Universal constructions and the universal property of quotients

This is going to be a short introduction to the concept of how we may create a universal construction of a certain concept through defining a universal property. We’ll demonstrate by defining the universal property of quotients, to characterize the concept of taking a quotient, e.g. a quotient group in group theory or quotient space in linear algebra.

Notions

From Wikipedia:

In mathematics, more specifically in category theory, a universal property 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.

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 coset or fiber), but any construction satisfying a given universal property is going to give an equivalent result. All there is to know about the construction is already contained in the universal property.

The universal morphism

One way of formulating the universal property is as follows. Let and be categories, a functor between them, and be objects in , and be an object in .

Then a universal morphism from to is the unique pair , which has the following universal property:

For every morphism , there is a unique morphism in such that the following diagram commutes.

Here is a concrete example. Let , the category with groups as objects and group homomorphisms as morphisms. Now the functor is simply the identity functor. We let and , where is an arbitrary group and ( is normal in ).

Now let , where is another group and let be any group homomorphism (morphism) between them. Then the universal morphism is . The universal property says that there is a unique morphism such that the diagram above commutes, that is .

You may notice that this is a classic result about quotient groups in group theory, and part of the proof of the first isomorphism theorem! Here is the canonical map sending each to its congruence class in . The diagram above closely resembles this one which appears in many standard algebra textbooks (e.g. Dummit and Foote).

OK, so what? The point is that this universal property characterizes any construction of the quotient group up to isomorphism, which means any construction of a quotient that satisfies the property is essentially the same. In Dummit and Foote, quotients are formulated using fibers, while a more standard approach is to use cosets, 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.

From another point of view, the universal property is the definition itself. Thus, the diagram above could be seen as the definition of the quotient group.

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 Yoneda lemma (perhaps among the most fundamental results in category theory) and how specifying all the maps in and out of an object uniquely determines it.

The comma category

An alternative, equivalent, and more abstract formulation of the universal property involves defining a comma category 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).

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).

Let be an object in a category . Then the coslice category is denoted and represents the category of all the arrows originating from ¹. Objects in look like pairs , where is an object in and is an arrow .

Morphisms from are arrows such that the following diagram commutes.

That is, .

Now we introduce the comma category, which essentially generalizes this notion. Let , , be categories, and let and be functors as such:

The comma category is denoted and has objects which are triples of the form where is an object in , is an object in , and is a morphism in , specifically the arrow .

Morphisms in between and are pairs , where and are morphisms in and respectively, such that and , and the following diagram commutes.

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 .

Set , where is the category with exactly one object and one morphism. Then the functor simply maps for some object in . The diagram becomes

which simply reduces to

Now note that all objects in are going to look like , where is the only object in , is some object in , and is a morphism from , where is just some fixed object in . For brevity, we’ll write to mean from now on.

Morphisms between and are still pairs of the form where is a morphism in , and is thus necessarily the identity morphism, while is a morphism in . So for brevity we’ll just refer to morphisms by just .

Now consider the object , where is an object in and we may note that here is the same universal morphism from earlier. Moreover, suppose that this object is initial, that is, there is a unique arrow from to every object in . This is sufficient to once again define the universal property. For every object , we have a unique morphism (due to our initial object) such that the following diagram commutes.

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.)

Now consider our previous example of the universal property of the quotient group. Once again, we are only working in the category of groups, so we may set . The functor is then the identity endofunctor on . The object is just some group, say . With the universal morphism , we define the object to be the initial object. Then take an arbitrary group , and a map . Then the pair is an object as well, so there exists a unique morphism such that . That is, this diagram commutes

This is the exact same diagram from the previous section, and is again the canonical map .

Finally, we note that in this degenerate case where , , our comma category essentially reduced to the coslice category introduced prior, so in fact to describe the universal property of quotients we only needed the notion of coslice². This diagram, under these conditions,

reduces to

where , , and are all objects in a single category . Objects are of the form , and morphisms are of the form , which is essentially exactly how the coslice category was defined, and the previous commutative diagram is just an inverted and relabeled version of the one we drew for the coslice.

Conclusion

The universal property we showed for quotient groups demonstrates how category theory can trivially characterize certain constructions. Notice that, just by replacing with (the category of vector spaces over a field with linear maps as morphisms), we obtain an equivalent univeral property of quotient spaces. This gives a hint for how quotients and the isomorphism theorems may be formulated in general via the idea of universal algebra.