.09899998 0.16499999 -0.29699993 0.16499999 c -0.3080001 0 -0.4949999 0.022000015 -0.5499997 0.055000007 c -0.055000305 0.033000022 -0.09899998 0.132 -0.11000013 0.29700002 l -0.65999985 6.798 c 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Objects in <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 3.2375555555555557em; height: 0.683em;" viewBox="0 0 35.61311111111111 7.513000000000001" width="35.61311111111111pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#gC2B624863CDE6828147FA47EC1496FCA" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 4.279 7.513000000000001)"><use xlink:href="#g93F710FCBC47C017329EE0E62637FE62" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 14.528555555555556 7.513000000000001)"><use xlink:href="#g70CD59EFF0AA5AC6D13B80D8AABF5124" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 23.084111111111113 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-1.1549997 -0.44000006 -1.4079998 -1.309 c -0.055000067 -0.19799995 -0.17600012 -0.7590003 -0.352 -1.6830001 h -0.71500015 c -0.24199986 0 -0.36299992 -0.011000156 -0.36299992 -0.2420001 c 0 -0.12099981 0.11000013 -0.17600012 0.34100008 -0.17600012 h 0.65999985 l -0.803 -4.257 c -0.12099981 -0.627 -0.23099995 -1.089 -0.32999992 -1.3970001 c -0.13199997 -0.40699995 -0.319 -0.61599994 -0.561 -0.61599994 c -0.16499996 0 -0.29700005 0.04399991 -0.41799998 0.12099993 c 0.352 0.055000067 0.528 0.26400006 0.528 0.61600006 c 0 0.28599995 -0.143 0.42899996 -0.44000006 0.42899996 c -0.37399995 0 -0.63799995 -0.32999998 -0.63799995 -0.70399994 c 0 -0.4840001 0.45100003 -0.79199994 0.968 -0.79199994 c 0.27499998 0 0.528 0.109999895 0.7370001 0.34099984 c 0.35199976 0.36300004 0.62699986 0.88 0.8249998 1.5730001 c 0.12100005 0.429 0.23099995 0.847 0.3080001 1.265 l 0.638 3.4209998 h 0.90199995 c 0.25299978 0 0.36299992 0.011000156 0.36299992 0.26400042 c 0 0.09899998 -0.11000013 0.1539998 -0.32999992 0.1539998 h -0.8470001 c 0.065999985 0.45099974 0.37400007 2.112 0.47300005 2.321 c 0.11000013 0.23099995 0.26399994 0.34100008 0.4619999 0.34100008 c 0.16499996 0 0.3080001 -0.04400015 0.42900038 -0.12100029 c -0.34100008 -0.07700014 -0.5170002 -0.2750001 -0.5170002 -0.6160002 c 0 -0.28599977 0.14300013 -0.4289999 0.44000006 -0.4289999 c 0.37400007 0 0.638 0.32999992 0.638 0.704 Z "/></symbol></defs></svg></span>.</p>
    <p>Now we introduce the comma category, which essentially generalizes this notion. Let <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 0.857em; height: 0.683em;" viewBox="0 0 9.427 7.513000000000001" width="9.427pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g9F48376B82E0DD159BB22EE5CFB26924" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g9F48376B82E0DD159BB22EE5CFB26924" overflow="visible"><path d="M 0 0m 4.389 2.563 h 2.321 l -0.40700006 -1.661 c -0.05499983 -0.24200004 -0.08799982 -0.42900002 -0.08799982 -0.55 c 0.032999992 -0.341 0.12099981 -0.51699996 0.47299957 -0.51699996 c 0.20900011 0 0.4840002 0.09899999 0.8250003 0.28599998 c 0.4289999 0.231 0.8250003 0.517 1.1659999 0.847 l -0.15400028 0.25299996 c -0.2859993 -0.19799995 -0.47299957 -0.32999998 -0.5609999 -0.385 c -0.1539998 -0.09899998 -0.28599977 -0.14299995 -0.4069996 -0.14299995 c -0.3080001 0 -0.31900024 0.36299998 -0.25300026 0.74799997 c 0.12100029 0.6819999 0.28600025 1.441 0.4949999 2.288 l 0.7919998 3.157 c 0.044000626 0.17599964 0.11000061 0.29699993 0.19799995 0.36299992 l 0.19799995 0.10999966 v 0.2750001 c -1.0669994 -0.09899998 -1.4849997 -0.28599977 -2.1889997 -0.704 c -0.38499975 -0.21999979 -0.83599997 -0.7149997 -1.375 -1.474 c -0.1869998 -0.26399994 -0.5609999 -0.84699965 -1.1219997 -1.7489998 c -0.8140001 -1.309 -1.2980001 -2.0570002 -1.4410002 -2.244 c -0.5389998 -0.69299996 -0.9569999 -1.045 -1.254 -1.045 c -0.13199997 0 -0.286 0.13200003 -0.26399994 0.25300002 c 0.04400003 0.16499996 0.25300002 0.53900003 0.12099993 0.671 c -0.065999985 0.110000014 -0.176 0.16499996 -0.32999992 0.16499996 c -0.34100002 0 -0.561 -0.176 -0.66 -0.528 c -0.15399998 -0.649 0.16500002 -1.089 0.759 -1.089 c 1.232 0 2.5300002 1.683 3.157 2.6729999 Z m 3.3769999 4.488 l 0.04400015 -0.07700014 l -0.96799994 -3.85 h -2.1009998 c 1.0669999 1.98 1.8809996 3.19 2.4199996 3.619 c 0.20900011 0.16499996 0.40700006 0.26399994 0.605 0.3080001 Z "/></symbol></defs></svg></span>, <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 0.778em; height: 0.683em;" viewBox="0 0 8.558 7.513000000000001" width="8.558pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g89FAAF2951AC0FB56DC9442765E79CBF" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g89FAAF2951AC0FB56DC9442765E79CBF" overflow="visible"><path d="M 0 0m 6.347 4.268 v 0.021999836 c 0.9789996 0.47300005 1.5619998 1.0669999 1.7270002 1.7709999 c 0.032999992 0.12100029 0.043999672 0.23099995 0.043999672 0.32999992 c 0 0.23099995 -0.065999985 0.42900038 -0.19799995 0.59400034 c -0.18700027 0.25299978 -0.572 0.40700006 -1.1659999 0.47300005 c -0.2420001 0.032999992 -0.68200016 0.043999672 -1.3200002 0.043999672 l 0.04400015 0.23099995 l -0.12100029 0.11000013 l -0.7589998 -0.36299992 c -0.957 -0.14300013 -1.76 -0.4619999 -2.387 -0.96799994 c -0.47299993 -0.37400007 -0.77 -0.7920003 -0.88 -1.2540002 c -0.07700002 -0.3080001 -0.054999948 -0.5169997 0.04400003 -0.638 c 0.04400003 -0.08799982 0.110000014 -0.13199997 0.20899999 -0.13199997 c 0.34099996 0 0.5500001 0.18700027 0.64900005 0.5500002 c 0.010999918 0.032999992 0.010999918 0.13199997 0.010999918 0.3080001 l 0.032999992 0.35199976 c 0.065999985 0.2420001 0.25300002 0.4840002 0.54999995 0.71500015 c 0.47300005 0.37400007 1.023 0.59399986 1.6499999 0.671 l -0.7919998 -3.157 c -0.18700004 -0.74800014 -0.3959999 -1.342 -0.605 -1.793 c -0.37399983 -0.79200006 -1.111 -1.4850001 -2.178 -1.4850001 l -0.50600004 -0.65999997 h 3.025 c 1.243 0 2.475 0.47300002 3.19 1.056 c 0.78100014 0.627 1.1659999 1.2760001 1.1659999 1.9580001 c 0 0.89100003 -0.572 1.089 -1.4299998 1.2650001 Z m 0.704 2.409 c 0.10999966 -0.14300013 0.16499996 -0.31900024 0.16499996 -0.5279999 c 0 -0.11000013 -0.022000313 -0.23100042 -0.055000305 -0.3630004 c -0.21999979 -0.8909998 -1.276 -1.4959998 -2.3209996 -1.4959998 c -0.11000013 0 -0.22000027 0.011000156 -0.19799995 0.11000013 l 0.69299984 2.7279997 c 0.9239998 0 1.4959998 -0.1539998 1.7160001 -0.45099974 Z m -0.41800022 -3.168 c 0.14300013 -0.18700004 0.22000027 -0.41799998 0.22000027 -0.671 c 0 -0.12100005 -0.022000313 -0.26400018 -0.055000305 -0.40700006 c -0.12099981 -0.528 -0.3959999 -0.96800005 -0.803 -1.331 c -0.48399973 -0.41800004 -1.144 -0.462 -2.013 -0.462 h -1.76 l -0.010999918 0.032999992 c 0.54999995 0.34099996 1.0339999 0.814 1.4519999 1.441 c 0.23100019 0.32999992 0.5060003 0.924 0.82500005 1.7709999 c 0.31900024 0.032999992 0.5500002 0.04400015 0.704 0.04400015 c 0.737 0 1.21 -0.14300013 1.441 -0.41799998 Z "/></symbol></defs></svg></span>, <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 0.654em; height: 0.683em;" viewBox="0 0 7.194 7.513000000000001" width="7.194pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g93F710FCBC47C017329EE0E62637FE62" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g93F710FCBC47C017329EE0E62637FE62" overflow="visible"><path d="M 0 0m 2.519 -0.16499999 c 0.90199995 0 1.6500001 0.24199998 2.244 0.72599995 c 0.1869998 0.143 0.36299992 0.352 0.5500002 0.627 l -0.40700006 0.561 l -0.19799995 -0.09899998 c -0.17600012 -0.40699995 -0.38500023 -0.715 -0.6160002 -0.90199995 c -0.385 -0.29700002 -0.78100014 -0.45100003 -1.21 -0.45100003 c -1.012 0 -1.485 0.935 -1.485 2.0240002 c 0 1.188 0.41799998 2.3979998 0.8579999 3.212 c 0.605 1.1329999 1.463 1.6939998 2.563 1.6939998 c 0.71500015 0 1.0669999 -0.28599977 1.0669999 -0.84699965 c 0 -0.44000006 -0.14299965 -0.8470001 -0.4289999 -1.2210002 c -0.3080001 -0.40700006 -0.671 -0.605 -1.0889997 -0.605 c -0.26399994 0 -0.39600015 0.07700014 -0.39600015 0.23099995 c 0 0.09899998 0.18700004 0.572 0.18700004 0.671 c 0 0.18700027 -0.12099981 0.2750001 -0.37400007 0.2750001 c -0.41799998 0 -0.671 -0.31899977 -0.671 -0.803 c 0 -0.5170002 0.39600015 -0.78100014 1.1989999 -0.78100014 c 1.2210002 0 2.1230001 0.8910003 2.376 1.914 c 0.04400015 0.16499996 0.06600046 0.32999992 0.06600046 0.47300005 c 0 0.7589998 -0.5500002 1.1329999 -1.6500001 1.1329999 c -1.6170001 0 -2.8490002 -0.737 -3.7070003 -2.2219996 c -0.45099998 -0.7920003 -0.95699996 -2.1780002 -0.95699996 -3.344 c 0 -1.3640001 0.73700005 -2.266 2.079 -2.266 Z "/></symbol></defs></svg></span> be categories, and let <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 0.673em; height: 0.683em;" viewBox="0 0 7.4030000000000005 7.513000000000001" width="7.4030000000000005pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g44684724E29AE89E9A254634DD6C1C20" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g44684724E29AE89E9A254634DD6C1C20" overflow="visible"><path d="M 0 0m 1.4629999 1.727 c 0 0.26399994 0.03300011 0.48399997 0.08800006 0.6600001 c 0 0.109999895 -0.054999948 0.16499996 -0.176 0.16499996 c -0.055000067 0 -0.09899998 -0.022000074 -0.12100005 -0.04400015 c -0.054999948 -0.05499983 -0.6819999 -2.4199998 -0.6819999 -2.596 c 0 -0.09900001 0.054999948 -0.154 0.16499996 -0.154 c 0.055000007 0 0.13200003 0.054999992 0.231 0.176 l 0.49499995 0.583 c 0.3850001 -0.50600004 1.001 -0.759 1.837 -0.759 c 0.7260001 0 1.3640001 0.297 1.9360001 0.88 c 0.572 0.58299994 0.8579998 1.232 0.8579998 1.958 c 0 0.51699996 -0.17600012 0.957 -0.53900003 1.309 c -0.16499996 0.14299989 -0.38499975 0.26400018 -0.65999985 0.36300015 c -0.25299978 0.05499983 -0.4949999 0.12099981 -0.737 0.1869998 l -0.7260001 0.19799995 c -0.36299992 0.09899998 -0.638 0.50600004 -0.638 0.9460001 c 0 0.47300005 0.18700004 0.8800001 0.572 1.2319999 c 0.385 0.35200024 0.8139999 0.5279999 1.287 0.5279999 c 1.0120001 0 1.5180001 -0.5499997 1.5180001 -1.6389999 c 0 -0.19799995 -0.04400015 -0.41799974 -0.04400015 -0.605 v -0.032999992 c 0.032999992 -0.076999664 0.0880003 -0.12099981 0.17600012 -0.12099981 c 0.09899998 0 0.16499996 0.08799982 0.20900011 0.25299978 l 0.5829997 2.387 c 0 0.09899998 -0.05499983 0.15400028 -0.16499996 0.15400028 c -0.05499983 0 -0.13199997 -0.055000305 -0.23099995 -0.17600012 l -0.47300005 -0.572 c -0.3080001 0.4949999 -0.8249998 0.74800014 -1.5619998 0.74800014 c -0.6930001 0 -1.3200002 -0.26399994 -1.8700001 -0.78100014 c -0.572 -0.5279999 -0.8579999 -1.1329999 -0.8579999 -1.8260002 c 0 -0.7919998 0.5170001 -1.4189997 1.166 -1.5949998 l 1.155 -0.29699993 c 0.59399986 -0.16499996 0.96799994 -0.37400007 0.96799994 -1.1110001 c 0 -0.50600004 -0.19799995 -0.957 -0.5829997 -1.3529999 c -0.38500023 -0.396 -0.8250003 -0.605 -1.3200002 -0.605 c -1.0780001 0 -1.8590001 0.47299996 -1.8590001 1.54 Z "/></symbol></defs></svg></span> and <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 0.732em; height: 0.683em;" viewBox="0 0 8.052 7.513000000000001" width="8.052pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g7CECFC2B2FFEE93E2441D572EB1D788D" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g7CECFC2B2FFEE93E2441D572EB1D788D" overflow="visible"><path d="M 0 0m 3.784 6.941 c 0 -0.032999992 -0.010999918 -0.11000013 -0.04399991 -0.21999979 l -1.4519999 -5.808 c -0.04400015 -0.16499996 -0.08800006 -0.264 -0.12100005 -0.30799997 c -0.09899998 -0.12100002 -0.47300005 -0.17600003 -1.133 -0.17600003 c -0.34100002 0 -0.49500006 0.033000022 -0.49500006 -0.253 c 0 -0.121 0.07700002 -0.176 0.22000003 -0.176 c 0.56099993 0 1.3529999 0.044 1.826 0.033 l 0.90199995 -0.011 c 0.15400004 0 0.7589998 -0.022 0.9460001 -0.022 c 0.1869998 0 0.2750001 0.088 0.2750001 0.264 c 0 0.110000014 -0.13199997 0.16499999 -0.40700006 0.16499999 c -0.572 0 -0.9020002 0.022000015 -1.0010002 0.07699999 c -0.032999992 0.022000015 -0.055000067 0.066000044 -0.055000067 0.13200003 l 1.4850001 5.995 c 0.032999992 0.15400028 0.065999985 0.25300026 0.09899998 0.28600025 c 0.04400015 0.065999985 0.3080001 0.09899998 0.7919998 0.09899998 c 0.605 0 1.0230002 -0.04400015 1.2320004 -0.14300013 c 0.20899963 -0.09899998 0.31899977 -0.32999992 0.31899977 -0.704 c 0 -0.19799995 -0.032999992 -0.4840002 -0.08799982 -0.8579998 c -0.022000313 -0.0880003 -0.032999992 -0.15400028 -0.032999992 -0.20900011 c 0 -0.12100029 0.05499983 -0.18700027 0.17599964 -0.18700027 c 0.09899998 0 0.16500044 0.09899998 0.19800043 0.28600025 l 0.29699993 1.8919997 c 0.01099968 0.055000305 0.021999836 0.12100029 0.021999836 0.18700027 c 0 0.11000013 -0.11000013 0.16499996 -0.34100008 0.16499996 h -6.028 c -0.286 0 -0.30799997 -0.032999992 -0.39600003 -0.2420001 l -0.649 -1.914 c -0.032999992 -0.09899998 -0.054999977 -0.16499996 -0.065999985 -0.20900011 c 0 -0.10999966 0.055000007 -0.16499996 0.176 -0.16499996 c 0.088 0 0.16500002 0.0880003 0.21999997 0.25300026 c 0.27500004 0.803 0.5500001 1.2979999 0.803 1.507 c 0.286 0.23099995 0.8360001 0.34100008 1.639 0.34100008 h 0.4289999 c 0.12100005 0 0.25300002 0.01099968 0.25300002 -0.07700014 Z "/></symbol></defs></svg></span> be functors as such:</p>
    <figure class="math block-math">
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    </figure>
    <p>We can use this comma category to define a universal property. First, note that we are not going to need all three categories, and the proof on Wikipedia uses a different definition of comma category involving only two. For our more general definition, we’re simply going to “collapse” one category, namely <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 0.857em; height: 0.683em;" viewBox="0 0 9.427 7.513000000000001" width="9.427pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g9F48376B82E0DD159BB22EE5CFB26924" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g9F48376B82E0DD159BB22EE5CFB26924" overflow="visible"><path d="M 0 0m 4.389 2.563 h 2.321 l -0.40700006 -1.661 c -0.05499983 -0.24200004 -0.08799982 -0.42900002 -0.08799982 -0.55 c 0.032999992 -0.341 0.12099981 -0.51699996 0.47299957 -0.51699996 c 0.20900011 0 0.4840002 0.09899999 0.8250003 0.28599998 c 0.4289999 0.231 0.8250003 0.517 1.1659999 0.847 l -0.15400028 0.25299996 c -0.2859993 -0.19799995 -0.47299957 -0.32999998 -0.5609999 -0.385 c -0.1539998 -0.09899998 -0.28599977 -0.14299995 -0.4069996 -0.14299995 c -0.3080001 0 -0.31900024 0.36299998 -0.25300026 0.74799997 c 0.12100029 0.6819999 0.28600025 1.441 0.4949999 2.288 l 0.7919998 3.157 c 0.044000626 0.17599964 0.11000061 0.29699993 0.19799995 0.36299992 l 0.19799995 0.10999966 v 0.2750001 c -1.0669994 -0.09899998 -1.4849997 -0.28599977 -2.1889997 -0.704 c -0.38499975 -0.21999979 -0.83599997 -0.7149997 -1.375 -1.474 c -0.1869998 -0.26399994 -0.5609999 -0.84699965 -1.1219997 -1.7489998 c -0.8140001 -1.309 -1.2980001 -2.0570002 -1.4410002 -2.244 c -0.5389998 -0.69299996 -0.9569999 -1.045 -1.254 -1.045 c -0.13199997 0 -0.286 0.13200003 -0.26399994 0.25300002 c 0.04400003 0.16499996 0.25300002 0.53900003 0.12099993 0.671 c -0.065999985 0.110000014 -0.176 0.16499996 -0.32999992 0.16499996 c -0.34100002 0 -0.561 -0.176 -0.66 -0.528 c -0.15399998 -0.649 0.16500002 -1.089 0.759 -1.089 c 1.232 0 2.5300002 1.683 3.157 2.6729999 Z m 3.3769999 4.488 l 0.04400015 -0.07700014 l -0.96799994 -3.85 h -2.1009998 c 1.0669999 1.98 1.8809996 3.19 2.4199996 3.619 c 0.20900011 0.16499996 0.40700006 0.26399994 0.605 0.3080001 Z "/></symbol></defs></svg></span>.</p>
    <p>Set <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 2.7655555555555558em; height: 0.683em;" viewBox="0 0 30.421111111111117 7.513000000000001" width="30.421111111111117pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g9F48376B82E0DD159BB22EE5CFB26924" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 12.482555555555557 7.513000000000001)"><use xlink:href="#g1B387A2F9014F570A058F676AC9B160C" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 24.096111111111114 7.513000000000001)"><use xlink:href="#g19853C21808F7EA54414097C2F2B9538" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g9F48376B82E0DD159BB22EE5CFB26924" overflow="visible"><path d="M 0 0m 4.389 2.563 h 2.321 l -0.40700006 -1.661 c -0.05499983 -0.24200004 -0.08799982 -0.42900002 -0.08799982 -0.55 c 0.032999992 -0.341 0.12099981 -0.51699996 0.47299957 -0.51699996 c 0.20900011 0 0.4840002 0.09899999 0.8250003 0.28599998 c 0.4289999 0.231 0.8250003 0.517 1.1659999 0.847 l -0.15400028 0.25299996 c -0.2859993 -0.19799995 -0.47299957 -0.32999998 -0.5609999 -0.385 c -0.1539998 -0.09899998 -0.28599977 -0.14299995 -0.4069996 -0.14299995 c -0.3080001 0 -0.31900024 0.36299998 -0.25300026 0.74799997 c 0.12100029 0.6819999 0.28600025 1.441 0.4949999 2.288 l 0.7919998 3.157 c 0.044000626 0.17599964 0.11000061 0.29699993 0.19799995 0.36299992 l 0.19799995 0.10999966 v 0.2750001 c -1.0669994 -0.09899998 -1.4849997 -0.28599977 -2.1889997 -0.704 c -0.38499975 -0.21999979 -0.83599997 -0.7149997 -1.375 -1.474 c -0.1869998 -0.26399994 -0.5609999 -0.84699965 -1.1219997 -1.7489998 c -0.8140001 -1.309 -1.2980001 -2.0570002 -1.4410002 -2.244 c -0.5389998 -0.69299996 -0.9569999 -1.045 -1.254 -1.045 c -0.13199997 0 -0.286 0.13200003 -0.26399994 0.25300002 c 0.04400003 0.16499996 0.25300002 0.53900003 0.12099993 0.671 c -0.065999985 0.110000014 -0.176 0.16499996 -0.32999992 0.16499996 c -0.34100002 0 -0.561 -0.176 -0.66 -0.528 c -0.15399998 -0.649 0.16500002 -1.089 0.759 -1.089 c 1.232 0 2.5300002 1.683 3.157 2.6729999 Z m 3.3769999 4.488 l 0.04400015 -0.07700014 l -0.96799994 -3.85 h -2.1009998 c 1.0669999 1.98 1.8809996 3.19 2.4199996 3.619 c 0.20900011 0.16499996 0.40700006 0.26399994 0.605 0.3080001 Z "/></symbol><symbol id="g1B387A2F9014F570A058F676AC9B160C" overflow="visible"><path d="M 0 0m 7.678 4.037 h -6.798 c -0.176 0 -0.264 -0.0880003 -0.264 -0.25300026 c 0 -0.16499996 0.088 -0.25300002 0.264 -0.25300002 h 6.798 c 0.17600012 0 0.26399994 0.08800006 0.26399994 0.25300002 c 0 0.13199997 -0.12099981 0.25300026 -0.26399994 0.25300026 Z m 0 -2.0680003 h -6.798 c -0.176 0 -0.264 -0.08799994 -0.264 -0.25300002 c 0 -0.16499996 0.088 -0.25300002 0.264 -0.25300002 h 6.798 c 0.17600012 0 0.26399994 0.08800006 0.26399994 0.25300002 c 0 0.143 -0.12099981 0.25300002 -0.26399994 0.25300002 Z "/></symbol><symbol id="g19853C21808F7EA54414097C2F2B9538" overflow="visible"><path d="M 0 0m 5.434 0 v 0.517 h -1.54 v 6.336 c 0 0.24199963 0 0.35199976 -0.286 0.35199976 c -0.12100005 0 -0.14300013 0 -0.2420001 -0.07700014 c -0.8469999 -0.62699986 -1.9799999 -0.62699986 -2.211 -0.62699986 h -0.21999997 v -0.5169997 h 0.21999997 c 0.176 0 0.781 0.01099968 1.4300001 0.21999979 v -5.687 h -1.529 v -0.517 c 0.48399997 0.033 1.6500001 0.033 2.189 0.033 c 0.53900003 0 1.7049999 0 2.1890001 -0.033 Z "/></symbol></defs></svg></span>, where <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 0.575em; height: 0.683em;" viewBox="0 0 6.325 7.513000000000001" width="6.325pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g19853C21808F7EA54414097C2F2B9538" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g19853C21808F7EA54414097C2F2B9538" overflow="visible"><path d="M 0 0m 5.434 0 v 0.517 h -1.54 v 6.336 c 0 0.24199963 0 0.35199976 -0.286 0.35199976 c -0.12100005 0 -0.14300013 0 -0.2420001 -0.07700014 c -0.8469999 -0.62699986 -1.9799999 -0.62699986 -2.211 -0.62699986 h -0.21999997 v -0.5169997 h 0.21999997 c 0.176 0 0.781 0.01099968 1.4300001 0.21999979 v -5.687 h -1.529 v -0.517 c 0.48399997 0.033 1.6500001 0.033 2.189 0.033 c 0.53900003 0 1.7049999 0 2.1890001 -0.033 Z "/></symbol></defs></svg></span> is the category with exactly one object and one morphism. 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The diagram becomes</p>
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    </figure>
    <p>which simply reduces to</p>
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class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 4.3840111111111115em; height: 0.683em;" viewBox="0 0 48.22412222222222 7.513000000000001" width="48.22412222222222pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g6ACFD3BC236DDF576FBD137BE52B79D6" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 8.577555555555556 7.513000000000001)"><use xlink:href="#g3C74867A0B23EAA8D2A33B2EBDF4296C" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 14.69111111111111 7.513000000000001)"><use xlink:href="#gB17F7D27B310B0FFC0EBEACCA214E57" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 24.082666666666665 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universal morphism from earlier. Moreover, suppose that this object is <em>initial</em>, that is, there is a unique arrow from <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 2.578666666666667em; height: 0.683em;" viewBox="0 0 28.365333333333332 7.513000000000001" width="28.365333333333332pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#gC2B624863CDE6828147FA47EC1496FCA" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 4.279 7.513000000000001)"><use xlink:href="#g1C1633DA384D29955B17072D7B057DB2" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 12.903 7.513000000000001)"><use xlink:href="#g955CD2C5F7FE607D0A2CBDAC4310C55A" x="0" y="0" fill="#000000" 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0.022000313 -0.065999985 0.032999992 -0.07699996 0.032999992 c -0.09900004 0 -0.15400004 -0.05499935 -0.15400004 -0.15399933 c 0 -0.044000626 0.022000015 -0.0880003 0.055000007 -0.12100029 c 0.57199997 -0.44000006 1.0339999 -1.1659999 1.375 -2.1670003 c 0.29700017 -0.86899996 0.45099998 -1.7269998 0.45099998 -2.574 v -0.9239998 c 0 -0.8470001 -0.1539998 -1.7050002 -0.45099998 -2.5740001 c -0.34099996 -1.0009999 -0.803 -1.727 -1.375 -2.167 c -0.032999992 -0.04399991 -0.055000007 -0.08799982 -0.055000007 -0.12099981 c 0 -0.099000216 0.055000007 -0.15400004 0.15400004 -0.15400004 c 0.010999978 0 0.04399997 0.010999918 0.07699996 0.032999992 Z "/></symbol></defs></svg></span> to every object in <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 2.4605555555555556em; height: 0.683em;" viewBox="0 0 27.066111111111113 7.513000000000001" width="27.066111111111113pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g44684724E29AE89E9A254634DD6C1C20" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 10.458555555555556 7.513000000000001)"><use xlink:href="#g70CD59EFF0AA5AC6D13B80D8AABF5124" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 19.014111111111113 7.513000000000001)"><use xlink:href="#g7CECFC2B2FFEE93E2441D572EB1D788D" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g44684724E29AE89E9A254634DD6C1C20" overflow="visible"><path d="M 0 0m 1.4629999 1.727 c 0 0.26399994 0.03300011 0.48399997 0.08800006 0.6600001 c 0 0.109999895 -0.054999948 0.16499996 -0.176 0.16499996 c -0.055000067 0 -0.09899998 -0.022000074 -0.12100005 -0.04400015 c -0.054999948 -0.05499983 -0.6819999 -2.4199998 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This is sufficient to once again define the universal property. 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    <p>If you recall our previous definition of the universal property, this is exactly that! That is, setting the initial object of our comma category leads to an equivalent diagram and the same existence of unique morphism as our original definition. (Technically, we only proved that setting this initial object implies the universal property, we also need to prove the converse to show the definitions are equivalent, but I won’t do that here.)</p>
    <p>Now consider our previous example of the universal property of the quotient group. 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The functor <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 0.732em; height: 0.683em;" viewBox="0 0 8.052 7.513000000000001" width="8.052pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g7CECFC2B2FFEE93E2441D572EB1D788D" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g7CECFC2B2FFEE93E2441D572EB1D788D" overflow="visible"><path d="M 0 0m 3.784 6.941 c 0 -0.032999992 -0.010999918 -0.11000013 -0.04399991 -0.21999979 l -1.4519999 -5.808 c -0.04400015 -0.16499996 -0.08800006 -0.264 -0.12100005 -0.30799997 c -0.09899998 -0.12100002 -0.47300005 -0.17600003 -1.133 -0.17600003 c -0.34100002 0 -0.49500006 0.033000022 -0.49500006 -0.253 c 0 -0.121 0.07700002 -0.176 0.22000003 -0.176 c 0.56099993 0 1.3529999 0.044 1.826 0.033 l 0.90199995 -0.011 c 0.15400004 0 0.7589998 -0.022 0.9460001 -0.022 c 0.1869998 0 0.2750001 0.088 0.2750001 0.264 c 0 0.110000014 -0.13199997 0.16499999 -0.40700006 0.16499999 c -0.572 0 -0.9020002 0.022000015 -1.0010002 0.07699999 c -0.032999992 0.022000015 -0.055000067 0.066000044 -0.055000067 0.13200003 l 1.4850001 5.995 c 0.032999992 0.15400028 0.065999985 0.25300026 0.09899998 0.28600025 c 0.04400015 0.065999985 0.3080001 0.09899998 0.7919998 0.09899998 c 0.605 0 1.0230002 -0.04400015 1.2320004 -0.14300013 c 0.20899963 -0.09899998 0.31899977 -0.32999992 0.31899977 -0.704 c 0 -0.19799995 -0.032999992 -0.4840002 -0.08799982 -0.8579998 c -0.022000313 -0.0880003 -0.032999992 -0.15400028 -0.032999992 -0.20900011 c 0 -0.12100029 0.05499983 -0.18700027 0.17599964 -0.18700027 c 0.09899998 0 0.16500044 0.09899998 0.19800043 0.28600025 l 0.29699993 1.8919997 c 0.01099968 0.055000305 0.021999836 0.12100029 0.021999836 0.18700027 c 0 0.11000013 -0.11000013 0.16499996 -0.34100008 0.16499996 h -6.028 c -0.286 0 -0.30799997 -0.032999992 -0.39600003 -0.2420001 l -0.649 -1.914 c -0.032999992 -0.09899998 -0.054999977 -0.16499996 -0.065999985 -0.20900011 c 0 -0.10999966 0.055000007 -0.16499996 0.176 -0.16499996 c 0.088 0 0.16500002 0.0880003 0.21999997 0.25300026 c 0.27500004 0.803 0.5500001 1.2979999 0.803 1.507 c 0.286 0.23099995 0.8360001 0.34100008 1.639 0.34100008 h 0.4289999 c 0.12100005 0 0.25300002 0.01099968 0.25300002 -0.07700014 Z "/></symbol></defs></svg></span> is then the identity endofunctor on <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; 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0 -0.6270001 0.286 -0.6270001 0.84699994 c 0 0.4510001 0.19799995 1.1769999 0.58299994 2.1890001 c 0.12100005 0.32999992 0.18700004 0.5610001 0.18700004 0.6930001 c 0 0.5499997 -0.385 0.93499994 -0.93499994 0.93499994 c -0.495 0 -0.88 -0.2750001 -1.155 -0.8249998 c -0.20900002 -0.42900014 -0.319 -0.7260003 -0.319 -0.8800001 c 0 -0.09899998 0.055000007 -0.15400004 0.176 -0.15400004 c 0.143 0 0.16499996 0.065999985 0.20899999 0.22000003 c 0.25300002 0.8800001 0.605 1.3199997 1.056 1.3199997 c 0.15400004 0 0.23099995 -0.10999966 0.23099995 -0.31899977 c 0 -0.16499996 -0.065999985 -0.3959999 -0.18699992 -0.7149999 c -0.41799998 -1.1110001 -0.627 -1.881 -0.627 -2.288 c 0 -0.84699994 0.528 -1.342 1.375 -1.342 c 0.47300005 0 0.90199995 0.22 1.2649999 0.671 c 0.15400004 -0.451 0.4840002 -0.671 0.99000025 -0.671 c 0.7589998 0 0.9899998 0.891 1.1659999 1.705 Z "/></symbol><symbol id="g1B387A2F9014F570A058F676AC9B160C" overflow="visible"><path d="M 0 0m 7.678 4.037 h -6.798 c -0.176 0 -0.264 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0 0.638 0.32999992 0.638 0.704 Z "/></symbol></defs></svg></span>. That is, this diagram commutes</p>
    <figure class="math block-math">
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    <p>This is the exact same diagram from the previous section, and <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 0.572em; height: 0.683em;" viewBox="0 0 6.292 7.513000000000001" width="6.292pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#gDB80CB4BA90681AE9DE4DC5FCDCA6EE" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="gDB80CB4BA90681AE9DE4DC5FCDCA6EE" overflow="visible"><path d="M 0 0m 5.973 1.584 c 0 0.09899998 -0.055000305 0.15400004 -0.17600012 0.15400004 c -0.09899998 0 -0.1539998 -0.07700002 -0.1869998 -0.23100007 c -0.055000305 -0.20899999 -0.11000013 -0.41799998 -0.18700027 -0.627 c -0.14300013 -0.451 -0.34100008 -0.682 -0.5829997 -0.682 c -0.19799995 0 -0.2970004 0.154 -0.2970004 0.46199995 c 0 0.18700004 0.07700014 0.561 0.22000027 1.1110001 l 0.29699993 1.166 c 0.09899998 0.3959999 0.1869998 0.671 0.25300026 1.0119998 l 0.065999985 0.29699993 c 0.021999836 0.07700014 0.032999992 0.13199997 0.032999992 0.16499996 c 0 0.22000027 -0.12100029 0.3300004 -0.3630004 0.3300004 c -0.24199963 0 -0.3959999 -0.13199997 -0.4619999 -0.40700006 l -0.8139999 -3.256 c -0.010999918 -0.032999992 -0.055000067 -0.109999955 -0.14300013 -0.24199998 c -0.17599988 -0.28599995 -0.6049998 -0.63799995 -1.0449998 -0.63799995 c -0.41799998 0 -0.6270001 0.286 -0.6270001 0.84699994 c 0 0.4510001 0.19799995 1.1769999 0.58299994 2.1890001 c 0.12100005 0.32999992 0.18700004 0.5610001 0.18700004 0.6930001 c 0 0.5499997 -0.385 0.93499994 -0.93499994 0.93499994 c -0.495 0 -0.88 -0.2750001 -1.155 -0.8249998 c -0.20900002 -0.42900014 -0.319 -0.7260003 -0.319 -0.8800001 c 0 -0.09899998 0.055000007 -0.15400004 0.176 -0.15400004 c 0.143 0 0.16499996 0.065999985 0.20899999 0.22000003 c 0.25300002 0.8800001 0.605 1.3199997 1.056 1.3199997 c 0.15400004 0 0.23099995 -0.10999966 0.23099995 -0.31899977 c 0 -0.16499996 -0.065999985 -0.3959999 -0.18699992 -0.7149999 c -0.41799998 -1.1110001 -0.627 -1.881 -0.627 -2.288 c 0 -0.84699994 0.528 -1.342 1.375 -1.342 c 0.47300005 0 0.90199995 0.22 1.2649999 0.671 c 0.15400004 -0.451 0.4840002 -0.671 0.99000025 -0.671 c 0.7589998 0 0.9899998 0.891 1.1659999 1.705 Z "/></symbol></defs></svg></span> is again the canonical map <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 0.595em; height: 0.683em;" viewBox="0 0 6.544999999999999 7.513000000000001" width="6.544999999999999pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#gE3440E5809249DFF9BBFDD61D8DFB2BF" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="gE3440E5809249DFF9BBFDD61D8DFB2BF" overflow="visible"><path d="M 0 0m 5.764 4.741 h -3.6299999 c -0.4510001 0 -0.8470001 -0.17600012 -1.1660001 -0.5170002 c -0.15399998 -0.16499996 -0.671 -0.86899996 -0.671 -1.0120001 c 0.044 -0.0769999 0.044 -0.14299989 0.176 -0.14299989 c 0.07700002 0 0.143 0.04399991 0.20899999 0.14299989 c 0.35200006 0.53900003 0.80300003 0.8140001 1.342 0.8140001 h 0.5610001 c -0.25300002 -0.957 -0.71500003 -2.134 -1.375 -3.5310001 c -0.055000067 -0.132 -0.08800006 -0.231 -0.08800006 -0.286 c 0 -0.22 0.12100005 -0.32999998 0.352 -0.32999998 c 0.20899999 0 0.36300004 0.121 0.462 0.352 c 0.19800007 0.627 0.34099996 1.1110001 0.4180001 1.441 l 0.6049998 2.354 h 1.1330001 c -0.29699993 -1.309 -0.45099998 -2.2220001 -0.45099998 -2.739 c 0 -0.5389999 0.12100005 -1.408 0.5280001 -1.408 c 0.23099995 0 0.48399973 0.22 0.48399973 0.45099998 c 0 0.055000007 -0.021999836 0.143 -0.065999985 0.25300002 c -0.20900011 0.517 -0.3080001 1.0999999 -0.3080001 1.7710001 c 0 0.51699996 0.065999985 1.0779998 0.18700027 1.6719999 h 1.1989999 c 0.38500023 0 0.572 0.13199997 0.572 0.40700006 c 0 0.25299978 -0.19799995 0.3080001 -0.47300005 0.3080001 Z "/></symbol></defs></svg></span>.</p>
    <p>Finally, we note that in this degenerate case where <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 2.7655555555555558em; height: 0.683em;" viewBox="0 0 30.421111111111113 7.513000000000001" width="30.421111111111113pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g89FAAF2951AC0FB56DC9442765E79CBF" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 11.613555555555557 7.513000000000001)"><use xlink:href="#g1B387A2F9014F570A058F676AC9B160C" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g><g class="typst-text" transform="matrix(1 0 0 -1 23.227111111111114 7.513000000000001)"><use xlink:href="#g93F710FCBC47C017329EE0E62637FE62" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g89FAAF2951AC0FB56DC9442765E79CBF" overflow="visible"><path d="M 0 0m 6.347 4.268 v 0.021999836 c 0.9789996 0.47300005 1.5619998 1.0669999 1.7270002 1.7709999 c 0.032999992 0.12100029 0.043999672 0.23099995 0.043999672 0.32999992 c 0 0.23099995 -0.065999985 0.42900038 -0.19799995 0.59400034 c -0.18700027 0.25299978 -0.572 0.40700006 -1.1659999 0.47300005 c -0.2420001 0.032999992 -0.68200016 0.043999672 -1.3200002 0.043999672 l 0.04400015 0.23099995 l -0.12100029 0.11000013 l -0.7589998 -0.36299992 c -0.957 -0.14300013 -1.76 -0.4619999 -2.387 -0.96799994 c -0.47299993 -0.37400007 -0.77 -0.7920003 -0.88 -1.2540002 c -0.07700002 -0.3080001 -0.054999948 -0.5169997 0.04400003 -0.638 c 0.04400003 -0.08799982 0.110000014 -0.13199997 0.20899999 -0.13199997 c 0.34099996 0 0.5500001 0.18700027 0.64900005 0.5500002 c 0.010999918 0.032999992 0.010999918 0.13199997 0.010999918 0.3080001 l 0.032999992 0.35199976 c 0.065999985 0.2420001 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essentially reduced to the coslice category introduced prior, so in fact to describe the universal property of quotients we only needed the notion of coslice<a id="loc-2" href="#loc-4" role="doc-noteref"><sup class="font-index text-iris text-[0.83em] hover:bg-love/15 transition-colors] ml-0.5 [vertical-align: baseline] relative -top-[0.33em]">2</sup></a>. This diagram, under these conditions,</p>
    <figure class="math block-math">
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    </figure>
    <p>reduces to</p>
    <figure class="math block-math">
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    <h2>Conclusion</h2>
    <p>The universal property we showed for quotient groups demonstrates how category theory can trivially characterize certain constructions. 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0.34100008 Z m -2.926 2.8049998 h 1.023 c -0.065999985 -0.16499996 -0.09899998 -0.38499975 -0.09899998 -0.65999985 v -5.3459997 c 0 -0.25300002 0.032999992 -0.44 0.08800006 -0.583 h -1.0120001 c 0.04400003 0.13200003 0.065999985 0.319 0.065999985 0.55 v 5.4890003 c 0 0.23099995 -0.021999955 0.41799974 -0.065999985 0.5499997 Z "/></symbol></defs></svg></span> with linear maps as morphisms), we obtain an equivalent univeral property of <em>quotient spaces</em>. This gives a hint for how quotients and the isomorphism theorems may be formulated in general via <a class="text-link external-link" href="https://en.wikipedia.org/wiki/Universal_algebra" target="_blank">the idea of universal algebra</a>.</p>
    <section role="doc-endnotes">
      <ol style="list-style-type: none">
        <li id="loc-3"><a href="#loc-1" role="doc-backlink"><sup>1</sup></a>The slice category, being its dual, is accordingly the category of all the arrows going to <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 0.75em; height: 0.683em;" viewBox="0 0 7.012500000000001 6.38605" width="7.012500000000001pt" height="6.38605pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 6.38605)"><use xlink:href="#g63064BB039639993890BA2EC886B5A0C" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g63064BB039639993890BA2EC886B5A0C" overflow="visible"><path d="M 0 0m 1.28095 0.02805 c 0.16830003 0 0.748 -0.02805 0.92565 -0.02805 c 0.14024997 0 0.20569992 0.0748 0.20569992 0.21505 c 0 0.08415 -0.05609989 0.14025001 -0.17764997 0.14960001 c -0.27114987 0.009350002 -0.4020499 0.10284999 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      </ol>
    </section>
]]></summary>
</entry>
<entry>
    <title>Postscript on HackMIT</title>
    <link href="https://web.youwen.dev/writing/postscript-on-hackmit" />
    <id>https://web.youwen.dev/writing/postscript-on-hackmit</id>
    <published>2025-09-22</published>
    <updated>2025-09-22T00:00:00Z</updated>
    <summary type="html"><![CDATA[    <p class="subhead">In 2025, I became likely the only student in high school—and also probably the youngest person—to win HackMIT in its twelve year history. Here, I detail my successful attempt to “hack” HackMIT and my contributions to the hackathon.</p>
    <p><em>This is a retrospective on HackMIT presented as an epic tale of wits and grit, contributed partly by guest writer Warren Futaba Lex. This essay is written from his perspective, not mine (Youwen’s), although I was on the team.</em></p>
    <p>I initially applied to HackMIT over the summer of 2025. I noted my experience in electrical engineering, but approached the application as a purely intellectual and philosophical exercise. I mostly detailed my fascination with the spirit of creativity and ingenuity embedded into the fabric of MIT and HackMIT in particular. I noted my careful study of the stories of Sam Bankman-Fried and other inventors, and how I sought to use the hackathon as an opportunity to contribute to Taiwanese–Azerbaijani technologies, a pseudonym for the dual-use technology industry and related megacorporations. Despite the numerous controversial and contentious opinions expressed in my application, it must have been favored by the HackMIT organizing committee, since I was immediately accepted as part of the early decision round. Two of my future teammates, students at MIT, were waitlisted and ultimately rejected, though were allowed to participate through a loophole in the HackMIT regulations. Whether I entered HackMIT by divine providence or a purely meritocratic process I shall never know, but I was determined to bring my ingenuity to bear on the world’s most pressing challenges for the hackathon.</p>
    <hr>
    <div class="w-full text-center text-xl text-subtle select-none">⁂</div>
    <p>Over the summer, I joined a loosely-associated intellectual collective known as functor.systems, a homotopy-coherent collective of free software hackers. Three other members of this organization, namely Youwen, Ananth, and a hitherto unknown quaternary interlocutor, decided to participate in the hackathon, and I decided to join them in an epic collaboration. I flew in to the MIT campus a few days prior to its start date to preview the full gamut of intellectual exercises on campus. After sitting through lectures in real analysis, fourier transforms, digital circuits, and political economy, I proceeded to synthesize my newfound knowledge into a guiding vision for my team.</p>
    <p>Immediately taking my theoretical political economy lessons and putting them to work (<em>mens et manus</em>), I decided to take a radical approach to hackathon labor allocation. Typical and primitive hackathon teams will decide on a singular project idea and commit the synchronized efforts of all four members towards the final product. I immediately identified this as a high point of inefficiency. At this key instant, our team’s triumphant fate was already sealed—it remained only a matter of implementing my machinations.</p>
    <p>Let me make a metaphor. Typical teams where all 4 members work on the same project are executing as asynchronous processes in a language such as JavaScript. They depend on each other’s state, namely, the current progress they’ve made. They may be forced to block until the requisite tasks are completed by each other. This drastically reduces efficiency.</p>
    <p>In our team, I decided to apply my lessons from political economy and structure our labor such that we operated at the utility-maximizing Pareto frontier. I developed three independent research “thrusts” for our team to focus on. First, Ananth was assigned to theoretical research and development to make full use of his training in the Zardini Lab developing abstractly nonsensical methods for engineering challenges as well as proficiency in Haskell. Second, Youwen was assigned to work on an experimental 3D user interface based on UV mapping as well as research the feasibility of deploying a swarm of artificial intelligence agents. The final research thrust was mine, and it was hardware development and deployment.</p>
    <p>In the previous process metaphor, rather than an asynchronous yet blocking set of processes a la JavaScript, we became fully concurrent and purely functional Haskellian processes. Rather than being forced to block on each other’s state, each innovator was free to rapidly deploy experimental designs at breakneck speeds without regard for each other. By giving each member autonomy over research and development, we could emulate the innovative environment of a research university, far more powerful than the corporatist structure of a traditional hackathon team.</p>
    <p>As an electrical engineer, incorporating my talents into a mostly software-based project was exceedingly difficult—however, from my extensive literature review, I discovered that there were critical innovations lacking in the augmented reality (AR) space, while the AI agents space was oversaturated with corporate slop. I proceeded to focus my attention on reverse engineering the hardware protocol for a pair of AR glasses that I managed to procure from corporate sponsors. After a few hours of hacking, I established a direct firmware interface with the device and was able to control it flawlessly over a local network. The amount of time it took to set this up was highly nontrivial, and completing a project of this scale under the time constraints was highly restrictive.</p>
    <p>Nevertheless, I was able to apply mathematical abstractions to further simplify my tasks and work through the numerous challenges I encountered. With some insight from my teammates and my attendance in a single lecture of an abstract algebra course while on campus, I used the permutation group to create a structured method for debugging the firmware interface. After consulting with our team’s resident category theorist, I developed a novel category–theoretic monadic interface to ensure fault-tolerant communication with the glasses, which allowed a provably optimal deployment of AR directions for maps, which formed the first of three major components in our project.</p>
    <p>At this time you may wonder what happened to our 4th unknown interlocutor. Where was his research thrust? Simply put, I had immediately identified him as well to be a suitable yappatron (as he was a double major in the MIT Sloan School of Management), so I sent him to bed early in order to recover full energy for our presentations the following day. This, too, was simply part of the Pareto frontier.</p>
    <p>Following a grueling 20-hour hackfest in a Hayden Library study room, we had made vast amounts of progress on each of our research thrusts. Firstly, as I mentioned, I had completed the deployment of AR glasses. This involved the deployment of our own servers within the MIT intranet. I had to procure an MIT domain from MIT IS&amp;T to facilitate the glasses’ operation (side note: did you know it was that easy to get an official MIT subdomain? visit <a class="text-link external-link" href="https://functor.mit.edu" target="_blank">functor.mit.edu</a>).</p>
    <p>Second, Ananth had made substantial progress on his theoretical research. He had developed a novel algorithm based on stochastic gradient descent in raw Haskell that would compute a mapping from a 2 dimensional geographic map to an <span role="math" class="math inline-math"><svg class="typst-frame" style="overflow: visible; width: 0.6em; height: 0.683em;" viewBox="0 0 6.6 7.513000000000001" width="6.6pt" height="7.513000000000001pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:h5="http://www.w3.org/1999/xhtml"><g><g class="typst-text" transform="matrix(1 0 0 -1 0 7.513000000000001)"><use xlink:href="#g4C8FDA4DDD7AC9548F8020818392A88E" x="0" y="0" fill="#000000" fill-rule="nonzero"/></g></g><defs id="glyph"><symbol id="g4C8FDA4DDD7AC9548F8020818392A88E" overflow="visible"><path d="M 0 0m 5.907 1.507 c -0.25300026 -0.86899996 -0.6160002 -1.309 -1.0669999 -1.309 c -0.14300013 0 -0.22000027 0.11 -0.22000027 0.319 c 0 0.15399998 0.065999985 0.407 0.19799995 0.74799997 c 0.44000006 1.199 0.6600003 1.991 0.6600003 2.3979998 c 0 0.7700002 -0.5170002 1.1990001 -1.2870002 1.1990001 c -0.6489999 0 -1.21 -0.28599977 -1.661 -0.86899996 c -0.08800006 0.48399973 -0.47300005 0.86899996 -1.0339999 0.86899996 c -0.61600006 0 -0.85800004 -0.572 -1.0120001 -1.0669999 c -0.109999985 -0.352 -0.16499999 -0.5610001 -0.16499999 -0.638 c 0 -0.09899998 0.055000007 -0.15400004 0.176 -0.15400004 c 0.055000007 0 0.088 0.010999918 0.12099999 0.032999992 c 0.055000007 0.09899998 0.088 0.17599988 0.09899998 0.25300002 c 0.19800001 0.83599997 0.45100003 1.2539997 0.74799997 1.2539997 c 0.19800007 0 0.29700005 -0.1539998 0.29700005 -0.4619999 c 0 -0.14299989 -0.054999948 -0.43999982 -0.176 -0.8909998 l -0.627 -2.497 c -0.032999992 -0.143 -0.09900004 -0.42900002 -0.09900004 -0.48400003 c 0 -0.22 0.12099999 -0.32999998 0.36299998 -0.32999998 c 0.20899999 0 0.36300004 0.11 0.44000006 0.32999998 c 0.021999955 0.055000007 0.08799994 0.32999998 0.20899999 0.803 l 0.23100007 0.979 l 0.32999992 1.254 c 0.12100005 0.25300002 0.3080001 0.50600004 0.53900003 0.77 c 0.319 0.35200024 0.7149999 0.5279999 1.188 0.5279999 c 0.36299992 0 0.53900003 -0.24199963 0.53900003 -0.7149997 c 0 -0.41799998 -0.23099995 -1.2540002 -0.704 -2.5080001 c -0.07700014 -0.19799995 -0.11000013 -0.36299992 -0.11000013 -0.5059999 c 0 -0.53900003 0.40700006 -0.935 0.93499994 -0.935 c 0.4840002 0 0.86899996 0.275 1.144 0.825 c 0.20900011 0.44000006 0.31900024 0.737 0.31900024 0.88 c 0 0.09899998 -0.055000305 0.15400004 -0.17600012 0.15400004 c -0.032999992 0 -0.19799995 -0.0990001 -0.19799995 -0.23100007 Z "/></symbol></defs></svg></span>-dimensional surface where the metric distance corresponded to an arbitrary metric. For example, we could compute a “travel-time map” in 3 dimensions (so that humans can interpret it) and warp a 2D map into a 3D version such that the distance between points on the map represents the amount of time it takes to walk/public transit/drive between them rather than physical distance.</p>
    <p>There was only a slight problem with Ananth’s project: the code had never actually been ran in the 19 hours prior, as he had been far too busy furiously hacking away. Upon running the code in the 20th hour, all Haskellian safeguard broke down and the code began to spew unintelligible and mangled data. Ananth began despairing and declaring doom for our team. He even suggested the ridiculous idea of “going to bed.”</p>
    <p>Unacceptable. Were we about to back down at the last moment? We stood at the precipice of greatness—we were this close to the summit. I had already sacrificed so much (2 days worth of SRVHS classes) to be here in Cambridge. MY NAME IS WARREN “KAITOTLEX” KAITO FUTABA T. LAWRENCE LEX LIN AND I WOULD NOT TAKE ANYTHING LESS THAN FULL AND UNABATED VICTORY. I WOULD REND THE CODE ASUNDER AND FORGE IT BACK TOGETHER INTO A BEAUTIFUL SYMPHONY OF TYPES AND LAMBDAS.</p>
    <p>In any case, my division of labor approach was instrumental in our final hour. With just a couple hours left to judging, we had to quickly pivot away from Ananth’s research towards a system that actually worked: Youwen’s swarm of AI agents. While I had been hacking away on the AR glasses and Ananth had been fruitlessly wrangling Haskell types, Youwen had successfully completed the deployment of multiple AI agents that would traverse the streets of Cambridge and go sailing in the Charles. This was essentially our side project in case the main one went awry.</p>
    <p>With no choice, I began the integration of our independent research operations. This was the true genius of my plan. At the end, all of our independent research would be synthesized into the one true project that would be even greater than the sum of its parts. I retrofitted my AR glasses to display the thoughts of the AI agents so that you could view the status and thoughts of the agents at all times. We then scrapped Ananth’s project for parts and jerry-rigged it to integrate loosely with the agents to feed the high dimensional embeddings of various metrics into the agents to further embody them. With this, we had finally created our end project: a fully autonomous swarm of AI agents that would rely on our novel space-time embedding system to traverse the world and feed data back to the user in their AR glasses as they walked around.</p>
    <p>This final product was far too advanced, despite it being simply our side project. That is, this final project we submitted was simply a random throwaway program that our functor.systems hackathon team created in parallel while the main project was being developed. Our operation at the Pareto frontier was just too powerful such that even our side project still easily cleared out every single other project. Our mysterious fourth teammate returned from his slumber to present our work and after a few rounds of judging we were declared victorious, winning the Grand Prize and highest honor in the entire event. I officially became the first high schooler in (known) history to ever walk away with a HackMIT grand prize, one of, if not the most prestigious awards in collegiate hacking. Warren “Futaba” Lex, signing off.</p>
]]></summary>
</entry>
<entry>
    <title>How to set up Kindle Scribe for reading Chinese</title>
    <link href="https://web.youwen.dev/hacking/reading-chinese-on-kindle-scribe" />
    <id>https://web.youwen.dev/hacking/reading-chinese-on-kindle-scribe</id>
    <published>2025-09-13</published>
    <updated>2025-09-13T00:00:00Z</updated>
    <summary type="html"><![CDATA[    <p>Kindles are no longer sold in Mainland China, but it is still surprisingly pleasant for reading Chinese (and CJK in general!).</p>
    <p>This article will specifically target the Scribe, but it should also work on the latest other models.</p>
    <p>Here’s the deal: the Kindle is definitely designed for a Latin-reading audience. It comes with a large selection of Latin fonts and no CJK fonts. Many Chinese books are not available on the e-bookstore. The latter is an easy fix; you can obtain Chinese ebooks however you like from other storefronts and load them onto the Kindle through Calibre or your preferred methods. However, the issue remains of font selection. By default, CJK characters <em>are</em> displayed, but switches weirdly between sans and serif (Song/Ming) glyphs. Also, if you don’t like the builtin system font, tough luck.</p>
    <p>Luckily, the Kindle actually allows you to load custom fonts, a little-known feature. All you have to do is download the fonts onto your computer and drop them into the <code>fonts/</code> directory in the Kindle filesystem.</p>
    <div class="py-2 px-4 text-[0.8em] rounded-md border-1 border-slate-200 dark:border-zinc-700 bg-slate-50 dark:bg-overlay leading-[1.5em]">
      <div><span style="font-variant-caps: all-small-caps"><span class="inline-block all-smallcaps">By the way</span></span></div>
      <div class="!mb-0 mt-2 prose-p:my-1 prose-p:mt-1">You can also use the method below to load your favorite Latin fonts! I loaded my preferred font, <a class="text-link external-link" href="https://mbtype.com/fonts/valkyrie/" target="_blank">Valkyrie</a> by Matthew Butterick, also the typeface for body text on this website, and now I can enjoy it when reading all of my books.</div>
    </div>
    <h2>Loading fonts</h2>
    <p>For Windows users: this is easy. Just plug in the Kindle and it should show up shortly in File Explorer.</p>
    <p>For Linux users—not so easy. Older Kindles acted as USB Mass Storage devices, so they would show up as a USB Drive. The Scribe, and presumably other newer devices, use the <a class="text-link external-link" href="https://en.wikipedia.org/wiki/Media_Transfer_Protocol" target="_blank">Media Transfer Protocol (MTP)</a>. This means that <code>lsblk</code> or <code>fdisk -l</code> will not bring up the Scribe’s filesystem and you can’t just use <code>mount(8)</code> to access it, unlike what the top results on Google would suggest.</p>
    <p>All you will need is a way to access an MTP filesystem. If you are using <span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">GNOME</span></span>, this should require no action on your part. The <span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">gnome</span></span> <a class="text-link external-link" href="https://en.wikipedia.org/wiki/GVfs" target="_blank">GVfs</a> should support MTP out of the box, and you should be able to find it in Nautilus (also called “Files”). On my end, I’m running NixOS with Hyprland, but I have GVfs and Nautilus installed anyways, so that’s what I used.</p>
    <p>Whichever way you chose to open the file system, simply peruse the Kindle’s files until you quickly find the <code>fonts</code> folder, then drop in your font of choice. I recommend the Noto CJK family, and the Song/Ming style. By the way, variable fonts are supported too!</p>
    <p>To obtain Noto CJK, a free font developed by Google, you can grab it from the <a class="text-link external-link" href="https://github.com/notofonts/noto-cjk" target="_blank">GitHub repository</a>. Noto Serif CJK Simplified Chinese in particular can be found in <a class="text-link external-link" href="https://github.com/notofonts/noto-cjk/tree/main/Serif/Variable/OTF" target="_blank">this directory</a>. Click the file “NotoSerifCJKsc-VF.otf” and then hit the “Download Raw” icon in the top right (it looks like an arrow pointing down into a bracket). (Note the other files are Noto Serif fonts for Traditional Chinese, Japanese, Korean, and Hong Kong Traditional Chinese, which can be installed similarly.)</p>
    <p>After downloading it, simply move the OTF file into the <code>fonts/</code> directory on the Kindle, disconnect it from your computer, and then it will show up alongside the builtin fonts in the menu!</p>
]]></summary>
</entry>
<entry>
    <title>Anatomy of a NixOS module</title>
    <link href="https://web.youwen.dev/hacking/anatomy-of-a-nixos-module" />
    <id>https://web.youwen.dev/hacking/anatomy-of-a-nixos-module</id>
    <published>2025-08-31</published>
    <updated>2025-08-31T00:00:00Z</updated>
    <summary type="html"><![CDATA[A not-quite-comprehensive primer on the NixOS module system.]]></summary>
</entry>
<entry>
    <title>Three Isomorphism Theorems in Linear Algebra</title>
    <link href="https://web.youwen.dev/math/three-isomorphism-theorems" />
    <id>https://web.youwen.dev/math/three-isomorphism-theorems</id>
    <published>2025-06-19</published>
    <updated>2025-06-19T00:00:00Z</updated>
    <summary type="html"><![CDATA[A proof of the first three isomorphism theorems for linear algebra in particular.]]></summary>
</entry>
<entry>
    <title>A critique of AI in education</title>
    <link href="https://web.youwen.dev/writing/ai-in-education" />
    <id>https://web.youwen.dev/writing/ai-in-education</id>
    <published>2025-06-15</published>
    <updated>2025-06-15T00:00:00Z</updated>
    <summary type="html"><![CDATA[    <figure class>
      <img src="https://pyxis.nymag.com/v1/imgs/bed/906/0c7c764b8c807add6e1a8f638b43ce2029-chat-gpt-college-lede.2x.rsocial.w600.gif" alt="hi chat" class="rounded-md mx-auto shadow-sm dark:shadow-none shadow-gray-900" loading="lazy">
      <figcaption class="text-[0.88em] text-center text-subtle">Graphic via <a class="text-link external-link" href="https://nymag.com/intelligencer/article/openai-chatgpt-ai-cheating-education-college-students-school.html" target="_blank"><em>New York Magazine</em>.</a></figcaption>
    </figure>
    <h2>Abstract</h2>
    <p>The entire project of public education is in crisis. Falling literacy rates and steady declines in almost every metric by which we measure student attainment are a common theme across the United States. The latest solution from Big Tech—after a long track record of EdTech overpromising and underdelivering educational outcomes—seems to be AI chatbots. Marketed as research aides and instantly accessible personal tutors that will pick up the slack left by underfunded schools and demotivated teachers, educational institutions—both at the K-12 and college level—seem to have no real answer to the rampant cheating induced by these programs. So far, these chatbots have only seemed to exacerbate the worst of our problems.</p>
    <p>In this project, we explore the ramifications of the proliferation of AI chatbots across high schools and colleges, and attempt to contextualize the demand for this technology through an examination of the perverse incentives we’ve placed at the heart of our education system. We develop an argument that chatbots, while a uniquely difficult problem, are <em>also</em> a symptom of the broader structural issues embedded into our educational institutions. We place particular focus on two interviews conducted with college students who recently graduated high school amidst of the rise of AI, in addition to the experiences of students, teachers, and administrators. We also examine formal academic sources to identify broader trends and contextualize individual experience with data.</p>
    <p>By drawing from multiple personal perspectives from diverse backgrounds, solidly buttressed against broader research and evidence, we hope to develop a theory of not just <em>how</em>, but <em>why</em> AI is reshaping education, and what can be done about it.</p>
    <figure class>
      <img src="https://cdn.youwen.dev/zotgpt.png" alt="ZotGPT" class="rounded-md mx-auto shadow-sm dark:shadow-none shadow-gray-900" loading="lazy">
      <figcaption class="text-[0.88em] text-center text-subtle">Do UCI students really need this?</figcaption>
    </figure>
    <h2>Getting up to speed</h2>
    <p>Let’s not pretend that ChatGPT is still failing basic arithmetic and failing to count Rs in strawberry—the most sophisticated chatbots now produce outputs more coherent and compelling than the majority of K-12 students, and arguably many college students. They take no effort to use, and are increasingly untraceable. So why would anyone write anything themselves, when an algorithm can churn out better and faster prose at the literal press of a button? Is there any reason why we <em>shouldn’t</em> teach every kid how to prompt ChatGPT rather than toil away writing lukewarm essays by hand?</p>
    <p>Last fall, Roy Lee transferred to Columbia University, after years of hard work at community college. He then proceeded to generate every assignment he received using ChatGPT, and he would “insert 20 percent of [his] humanity, [his] voice, into it.” He declared that “[most] assignments in college are not relevant” and that they are “hackable” by AI <span id="loc-1">(</span><a href="#loc-7" role="doc-biblioref">Walsh</a>).</p>
    <div class="py-2 px-4 text-[0.8em] rounded-md border-1 border-slate-200 dark:border-zinc-700 bg-slate-50 dark:bg-overlay leading-[1.5em]">
      <div><span style="font-variant-caps: all-small-caps"><span class="inline-block all-smallcaps">By the way</span></span></div>
      <div class="!mb-0 mt-2 prose-p:my-1 prose-p:mt-1">Funny story—it just so happens that I actually used to know Roy. I took dual enrollment classes at the community college he attended in the Bay Area before he transferred to Columbia. I was aware about his interview cheating tool months before it went viral. It’s funny how these things work out.</div>
    </div>
    <p>How have we reached a point where even some of the best students in the country, admitted to the most prestigious schools, view education with such contempt and apathy, that they are eager to bypass the entire project using a chatbot?</p>
    <p>Lee is perhaps abnormally flagrant in his use of AI. But a study finds that a quarter of US teens use ChatGPT for schoolwork—the actual number is almost certainly higher <span id="loc-2">(</span><a href="#loc-8" role="doc-biblioref">Olivia S.</a>).</p>
    <figure class>
      <img src="https://cdn.youwen.dev/chatgpt-iq.png" alt="ChatGPT what is my IQ?" class="rounded-md mx-auto shadow-sm dark:shadow-none shadow-gray-900" loading="lazy">
      <figcaption class="text-[0.88em] text-center text-subtle">What a thoughtful tool that will never mislead students.</figcaption>
    </figure>
    <p>Here’s the two pieces of the puzzle we’ll look at. One—is introducing AI chatbots into education harmful? Two—if it is, what incentives in our education system are pushing students to rely on AI, even despite them possibly knowing the detrimental effects? The story is more complex than simply student laziness or underattainment, since even hardworking and high-achieving students like Lee are caught up in it.</p>
    <p>Before we dive deeper, let’s clear something else up as well.</p>
    <p>When chatbots first released, many ridiculed them and proclaimed that they were useless toys which could not produce any real meaningful outputs. Tech companies got to work for 2 years—improving their capabilities to the point where genuine nontrivial tasks could be performed to moderate or great success. ChatGPT now is an entirely different story from the ChatGPT 3 years ago, in terms of effectiveness in cheating on assignments.</p>
    <p>I bring this up to highlight an important point about chatbots, and generative AI in general, which we should keep in mind. We cannot criticize them solely, or even primarily, on account of them producing poor outputs. Because the companies producing them <em>will</em> fix the bugs, and they’ll keep making the tech better. (Previously, ChatGPT essays were hilariously easy to spot and universally nonsensical, even for high school writing. Now, a properly prompted chatbot essay from a cutting-edge language model may exceed many high school students’ writing in quality.) Throughout this project, we will be more focused on critique on the basis of how the use of AI tools itself is detrimental to students, and view the rise of cheating with AI as an acceleration of an existing force in our educational institutions that has already been at work for years. We won’t consider any criticisms based on the poor quality of AI output, because I think those are by far the weakest.</p>
    <p>So here’s the goal—by the end, we should begin seeing why we should be cautious when introducing AI into education, as well as how our incentives for education need to change, such that even if chatbots and generative AI continue advancing in capability, education continues to be valued.</p>
    <h2>A couple conversations on education and AI</h2>
    <p>The best place to get a perspective on the state of education is presumably to talk to the students themselves. Our goal—find out more about the forces behind why a student like Roy Lee might head to one of the world’s preeminent educational institutions—and refuse to even study.</p>
    <p>Towards this end, I spoke with two friends from high school, now in college, and asked them what they thought. First, let’s talk to Isaac, who now attends UC Berkeley. (Both consented to the publication of their interviews.)</p>
    <figure class>
      <img src="https://cdn.youwen.dev/isaac-mission-peak.jpg" alt="isaac (right)" class="rounded-md mx-auto shadow-sm dark:shadow-none shadow-gray-900 max-w-80" loading="lazy">
      <figcaption class="text-[0.88em] text-center text-subtle">Isaac (right) and a few other friends, at the summit of Mission Peak.</figcaption>
    </figure>
    <div class="py-2 px-4 text-[0.8em] rounded-md border-1 border-slate-200 dark:border-zinc-700 bg-slate-50 dark:bg-overlay leading-[1.5em]">
      <div><span style="font-variant-caps: all-small-caps"><span class="inline-block all-smallcaps">By the way</span></span></div>
      <div class="!mb-0 mt-2 prose-p:my-1 prose-p:mt-1">
        <p>Before reading, you may want to listen to the actual recording of the interview with Isaac—sorry, the one with Ananth is not available, the quality was terrible.</p>
        <audio controls class="mt-4"><source src="https://cdn.youwen.dev/isaac-interview-edited.mp3" type="audio/mpeg"></audio>
      </div>
    </div>
    <h3>Why go to college &amp; get educated?</h3>
    <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">
      <p><strong>Youwen:</strong> <em>What do you think the purpose of college is right now for the average student?</em><br><br><strong>Isaac:</strong> Yeah, so I think generally speaking, most people would agree that the large majority of people go to college to get a job and set themselves up for success in later life.</p>
      <p>I’ve had some experience, that have opened my eyes that. College really isn’t just this I guess stepping stone you would say into future or into joining the workforce. But as for what I think college should be, I don’t think it’s unfair to treat college as a something or a stepping stone or something that’s preparing you for work.</p>
      <p>Just with how expensive it is, people are paying like, $200,000 to get a degree there. There should be some like tangible, materialistic benefit to getting a college degree, other than just like a intellectual development. But I do think that’s like a very major aspect also of college.</p>
    </blockquote>
    <p>Isaac makes a good point here. In principle, college shouldn’t just a place to get a degree for a high paying job later on, but rather a place to find and improve yourself through education. However, the realities of our society also heavily incentivize treating college as an investment on a later return. In particular, the emphasis is on <em>results</em>—test scores, GPA, extracurriculars to neatly fit on a resume—and not the process of learning itself. Throughout this project I will be largely critical of this, with the understanding that it is not a system any individual can opt out of.</p>
    <p>Now let’s speak with Ananth, who is now a rising sophomore at MIT. I asked him the same question.</p>
    <figure class>
      <img src="https://zardini.mit.edu/img/people/AnanthVenkatesh.jpeg" alt="ananth venkatesh" class="rounded-md mx-auto shadow-sm dark:shadow-none shadow-gray-900 max-w-64" loading="lazy">
      <figcaption class="text-[0.88em] text-center text-subtle">Headshot of Ananth. Courtesy of the <a class="text-link external-link" href="https://zardini.mit.edu/people/ananth-venkatesh/" target="_blank">MIT Zardini Lab</a>.</figcaption>
    </figure>
    <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">
      <p><strong>Youwen:</strong> <em>What are your educational goals? And by that what do you hope to get out of college?</em><br><br><strong>Ananth:</strong> I’m primarily interested in gaining some experience with research in these areas.</p>
      <p>And specifically different ways of thinking about very abstract problems in academia.</p>
    </blockquote>
    <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">
      <p><strong>Youwen:</strong> <em>That sounds very interesting. Broadly speaking, tell me what you see as the purpose of like college and education in general?</em><br><br><strong>Ananth:</strong> The broader purpose of college is to prepare the greatest number of people to lead careers in academia and to do research in fundamental science and other disciplines that are going to truly result in some groundbreaking innovation.</p>
      <p>So I think unfortunately, a lot of American universities prioritize career readiness, people getting jobs, other trivialities over what should really be the fundamental goal of any institution of higher education, which is to prepare people to do research and really think about academic disciplines in new ways, because that is really the ultimate goal of humanity.</p>
      <p>And in doing so, we basically are able to achieve the most progress and the greatest innovation. So that should really be the goal of any policies surrounding higher education.</p>
    </blockquote>
    <p>Ananth’s view is interesting, and arguably rather uncommon. He says that a lot of students have picked up the idea that college is some sort of a job training center rather than an institution of higher learning.</p>
    <p>This seems to be corroborated by our other accounts. James D. Walsh, in <em>New York Magazine</em>’s <em>Intelligencer</em>, talks to a few students about AI. One of them was our good friend Roy Lee from earlier, who explicitly sees college as nothing more than a place to “meet your wife and [startup] co-founder.”</p>
    <p>Ultimately, it’s this culture where the actual education matters less than the <em>outcomes</em> you can get from going to a prestigious university which heavily incentivizes students to turn to AI to cheat on assignments.</p>
    <h3>What do people think about AI?</h3>
    <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]"><strong>Youwen:</strong> <em>What has been the general attitude amongst classmates and your peers towards AI?</em><br><br><strong>Isaac:</strong> The large sentiment from coming to college and in high school is that people just don’t really care. Like, it’s easy and it’s accessible and I think people don’t really feel they’re hurting anybody from using it.</blockquote>
    <p>Studies corroborate this—students view AI as either positive or neutrally <span id="loc-3">(</span><a href="#loc-9" role="doc-biblioref">Zieve-Cohen et al.</a>; <a href="#loc-8" role="doc-biblioref">Olivia S.</a>).</p>
    <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]"><strong>Youwen:</strong> <em>Tell me about people incorporating AI into their work. How do you square that with this other goal of college you talked about earlier, where it’s not just to complete assignments, but to develop yourself?</em><br><br><strong>Isaac:</strong> It’s just, it’s less fulfilling. Because of the resources available and the use of AI. Yeah I guess in general, I think the meaningfulness of it has just been diminished by people using AI.</blockquote>
    <h3>Why are students turning to AI?</h3>
    <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]"><strong>Youwen:</strong> <em>Why do you think so many students are turning to AI to complete their assignments or even to just assist them?</em><br><br><strong>Isaac:</strong> And then if you see all your peers using it too, it just, it strongly incentivizes you to participate as well. So yeah, I would say just the, it’s easy, it’s accessible, and seeing your peers use it also is a motivator.</blockquote>
    <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">
      <p><strong>Youwen:</strong> <em>Do you feel affected by it?</em><br><br><strong>Isaac:</strong> Yeah, it’s definitely tempting.</p>
      <p>[In] one of the most recent classes I had, we had this final paper and I had this buddy I was talking to because they gave us two months to write it. And I spent a lot of time on the paper and we were talking about it and it was like the day before it was due, he was like, “oh yeah, I’m just gonna use chat to write it up.</p>
      <p>It’s frustrating ‘cause you put all that work in.</p>
    </blockquote>
    <p>A recent study shows a positive correlation between those who rely heavily on AI and the personality trait of learned helplessness—demonstrated precisely in this anecdote <span id="loc-4">(</span><a href="#loc-10" role="doc-biblioref">Azeem S.</a>). Isaac’s friend can no longer even write on their own, and they make no effort to change this.</p>
    <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">
      <p><strong>Youwen:</strong> <em>What’s the general attitude amongst classmates and your peers towards AI? Are they generally open to using it?</em><br><br><strong>Ananth:</strong> I think AI use is highly encouraged. People are generally using it to automate tedious tasks. I think especially in academia part of the appeal of AI agents is that we can just take a lot of these calculations or basically work that you had just assigned to someone else, and instead give it to an AI agency so you can clear your mind to think at a higher level of abstraction.</p>
      <p>And this is really a powerful new tool that we could be using to do research. So in general, I think there’s widespread acceptance of AI tools.</p>
    </blockquote>
    <p>What Ananth brings up is worth looking into. When you’re operating at very high levels, AI can be helpful in performing low-level repetitive tasks. Though frankly, many students are not yet operating at these sorts of high levels Ananth describes.</p>
    <p>Additionally, we still need to be careful when considering the ethical implications of AI, as things are often not as simple as they seem. For example, teachers are using AI to accelerate their work, creating lesson plans and grading assignments <span id="loc-5">(</span><a href="#loc-11" role="doc-biblioref">Goldstein</a>). This seems benign on the surface, but as we discuss further below, has profound ethical implications.</p>
    <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">
      <p><strong>Youwen:</strong> <em>Has you had any experience with, people using AI for [completing assignments] and how has it affected, if at all, your experience in classes?</em><br><br><strong>Ananth:</strong> I think practically everyone uses that some form of AI tools on most assignments. And I think generally they’re a tool, they’re not able to solve like entire problems.</p>
      <p>So they’re really only useful for getting like guiding principles or maybe regurgitating information from a lecture. But by themselves they have very limited capacity to solve, like intellectually challenging problems.</p>
    </blockquote>
    <p>Around 26% of students are using AI for assignments (<a href="#loc-8" role="doc-biblioref">Olivia S.</a>), which is likely an underestimate. So, not quite the “everyone” Ananth describes, but a significant chunk.</p>
    <h2>A thought experiment about ethical AI use</h2>
    <p>In the previous section, we briefly touched on the ethics of AI use, and we expound upon that here, in particular highlighting how “ethical AI use” is often <em>not</em> ethical upon further inspection.</p>
    <p>In <em>Camera Obscura</em>, AI litigator Matthew Butterick makes the argument that even as a tool, AI can be problematic. For instance, one use of AI that seems benign which is raised by both Ananth in his interview and by some of the other sources is to proofread/rewrite essays with better grammar and diction. To further analyze this situation, Butterick formulates this thought experiment: consider two teachers, Alice and Bob, who adopt different policies towards AI.</p>
    <p>Alice’s system allows students to draft their papers using AI, provided they are responsible for all fact-checking and ensuring the paper meets all of the expected guidelines.</p>
    <p>Bob instead introduces an “AI essay validator” that takes in students’ papers and creates a “validated” version that is changed in either subtle or major ways. Students must submit the validated versions of their paper.</p>
    <p>It turns out that Bob’s AI essay validator simply just secretly discards the original essay and generates a new one to replace it with (with ChatGPT, etc.).</p>
    <p>Most people would agree that Bob’s system is obviously unethical. Some students have their paper minimally changed, but others have entire core arguments reformulated and rewritten. Clearly, the system is suppressing the free expression of the students, and ensuring every paper is aligned with the biases baked into the chatbot powering the AI essay validator.</p>
    <p>But a lot of people would say that in contrast, Alice’s system <em>is</em> ethical. AI is a tool, and it’s not going away, so it doesn’t make sense to restrict students from using a tool, right? As long as they remain responsible for their work and any mistakes, they simply become more efficient writers.</p>
    <p>Butterick argues that, in fact, there is no morally relevant difference between Alice’s and Bob’s systems. True, Alice’s is more transparent, which is a boon. But in either case, students’ thoughts and free speech are effectively being suppressed in exactly the same way—by being filtered through an AI chatbot. In Alice’s case, it’s just being done voluntarily by students with her permission—but the outcome is effectively the same.</p>
    <p>So if you agree with the argument above—and accept that Bob is clearly acting unethically—then it’s clearly unethical to permit students to draft papers with AI as well.</p>
    <p>We already have a culture of exclusivity in academia built around “academic” language and writing, but AI gives everyone a way to write in an “academic” way. When there are real advantages for many students to adopt an unaffected academic tone over their own less polished writing, it’s clear why they would turn to chatbots to regurgitate their writing into a generic but polished form. But as we discussed above, this is effectively equivalent to filtering them through Teacher Bob’s fake essay validator, and detrimental to both the students and the free exchange of thought in general.</p>
    <h3>Tangible impacts</h3>
    <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">
      <p><strong>Youwen:</strong> <em>Do you see any kind of negative impacts on your peers when they are so reliant [on AI]?</em><br><br><strong>Isaac:</strong> I remember at the end of the semester, they [a friend in college] actually told me that they were like, they felt incapable of actually like writing an essay.</p>
      <p>I really do think if you become totally self-reliant on it, like that, it just becomes a crutch and it prohibits you from actually growing.</p>
    </blockquote>
    <p>Another student named Wendy interviewed by Walsh in <em>Intelligencer</em>, this time a freshman finance major, says that she is against “copy and pasting” and “cheating and plagiarism.” But she relies heavily on AI tools for essay writing. She has it down to a science—first, she tells the AI not to use advanced English because she is a freshman. Then, she provides some background on the class to make sure the writing is in the correct context, followed by the actual assignment instructions. Finally, she asks for an outline of the essay to write—the main points of each paragraph, the general structure and organization. Then all that remains is to write the essay herself, with the outline provided by the chatbot.</p>
    <p>Again, this is a process that feels responsible and ethical at first glance. Indeed, Wendy is not even downright contemptuous of education like Lee—she acknowledges that copy-pasting and plagiarism are wrong. But clearly she feels that her approach is responsible and uses AI constructively. However, her approach clearly just takes us back to a situation like Bob’s essay validator—only in this case, rather than having the entire essay generated, she generates all of the <em>ideas</em> and writes them into prose herself, turning writing from a research process into a fill-in-the-blank.</p>
    <figure class>
      <img src="https://brevity.wordpress.com/wp-content/uploads/2023/02/screen-shot-2023-02-08-at-2.44.01-pm.png" alt="mad libs" class="rounded-md mx-auto shadow-sm dark:shadow-none shadow-gray-900 max-w-80" loading="lazy">
      <figcaption class="text-[0.88em] text-center text-subtle">Wendy’s AI use reduces her involvement to essentially playing Mad Libs.</figcaption>
    </figure>
    <p>Absurdly, Wendy actually fully recognizes the importance of the research process.</p>
    <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">“Honestly,” she continued, “I think there is beauty in trying to plan your essay. You learn a lot. You have to think, Oh, what can I write in this paragraph? Or What should my thesis be? ”</blockquote>
    <p>But in high school english class, back when she wrote her own essays, she would get bad grades. Now, with the help of ChatGPT, she gets good grades effortlessly with her new method. “An essay with ChatGPT, it’s like it just gives you straight up what you have to follow. You just don’t really have to think that much.”</p>
    <p>In their study, <a href="#loc-9" role="doc-biblioref">Zieve-Cohen et al.</a> note that in some interviews they conducted with students, when asked about whether a student uses the school’s “Writing Center” for help, it carries a negative connotation (the writing center was for people with “problems with writing”), while asking if they used ChatGPT for assistance carried no such negative connotations.</p>
    <p>In Wendy’s case, she clearly care more about the final result (her grade in the course) than her actual learning. Wendy <em>knows</em> that it’d be better for her to do her own writing (ironically, the essay she wrote with the help of AI was literally about how learning is what makes us human), but getting a good grade is more important to her. With the students in <a href="#loc-9" role="doc-biblioref">Zieve-Cohen et al.</a>‘s study, they see working to improve their writing skills as a sign that there was a <em>problem</em> with their writing in the first place, while using ChatGPT to assist with drafting or brainstorming is totally fine.</p>
    <figure class>
      <img src="https://www.pewresearch.org/wp-content/uploads/sites/20/2023/11/SR_23.11.21_ai-roundup_2.png" alt="chart about american views of AI" class="rounded-md mx-auto shadow-sm dark:shadow-none shadow-gray-900" loading="lazy">
      <figcaption class="text-[0.88em] text-center text-subtle">Many Americans take Wendy’s view—that it’s acceptable to use AI as long as the final writing is done by the student.</figcaption>
    </figure>
    <p>In either case, rather than being incentivized to prioritize their actual learning (even when they are aware of it, like Wendy) and work to improve their skills in areas they are weak in, ChatGPT provides an easy out. Under a system where the only way through is getting good grades and a high <span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">gpa</span></span>, it is clear why students take the easy way through. As Ananth mentioned, if the focus is primarily on maintaining high grades for college admissions, or for job recruitment, then actual skills and education will always fall to the wayside in favor of whatever flawed metrics jobs and colleges targeting. The system has been like this, for a long time. With AI, this divide has only become far more pronounced (it used to take actual effort to cheat effectively!).</p>
    <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">
      <p><strong>Youwen:</strong> <em>Are there any unexpected consequences to the widespread adoption [of AI]?</em><br><br><strong>Ananth:</strong> I think a recent one is Grok AI, which was released by Elon Musk, has started talking about white genocide and the Boer genocide in South Africa. Which is very interesting because it’s been able to turn any conversation into a conversation about white genocide, really furthering the views of the fascist administration and bringing in these geopolitical issues into the Twitter mainstream, where you have all of these very interesting individuals engaging in political discussions with AI chatbot that seems to have grown a mind of its own after being indoctrinated by Elon Musk.</p>
      <p>I think that’s a very interesting example of where you can have new behaviors and especially unexpected behaviors emerge from some set of training.</p>
    </blockquote>
    <figure class>
      <img src="https://substackcdn.com/image/fetch/w_1456,c_limit,f_webp,q_auto:good,fl_progressive:steep/https%3A%2F%2Fsubstack-post-media.s3.amazonaws.com%2Fpublic%2Fimages%2F40228a08-5457-4884-b223-01bbd9cb44c4_2358x967.png" alt="grok white genocide" class="rounded-md mx-auto shadow-sm dark:shadow-none shadow-gray-900" loading="lazy">
      <figcaption class="text-[0.88em] text-center text-subtle">Pictured: Elon Musk’s Grok AI chatbot, after it was deliberately programmed to derail unrelated conversations into white genocide conspiracy theories. Can we really trust this thing to teach kids about history?</figcaption>
    </figure>
    <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">
      <p><strong>Youwen:</strong> <em>If you have AI not just influence people online, but it’s deployed into the education system, then if it’s sapping people of the ability to do critical thinking for people and they’re not making their own choices in school, do you feel that kind of bleeds out into society at large?</em><br><br><strong>Ananth:</strong> Yeah, certainly. I think definitely schools and education are a huge part of achieving achieving greater awareness for what the values and political system look like. Helping educate people about their role in society and civic duties. But. I think, yeah, you definitely need to restructure the education system to look at things from a more investigative research based manner where we look at, how do we actually improve the political system?</p>
      <p>People generally do not have any coherent political beliefs. So that leads to a lot of erosion of democracy that could significantly be improved if we had better education system.</p>
    </blockquote>
    <figure class>
      <img src="https://static01.nyt.com/images/2023/02/07/technology/disinfp-chatgpt-promos-promo/disinfp-chatgpt-promos-promo-videoSixteenByNineJumbo1600.jpg?year=2023&amp;h=901&amp;w=1600&amp;s=a4544c195933ddec6e3f4d16ccf41a1eccbc58a2437d6eb9aab45d3c5bc49cdd&amp;k=ZQJBKqZ0VN&amp;tw=1" alt="chatgpt instructed to write about Alex Jones conspiracy theories about Sandy Hook" class="rounded-md mx-auto shadow-sm dark:shadow-none shadow-gray-900" loading="lazy">
      <figcaption class="text-[0.88em] text-center text-subtle">ChatGPT makes it significantly easier for anyone to generate massive amounts of credible-looking misinformation.</figcaption>
    </figure>
    <h3>Solutions?</h3>
    <p>We talked for a bit about what solutions we could envision—but none seemed very compelling.</p>
    <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">
      <p><strong>Youwen:</strong> <em>What does [higher ed and K-12] need to do so that students are less incentivized to take the path of least resistance, which in this case is AI?</em><br><br><strong>Isaac:</strong> It’s challenging just ‘cause of how accessible it is.</p>
      <p>I tutor kids online and I had this one sixth grader that I tutored. Their parent or their mom actually came to me and she was talking about how she found out that he had recently discovered ChatGPT and was using it for all his assignments.</p>
      <p>And she had asked me to talk with him about it, and I did. And I think we had a productive conversation. I think hopefully he, he is, he’d be more discouraged to use AI in the future. And I think just having those conversations from figures of authority and the future generations’ lives that could be productive and possibly prevent like using AI as a crutch.</p>
    </blockquote>
    <p>Teachers have been trying their best to deal with AI by having productive conversations with their students and implementing changes in their lesson plans <span id="loc-6">(</span><a href="#loc-12" role="doc-biblioref">Waxman</a>). Some examples: responsibling disclosing the use of AI, prompting the AI to help generate ideas, not write the essays, and using AI for feedback on work. Referring back to our previous discussions, we can already see how these methods may be ethically dubious.</p>
    <p>At the end of the day, it seems that there is not really a foolproof method to allowing students to coexist with AI in the classroom.</p>
    <h3>Conclusion</h3>
    <p>So what exactly can we do? As a rule, schools should probably ban AI. I also believe that most won’t, because this problem is structural and the roots of which reach deep into educational institutions. Indeed, no university wants to be seen as the anti-progress institution that isn’t embracing this revolutionary new technology. Deeper than that, if we are to make any progress on the cultural front (i.e. making students actually care about their work such that they don’t feel compelled to complete it with AI), we need to reorient the view that getting your education is just a means to an end (getting into a good college, high paying job, etc.) into viewing education as a worthwhile end in and of itself.</p>
    <p>For starters—on a local level, K–12 teachers should probably start doing all their assignments in-class, on paper again. College professors are already returning to blue books and primarily in-person assessments. School districts and universities can adopt stricter policies to minimize activities which are easy for students to cheat on with AI. All of these are things that can be done in individual classrooms or schools.</p>
    <p>But the educational and cultural reform we are really looking for goes beyond any individual policies. For one—though I’ve been critical of the idea that education should be a means to a high paying job—we cannot expect people in a society as unequal and unforgiving as ours to not view education as primarily a social mobility ladder. So a real fix will likely come from a progressive movement larger than even the education system itself.</p>
    <p>In the meantime, within education in particular, we should work to reduce the stigma around failure and place more emphasis on improvement and progress. Earlier, we discussed Wendy and the other students who used AI because they didn’t want to try and genuinely improve their skills, out of fear of bad grades or social stigma.</p>
    <p>Thanks to AI, it’s now easier than ever for students to avoid taking risks and applying themselves to their education. In response, it’s more important than ever for the education system to be more lenient with mistakes, de-emphasize conformance with the status quo, and allow people to experiment and fall.</p>
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    </div>
    <h2>Annotated bibliography</h2>
    <div class="py-2 px-4 text-[0.8em] rounded-md border-1 border-slate-200 dark:border-zinc-700 bg-slate-50 dark:bg-overlay leading-[1.5em] my-3">
      <p class="border-b-1 border-b-foreground pb-2"><a href="#loc-7" role="doc-biblioref">Walsh, James D. “Everyone Is Cheating Their Way Through College: Chatgpt Has Unraveled the Entire Academic Project.” <em>Intelligencer (New York Magazine)</em>, May 2025, <a href="https://nymag.com/intelligencer/article/openai-chatgpt-ai-cheating-education-college-students-school.html">https://nymag.com/intelligencer/article/openai-chatgpt-ai-cheating-education-college-students-school.html</a></a></p>
      <p>This <em>New York Magazine</em> article is a long discussion of a few particular students’ experiences with ChatGPT and other chatbots. It begins with a student at Columbia University named Roy Lee, who expresses his disdain for the education system and college. He built a ChatGPT powered interview cheating tool that gives people an invisible AI assistant overlay in Zoom interviews to answer questions for them. In his eyes, school is nothing more than a place to network (“It’s the best place to meet your co-founder and your wife”) and assignments are “hackable” and worthless (since they can be done by ChatGPT). This sentiment really captures exactly the rot in the education system that I’m looking to describe—the incentives are such that everyone sees education as a career advancement opportunity. This starts at “where will we use this in real life?” in math class and ends with someone like Roy, declaring his entire core college education as “worthless” because it’s not helping him build his startup. The rest of the article also goes over a few really good anecdotes and interviews, which will be very useful in my project.</p>
      <p>Funny story—I actually used to know Roy in high school. I took dual enrollment classes at the community college he attended in the Bay Area before he transferred to Columbia. He texted me about his interview cheating tool months before it went viral—he originally wanted to recruit me to work on it. It’s funny how these things work out. Also, he was just as insufferable in person as the article portrays him.</p>
    </div>
    <div class="py-2 px-4 text-[0.8em] rounded-md border-1 border-slate-200 dark:border-zinc-700 bg-slate-50 dark:bg-overlay leading-[1.5em] my-3">
      <p class="border-b-1 border-b-foreground pb-2"><a id="loc-18" href="#loc-13" role="doc-biblioref">Classrooms, New. Mar. 2024, <a href="https://www.youtube.com/watch?v=nMxPTKQ4OB8">https://www.youtube.com/watch?v=nMxPTKQ4OB8</a></a></p>
      <p>This is an interview with a few former school administrators discussing the role of AI in education. It takes an positive and optimistic tone, talking about the many “exciting opportunities” AI can bring in the short term. They also advocate for longer-term structural change—to better integrate AI, to produce the “educational outcomes we want this technology to create.”</p>
      <p>I selected this interview because it’s a perspective from school admin, not teachers or students, which we’ve been looking at primarily. The interests of administrators are unfortunately often not entirely aligned in the best interest of students and teachers. Also, note that these former administrators now work for an EdTech-adjacent company called New Classrooms. Their perspective, then, is similar to that of tech companies, that we should integrate AI into education because it’s <em>obviously</em> good.</p>
    </div>
    <div class="py-2 px-4 text-[0.8em] rounded-md border-1 border-slate-200 dark:border-zinc-700 bg-slate-50 dark:bg-overlay leading-[1.5em] my-3">
      <p class="border-b-1 border-b-foreground pb-2"><a id="loc-19" href="#loc-14" role="doc-biblioref">Kelly, Mary Louise. <em>'Everybody Is Cheating': Why This Teacher Has Adopted an Open Chatgpt Policy</em>. 26 Jan. 2023, <a href="https://www.npr.org/2023/01/26/1151499213/chatgpt-ai-education-cheating-classroom-wharton-school">https://www.npr.org/2023/01/26/1151499213/chatgpt-ai-education-cheating-classroom-wharton-school</a></a></p>
      <p>This is an article/interview with a professor (Professor Mollick) who has adopted an “Open ChatGPT Policy.” He says that since students will use it anyways, he might as well integrate it officially into his syllabus.</p>
      <p>This seems to be a common occurrence across universities. (Indeed, Writing 60 seems to have done this too. It’s a sensible conclusion to draw.) But one particular remark I’d like to note is near the end:</p>
      <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">
        <p>Mollick agrees, but isn’t convinced that educators can ever truly stop cheating.</p>
        <p>“I don’t think human nature changes as a result of ChatGPT. I think capability did.”</p>
      </blockquote>
      <p>Mollick says that everyone cheating with ChatGPT is just human nature and there’s nothing we can do about it. I disagree. There are actions that can be taken now, as well as policy changes and fundamental restructurings that need to happen, but we needn’t accept the way things are going as human nature.</p>
    </div>
    <div class="py-2 px-4 text-[0.8em] rounded-md border-1 border-slate-200 dark:border-zinc-700 bg-slate-50 dark:bg-overlay leading-[1.5em] my-3">
      <p class="border-b-1 border-b-foreground pb-2"><a href="#loc-11" role="doc-biblioref">Goldstein, Dana. “Teachers Worry About Students Using A.I. But They Love It for Themselves..” <em>The New York Times</em>, Apr. 2025, <a href="https://www.nytimes.com/2025/04/14/us/schools-ai-teachers-writing.html">https://www.nytimes.com/2025/04/14/us/schools-ai-teachers-writing.html</a></a></p>
      <p>An article about educators experimenting with using AI to help accelerate their work. Jon Gold, a middle school history teacher, uses generative AI for lesson planning by feeding it curriculum pages. But, he says that he does not want students to use AI themselves to write the essays he assigns. Here we see a complicated dynamic—AI can legitimately help overworked and underpaid teachers do work much faster. But the quality is dubious, and, it would be far more sensible to just not underpay or overwork them.</p>
      <p>The article also shows a general attitude of educators being in favor of “AI literacy.” We see two formulations, broadly. One is to teach students how to use AI effectively. I think this is unnecessary, and as other sources show, detrimental. The other is that students should understand the limitations of the technology and how it can be wrong. I think this is important.</p>
    </div>
    <div class="py-2 px-4 text-[0.8em] rounded-md border-1 border-slate-200 dark:border-zinc-700 bg-slate-50 dark:bg-overlay leading-[1.5em] my-3">
      <p class="border-b-1 border-b-foreground pb-2"><a href="#loc-9" role="doc-biblioref">Zieve-Cohen, Mariana, et al. “Balancing Efficiency and Ethics: Student Perspectives on Chatgpt.” <em>The Peer Review</em>, vol. 9, no. 1, 2023, <a href="https://thepeerreview-iwca.org/issues/issue-9-1/balancing-efficiency-and-ethics-student-perspectives-on-chatgpt/">https://thepeerreview-iwca.org/issues/issue-9-1/balancing-efficiency-and-ethics-student-perspectives-on-chatgpt/</a></a></p>
      <p>A study conducted to analyze sentiment towards AI amongst students, through in-depth, structured interviews conducted over Zoom. 33% expressed negative attitudes, 40% expressed positive, and 26.7% were neutral. Students “demonstrated awareness and thoughtfulness regarding their use of ChatGPT.” It also mentions that in some interviews, when asked about whether a student uses the “Writing Center” for help, it carries a negative connotation as if the writing center was for people with “problems,” while ChatGPT had no such connotations. Perhaps this is worth looking further into.</p>
    </div>
    <div class="py-2 px-4 text-[0.8em] rounded-md border-1 border-slate-200 dark:border-zinc-700 bg-slate-50 dark:bg-overlay leading-[1.5em] my-3">
      <p class="border-b-1 border-b-foreground pb-2"><a id="loc-20" href="#loc-15" role="doc-biblioref">Butterick, Matthew. “Camera Obscura: The Case against AI in Classrooms.” <em>Chron</em>, <a href="https://matthewbutterick.com/chron/camera-obscura-the-case-against-ai-in-classrooms.html">https://matthewbutterick.com/chron/camera-obscura-the-case-against-ai-in-classrooms.html</a></a></p>
      <p>This is an opinion piece that also incorporates some teacher perspectives. The author is a lawyer who is currently litigating against ChatGPT and other AI tools for alleged copyright infringement. The article makes an argument for why AI shouldn’t be incorporated into the education system, arguing that we don’t just aim to teach facts to children in schools. Rather, the process of research and critically examining sources of information is more important. Even if AI could research and produce information that students need only fact check, students should still be researching on their own.</p>
    </div>
    <div class="py-2 px-4 text-[0.8em] rounded-md border-1 border-slate-200 dark:border-zinc-700 bg-slate-50 dark:bg-overlay leading-[1.5em] my-3">
      <p class="border-b-1 border-b-foreground pb-2"><a href="#loc-10" role="doc-biblioref">Azeem S., Abbas M. <em>Personality Correlates of Academic Use of Generative Artificial Intelligence and Its Outcomes: Does Fairness Matter?</em>. Mar. 2025, <a href="https://doi.org/10.1007/s10639-025-13489-6">https://doi.org/10.1007/s10639-025-13489-6</a></a></p>
      <p>This is an academic study on personality traits that arise/are associated with from reliance on GenAI tools. In particular, academic performance and academic self-efficacy were both negatively associated with use of GenAI while learned helplessness (that is, the induced inability to solve problems that the AI chatbots cannot solve) is associated positively with GenAI. This mainly confirms, empirically, facts that are easy to guess from the interviews and other sources.</p>
    </div>
    <div class="py-2 px-4 text-[0.8em] rounded-md border-1 border-slate-200 dark:border-zinc-700 bg-slate-50 dark:bg-overlay leading-[1.5em] my-3">
      <p class="border-b-1 border-b-foreground pb-2"><a id="loc-21" href="#loc-16" role="doc-biblioref">Lake, Robin. <em>AI Is Coming to U.S. Classrooms, But Who Will Benefit?</em>. May 2024, <a href="https://crpe.org/ai-is-coming-to-u-s-classrooms-but-who-will-benefit/">https://crpe.org/ai-is-coming-to-u-s-classrooms-but-who-will-benefit/</a></a></p>
      <p>A research publication from the Center for Reinventing Public Education at ASU. It mainly discusses the speed of adoption of AI by teachers in particular, and barriers to it. This provides some perhaps useful statistical data, but the main takeaway from it should be its cavalier attitude to AI adoption. Throughout, it only ever considers responses from teachers based on their perception of the improvements made by AI, not actual empirical data or controlled studies. Indeed, most of the other studies examined have shown that AI actually leads to various deleterious outcomes in students’ personalities and performance. This article, then, serves as a good model of the attitude around EdTech and AI—adopt first, ask questions later—lest you be left behind by the coming revolution.</p>
    </div>
    <div class="py-2 px-4 text-[0.8em] rounded-md border-1 border-slate-200 dark:border-zinc-700 bg-slate-50 dark:bg-overlay leading-[1.5em] my-3">
      <p class="border-b-1 border-b-foreground pb-2"><a id="loc-22" href="#loc-17" role="doc-biblioref">Blose, Abreanna. <em>As Chatgpt Enters the Classroom, Teachers Weigh Pros and Cons</em>. 12 Apr. 2023, <a href="https://www.nea.org/nea-today/all-news-articles/chatgpt-enters-classroom-teachers-weigh-pros-and-cons">https://www.nea.org/nea-today/all-news-articles/chatgpt-enters-classroom-teachers-weigh-pros-and-cons</a></a></p>
      <p>Some data from this article: 1 in 4 teachers have caught students cheating u sing ChatGPT or other chatbots. 43% of educators feel that AI will make their jobs more difficult while the rest predict it will make their lives easier.</p>
      <p>Some experiences from teachers are also provided showcasing how AI is materially helping them in the classroom.</p>
      <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">“With Chat, you can translate all of the supplementary materials into [students’] native language, which makes it far easier for them to complete their assignments,” says Shields.</blockquote>
      <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">
        <p>The chatbot also can rewrite responses at different reading levels. Shields explains that she sent tenth-grade assignments to ChatGPT, asking the chatbot to restructure the material at a sixth-grade level. Her students were then able to follow along and participate in class with their differentiated material.</p>
        <p>“It’s really easy now to make different types of assignments,” says Shields.</p>
      </blockquote>
      <p>But also concerns:</p>
      <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">I think ChatGPT is a crutch that will prevent students from actually needing to learn content. Although I can see its use for small tasks, like how to email a teacher with questions, it also prevents students from developing the soft skills that completing those small tasks allows. —K–12 teacher</blockquote>
      <p>This source mainly provides teacher anecdotes both in favor of AI and against. In particular, it does show that there are legitimate uses for the technology in education, but, even in the article some of the uses brought up are immediately arguably unethical.</p>
      <blockquote class="border-l-solid border-l-4 border-l-subtle px-4 w-fit text-[0.9em]">Shields suggests students evaluate information generated by ChatGPT. Addressing the validity of an AI-generated report will build skills in editing and fact-checking. She also believes teachers should reimagine their assignments.</blockquote>
      <p>There is an argument against this idea that students should be AI fact-checkers, in (<a href="#loc-15" role="doc-biblioref">Butterick</a>).</p>
    </div>
]]></summary>
</entry>
<entry>
    <title>Parallelizing this website for free</title>
    <link href="https://web.youwen.dev/hacking/parallelizing-this-website-for-free" />
    <id>https://web.youwen.dev/hacking/parallelizing-this-website-for-free</id>
    <published>2025-05-13</published>
    <updated>2025-05-13T00:00:00Z</updated>
    <summary type="html"><![CDATA[    <p>There are around 20-30 pages on this website now and builds have been getting somewhat slow (in the realm of  4 seconds total). Since I haven’t implemented any sort of hot reload, I need to rebuild the whole site to view any changes, and it’s actually a bit cumbersome to do development now. Using <code>rayon</code>, a data parallelism library, I parallelized the entire query and build step and achieved a ×6 speedup for basically zero effort.</p>
    <div class="py-2 px-4 text-[0.8em] rounded-md border-1 border-slate-200 dark:border-zinc-700 bg-slate-50 dark:bg-overlay leading-[1.5em]">
      <div><span style="font-variant-caps: all-small-caps"><span class="inline-block all-smallcaps">By the way</span></span></div>
      <div class="!mb-0 mt-2 prose-p:my-1 prose-p:mt-1">
        <p>This site is generated entirely by my custom static site generator written in Rust, and the pages are created using Typst.</p>
        <p>Typst is a typesetting system like LaTeX, and the Typst CLI is a program that compiles the Typst markup language into <span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">pdf</span></span> or <span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">html</span></span> documents.</p>
      </div>
    </div>
    <p>Here’s how the build process works at a high level:</p>
    <ol>
      <li value="1">Crawl the routes directory to build up an internal tree representation of the website, where the leaves represent webpages. Each leaf is a struct that holds information about the Typst document used to build it.</li>
      <li value="2">Traverse the tree and populate each leaf with metadata about the document it represents, by invoking <code>typst query</code> on its source file.</li>
      <li value="3">Do whatever intermediate data transformations and processing you need on the tree.</li>
      <li value="4">The actual build step—traverse the tree and invoking <code>typst build</code> on everything that needs to be built, then insert the output into an <span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">html</span></span> template (which contains the <code>&lt;head></code> metadata, navbar, and all the other common website bits) to create the final webpage.</li>
    </ol>
    <p>In the code, the tree is represented by series of nested <code>HashMap</code>s, where the leaves represent a webpage. Tree traversal is implemented depth-first with a combination of recursion and iteration. We call <code>.iter()</code> on the <code>HashMap</code>, then map over all the nodes in a level. If we see a leaf (representing a page to be built), we call <code>typst build</code> to turn it into a raw <span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">html</span></span> artifact, and then embed the content into one of our templates . If we see a nested subtree instead, then we recursively traverse it.</p>
    <p>This immediately presented an obvious place to find a lot of performance gains. It should be perfectly safe to parallelize querying metadata and building the <span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">html</span></span> artifacts, which are by far the most expensive operations.</p>
    <h2>Parallelism for free</h2>
    <p>Writing concurrent code is a notoriously difficult task (to get right) usually, and often the increased complexity is not worth the speed gains. But Rustaceans reading this may know of a library called <a class="text-link external-link" href="https://github.com/rayon-rs/rayon" target="_blank">rayon</a>, that can automatically parallelize anything using <code>foo.iter()</code> by changing it into <code>foo.par_iter()</code>. Usually it is very easy to add, so I decided to try it out.</p>
    <p>First I needed a small refactor to separate out the build step into two phases. In the original design, in order to build a given page, we call <code>typst build</code> on its source and then generate the final result embedded in our <span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">html</span></span> template in one step. What we’re going to do instead is first concurrently call <code>typst build</code> on every page and build every artifact first, followed by a second pass to generate all the final pages using our templates.</p>
    <p>The reason for splitting these operations up into two passes is that calling Typst to build our initial artifacts is by far the most time consuming and stands to benefit the most from concurrency. Actually generating the final pages is already extremely fast, so we’ll keep it serial for now. Not to mention, it’s also prone to race conditions because it involves creating lots of nested directories and files.</p>
    <p>A final consideration—although <code>rayon</code> handles data races very well thanks to “fearless concurrency,” it can’t guarantee safety from race conditions in real world IO. In particular, I realized that if I had two Typst files with the same filename but representing different routes, the file would get overwritten and things would get real wacky. (concrete example: both <code>/+index.typ</code> and <code>/cv/+index.typ</code> would get compiled to <code>+index.html</code>, so one of them would end up overwriting the other).</p>
    <p>This was easily fixed by hashing the canonical path of the Typst source file and appending it to the filename, guaranteeing that every distinct page gets a unique filename.</p>
    <pre><code>+index-10358123698156044784.html<br>+index-5389558411361854159.html</code></pre>
    <p>Now we should be safe from race issues! With these slight architectural changes, we should get the promised parallelism for free.</p>
    <h2>So it’s that easy?</h2>
    <p>Well, I also had to refactor the iterator so it used the functional patterns <code>rayon</code> expects rather than a simple <code>for</code> loop. But yeah, after it was working synchronously, I just needed to swap out one line of code.</p>
    <p>Here’s the refactored code of the original, slow function that does the tree traversal and builds the artifacts:</p>
    <pre><code data-lang="rust"><span style="color: #d73948">fn</span> <span style="color: #4b69c6">build_artifacts</span>(<br>    <span style="color: #d73948">&amp;</span>self,<br>    route_tree: <span style="color: #d73948">&amp;</span><span style="color: #d73948">mut</span> RouteTree,<br>    html_artifacts_path: <span style="color: #d73948">&amp;</span>Path,<br>) -> <span style="color: #4b69c6">Result</span>&lt;(), WorldError> {<br>    route_tree<br>        .<span style="color: #4b69c6">iter_mut</span>()<br>        .<span style="color: #4b69c6">try_for_each</span>(|(filename, node)| <span style="color: #d73948">match</span> node {<br>            RouteNode::Page(typst_doc) <span style="color: #d73948">=></span> {<br>                <span style="color: #16718d">event!</span>(Level::<span style="color: #b60157">INFO</span>, <span style="color: #198810">"</span><span style="color: #198810">Building artifact for {}</span><span style="color: #198810">"</span>, filename);<br><br>                <span style="color: #d73948">self</span>.<span style="color: #4b69c6">build_doc</span>(typst_doc, html_artifacts_path)<span style="color: #d73948">?</span>;<br><br>                <span style="color: #16718d">event!</span>(<br>                    Level::<span style="color: #b60157">DEBUG</span>,<br>                    <span style="color: #198810">"</span><span style="color: #198810">Built an HTML artifact from {filename}, at path {:?}</span><span style="color: #198810">"</span>,<br>                    typst_doc.source_path.<span style="color: #4b69c6">as_os_str</span>(),<br>                );<br><br>                <span style="color: #4b69c6">Ok</span>(())<br>            }<br>            RouteNode::Nested(hash_map) <span style="color: #d73948">=></span> <span style="color: #d73948">self</span>.<span style="color: #4b69c6">build_artifacts</span>(hash_map, html_artifacts_path),<br>            RouteNode::Redirect(<span style="color: #d73948">_</span>) <span style="color: #d73948">=></span> <span style="color: #4b69c6">Ok</span>(()),<br>        })<br>}</code></pre>
    <p>We’re going to make a one liner change to make it concurrent:</p>
    <pre><code data-lang="rust">route_tree<br>    .<span style="color: #4b69c6">iter_mut</span>()<br><span style="color: #74747c">//</span><span style="color: #74747c"> use the parallel iterator from `rayon`</span><br>route_tree<br>    .<span style="color: #4b69c6">par_iter_mut</span>()</code></pre>
    <p>And we’re done! In theory, our tree traversal is now going to run concurrently thanks to <code>rayon</code>.</p>
    <div class="py-2 px-4 text-[0.8em] rounded-md border-1 border-slate-200 dark:border-zinc-700 bg-slate-50 dark:bg-overlay leading-[1.5em]">
      <div><span style="font-variant-caps: all-small-caps"><span class="inline-block all-smallcaps">By the way</span></span></div>
      <div class="!mb-0 mt-2 prose-p:my-1 prose-p:mt-1">We do something very similar to parallelize the query step as well.</div>
    </div>
    <h2>But was it faster?</h2>
    <p>Did it work? It’s Rust, so of course it worked first try!</p>
    <p>Seriously, it really just worked. With our one-liner change, <code>rayon</code> now automatically spins up a thread pool and spawns many concurrent <code>typst</code> processes to perform the build.</p>
    <h3>Benchmarks</h3>
    <p>Here’s benchmark data using <a class="text-link external-link" href="https://github.com/sharkdp/hyperfine" target="_blank">hyperfine</a> on the actual site (executed on Apple Silicon, M1 Pro, running NixOS w/ Linux 6.13.5-asahi):</p>
    <ol>
      <li value="1">
        <p>Before (average total time elapsed,  4.6 seconds)</p>
        <table>
          <thead>
            <tr>
              <th>Command</th>
              <th>Mean [s]</th>
              <th>Min [s]</th>
              <th>Max [s]</th>
            </tr>
          </thead>
          <tbody>
            <tr>
              <td><code>site build --minify</code></td>
              <td>4.583 ± 0.102</td>
              <td>4.454</td>
              <td>4.738</td>
            </tr>
          </tbody>
        </table>
      </li>
      <li value="2">
        <p>After (average total time elapsed,  0.9 seconds)</p>
        <table>
          <thead>
            <tr>
              <th>Command</th>
              <th>Mean [ms]</th>
              <th>Min [ms]</th>
              <th>Max [ms]</th>
            </tr>
          </thead>
          <tbody>
            <tr>
              <td><code>site build --minify</code></td>
              <td>908.6 ± 26.0</td>
              <td>874.9</td>
              <td>943.2</td>
            </tr>
          </tbody>
        </table>
      </li>
    </ol>
    <p>For a more pronounced trial, I’ve generated 500 synthetic pages (long-form blog posts) and then benchmarked it again.</p>
    <ol>
      <li value="1">
        <p>Before, building 500 synthetic pages (average total time elapsed,  65 seconds)</p>
        <table>
          <thead>
            <tr>
              <th>Command</th>
              <th>Mean [ms]</th>
              <th>Min [s]</th>
              <th>Max [s]</th>
            </tr>
          </thead>
          <tbody>
            <tr>
              <td><code>site build --minify</code></td>
              <td>65.32 ± 1.320</td>
              <td>45.24</td>
              <td>71.393</td>
            </tr>
          </tbody>
        </table>
      </li>
      <li value="2">
        <p>After, building 500 synthetic pages (average total time elapsed,  10 seconds)</p>
        <table>
          <thead>
            <tr>
              <th>Command</th>
              <th>Mean [s]</th>
              <th>Min [s]</th>
              <th>Max [s]</th>
            </tr>
          </thead>
          <tbody>
            <tr>
              <td><code>site build --minify</code></td>
              <td>10.235 ± 0.051</td>
              <td>10.156</td>
              <td>10.316</td>
            </tr>
          </tbody>
        </table>
      </li>
    </ol>
    <h2>Conclusion</h2>
    <p>There is one concern that I have with my implementation. Because the tree traversal is recursive, it makes a new call to <code>rayon</code> on every single nested subdirectory. I’m not exactly sure how <code>rayon</code> handles this under the hood but my suspicion is that it will spin up a new thread pool for every single recursive invocation of <code>par_iter()</code>, which could potentially be very slow. Right now the site only contains 3–4 nested subdirectories so I’m not affected yet.</p>
    <p>The best way to fix this is to probably to transform the tree into a flat vector of pointers to each node, and then call <code>.par_iter()</code> once to operate on every page in parallel from one thread pool.</p>
    <p>That’s a problem for later. I’m quite pleased with how easily <code>rayon</code> was able to let me write multithreaded code without ever thinking about synchronization, locks, mutexes, and other junk. Once again, Rustacean technology was more powerful than I could’ve imagined in my wildest dreams.</p>
]]></summary>
</entry>
<entry>
    <title>Doing web development in Typst</title>
    <link href="https://web.youwen.dev/hacking/doing-web-development-in-typst" />
    <id>https://web.youwen.dev/hacking/doing-web-development-in-typst</id>
    <published>2025-05-10</published>
    <updated>2025-05-10T00:00:00Z</updated>
    <summary type="html"><![CDATA[    <p>Over the past month I’ve been experimenting with writing a website (specifically, this website) using <a class="text-link external-link" href="https://typst.app/" target="_blank">Typst</a>, a document preparation system and markup language. Most people know Typst as a LaTeX replacement, but it can also generate <span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">html</span></span>. I’ve found that it can probably act as a replacement for Markdown/Pandoc based static site generator systems too.</p>
    <h2>How</h2>
    <p>There are other pages on this site explaining this, but essentially I wrote a tiny <a class="text-link internal-link" href="/epilogue">static site generator</a> using Rust that calls the Typst <span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">cli</span></span> to produce <span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">html</span></span>, and then I embed it into a template (that contains navigation widgets, footer, <code>&lt;head></code>, etc). Each page is a Typst source file. The Typst source code for each page can emit metadata, similar to how <span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">yaml</span></span> frontmatter works in Markdown. I implemented it so the syntax for setting metadata on a web page is really similar to setting up a typical Typst document template. For example, here’s the metadata for this page:</p>
    <pre><code data-lang="typst"><span style="color: #d73948">#</span><span style="color: #d73948">show</span>: html-shim.<span style="color: #4b69c6">with</span>(<br>  date: <span style="color: #4b69c6">datetime</span>(<br>    day: <span style="color: #b60157">10</span>,<br>    year: <span style="color: #b60157">2025</span>,<br>    month: <span style="color: #b60157">5</span>,<br>  ),<br>  location: <span style="color: #198810">"Santa Barbara, California"</span>,<br>  title: <span style="color: #198810">"Doing web development in Typst"</span>,<br>  enable-comments: <span style="color: #d73948">true</span>,<br>)</code></pre>
    <p>Typst <span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">html</span></span> export is experimental, so I wrote this <a class="text-link external-link" href="https://github.com/youwen5/web/blob/main/typst/lib/html-shim/0.1.0/html-shim.typ" target="_blank"><span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">html</span></span> shim</a> that makes stuff work right on the web (like smallcaps, for example), renders math properly, and implements the aforementioned metadata system.</p>
    <h2>Ergonomics</h2>
    <p>Obviously writing content is super natural. You can just type plaintext paragraphs, and Typst will render it with nice semantic <span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">html</span></span>. With a little hacking, I got features like smallcaps working in the browser. And native math support is nice:</p>
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    </div>
    <p>But doing more complicated stuff is possible too. For example, those widgets on the <a class="text-link internal-link" href="/">main page</a> were written in pure Typst.</p>
    <div class="grid grid-cols-1 md:grid-cols-2 gap-6 font-sans font-light text-love">
      <div class="space-y-2 prose-lg">
        <div class="border-b-love border-b-1 py-1">B.S. Mathematics<br>University of California, Santa Barbara</div>
        <div class="border-b-love border-b-1 py-1">B.S. Computer Science<br>University of California, Santa Barbara</div>
      </div>
      <div class="space-y-[7.5px] prose-lg">
        <div class="border-b-love border-b-1 inline-flex justify-between w-full gap-2"><span>in Santa Barbara</span> <span><span style="font-variant-caps: all-small-caps"><span class="inline-block all-smallcaps">California, USA</span></span></span></div>
        <div class="border-b-love border-b-1 inline-flex justify-between w-full gap-2"><span>near San Francisco</span> <span><span style="font-variant-caps: all-small-caps"><span class="inline-block all-smallcaps">California, USA</span></span></span></div>
        <div class="border-b-love border-b-1 inline-flex justify-between w-full gap-2"><span>previously near Salt Lake City</span> <span><span style="font-variant-caps: all-small-caps"><span class="inline-block all-smallcaps">Utah, USA</span></span></span></div>
        <div class="border-b-love border-b-1 inline-flex justify-between w-full gap-2"><span>previously in Shanghai</span> <span><span style="font-variant-caps: all-small-caps"><span class="inline-block all-smallcaps">China</span></span></span></div>
      </div>
    </div>
    <p>It turns out that having access to a full programming language is really nice. To produce the above, I defined a <code>location-entry</code> function like this:</p>
    <pre><code data-lang="typst"><span style="color: #d73948">#</span><span style="color: #d73948">let</span> <span style="color: #4b69c6">location-entry</span>(area: <span style="color: #198810">"nowhere"</span>, country-or-state: <span style="color: #198810">"now here"</span>) <span style="color: #d73948">=</span> {<br>  html.<span style="color: #4b69c6">elem</span>(<br>    <span style="color: #198810">"div"</span>,<br>    attrs: (<br>      class: <span style="color: #198810">"border-b-love border-b-1 inline-flex justify-between w-full gap-2"</span>,<br>    ),<br>  )[<br>    <span style="color: #8b41b1">#</span><span style="color: #8b41b1">html</span>.<span style="color: #4b69c6">elem</span>(<span style="color: #198810">"span"</span>, area)<br>    <span style="color: #8b41b1">#</span><span style="color: #8b41b1">html</span>.<span style="color: #4b69c6">elem</span>(<span style="color: #198810">"span"</span>, country-or-state)<br>  ]<br>}</code></pre>
    <p>Then I can create a <code>div</code>, and then call my function repeatedly to populate it. Kind of like a React component, without the pain and complexity. Notice that the <span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">css</span></span> can all be done inline using Tailwind (which was surprisingly easy to set up).</p>
    <pre><code data-lang="typst"><span style="color: #8b41b1">#</span><span style="color: #8b41b1">html</span>.<span style="color: #4b69c6">elem</span>(<span style="color: #198810">"div"</span>, attrs: (class: <span style="color: #198810">"space-y-[7.5px] prose-lg"</span>))[<br>  <span style="color: #4b69c6">#</span><span style="color: #4b69c6">location-entry</span>(<br>    area: [in Santa Barbara],<br>    country-or-state: [<br>      <span style="color: #4b69c6">#</span><span style="color: #4b69c6">smallcaps</span>(all: <span style="color: #d73948">true</span>)[California, USA]<br>    ],<br>  )<br>  <span style="color: #4b69c6">#</span><span style="color: #4b69c6">location-entry</span>(<br>    area: [near San Francisco],<br>    country-or-state: [<br>      <span style="color: #4b69c6">#</span><span style="color: #4b69c6">smallcaps</span>(all: <span style="color: #d73948">true</span>)[California, USA]<br>    ],<br>  )<br>  <span style="color: #4b69c6">#</span><span style="color: #4b69c6">location-entry</span>(<br>    area: [previously near Salt Lake City],<br>    country-or-state: [<br>      <span style="color: #4b69c6">#</span><span style="color: #4b69c6">smallcaps</span>(all: <span style="color: #d73948">true</span>)[Utah, USA]<br>    ],<br>  )<br>  <span style="color: #4b69c6">#</span><span style="color: #4b69c6">location-entry</span>(<br>    area: [previously in Shanghai],<br>    country-or-state: [<br>      <span style="color: #4b69c6">#</span><span style="color: #4b69c6">smallcaps</span>(all: <span style="color: #d73948">true</span>)[China]<br>    ],<br>  )<br>]</code></pre>
    <p>Obviously you can’t create any interactive components like a fully fledged web framework, but you get this middle ground between the austerity and ease of writing of Markdown, and the power of a templating system. (I am aware of <span style="font-variant-caps: small-caps"><a class="text-link external-link" href="https://mdxjs.com/" target="_blank"><span class="inline-block smallcaps">mdx</span></a></span>, but then you have to use a fully-fledged web framework 🤢.) It’s surprisingly nice to use, despite <span style="font-variant-caps: small-caps"><span class="inline-block smallcaps">html</span></span> export being a total afterthought in Typst right now (which, again, is 99% a document preparation system, not a static site generator).</p>
    <p>Plus, I write all my documents and notes in Typst too. Now I can write web pages (like this one) with the same syntax, and share content.</p>
]]></summary>
</entry>

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