Welcome to the first series of posts on this blog! First, I want to
describe the motivation behind this series of posts and what it will
attempt to do. This past semester, I took a specialized course on
category theory. It was a small group with a fantastic professor, and it
was one of the most enjoyable courses I’ve taken. Even so, there were
some concepts that were difficult to follow, namely universal arrows^{1} and adjoint functors. I managed to
wrap my head around these topics enough to do well on the final exam,
but I still feel my understanding could be improved. So one goal of the
present series of posts is to explore those concepts more thoroughly,
trying to explain them in my own words and justify them rather than just
stating definitions and proving theorems. The other goal of this series
is to address the titular term “abelianization”. I’ve been self-studying
some more algebra this summer as my school only offers a one-semester
course, and thanks to Wikipedia, I learned that abelianization provides
an interesting example of universal arrows and adjoint functors. Thus,
the second goal of this “volley” of posts is to describe abelianization
in the context of category theory.

Given that these are the goals of this series, I have to make some pretty heavy assumptions of pre-existing understanding in order to write these posts without spending 75% or more of my time on buildup. On the category theory side, I assume familiarity with categories, basic categorical constructions, functors, and natural transformations. On the algebra side, I assume familiarity with basic group theory, including homomorphisms, kernels, normal subgroups, and quotient groups. Now, with the housekeeping out of the way, let’s get on to the math.

In this first post, I’ll introduce the concept of universal arrows, give some examples of them, and prove some basic theorems. Many times in category theory, or algebra more broadly, we encounter so-called “universal constructions,” which usually induce certain unique arrows according to certain properties. The principle behind the idea of a “universal arrow” is to provide a general framework for this concept.

To wrap our heads around this incredibly abstract definition, we’ll consider a familiar construction, the coproduct, as a universal arrow. In order to do this, we’ll need to define the diagonal functor.

Now, let and be objects in which have a coproduct . Let and be the coprojection arrows associated with the coproduct. We will show that the pair is a universal arrow from to . To demonstrate this, suppose we have an arrow in , noting that . We need to show that there is a unique arrow such that the following diagram commutes.

The definition of the coproduct already gives us this exact arrow: a unique arrow such that and . This in turn means that as desired. By following a similar line of logic, it should be easy to see how every universal arrow from to satisfies the definition of a coproduct for and . This sort of correspondence leads us nicely to the following property of universal arrows:

*Proof.* By the definition of a universal arrow, we already
have a unique arrow
such that
and another unique arrow
such that
. These arrows are uniquely specified by the following
diagrams.

We will show that and are inverses. To do this, note that the universality of guarantees a unique arrow such that . Clearly satisfies this property, since . But satisfies the same property, since

This means we must have . Similarly, the universality of guarantees a unique arrow such that . Again, clearly fills this role. But so does , since

This means , showing that is an isomorphism as desired. ◻

This theorem should help assure us that universal arrows are in fact a good way of describing the universal constructions we tend to encounter in mathematics, as uniqueness up to isomorphism is a hallmark of these objects. But part of the real magic of universal arrows is seeing how they show up all over the place. We have already seen how the coproduct arises as a universal arrow, but that’s a bit of a tame example. This next example will be more interesting, as we have to put in more work to fulfill the definition of a universal arrow.

Before we go on, I should warn that this example was not an original part of the plan for this post, and it might get a bit in the weeds on the linear algebra side of things. However, once I got started, I was too excited not to include it, so I hope you’ll indulge me.

Looking at this definition, it seems to scream “universal arrow”. It has the exact same shape as the universal arrow diagrams, and most sources describe this definition as the universal property of the tensor product. But at the same time, it kind of looks like a mess; we have linear maps and bilinear maps mixed together in the same commutative diagram, and there are no functors in sight. As it turns out, dear reader, finding how functors fit in and make this diagram into something cleaner will be incredibly beautiful. In this next section, I should acknowledge the help of Wikipedia’s discussion of the Tensor-Hom Adjunction in guiding my thought process. I know, mentioning adjunctions is spoilers for later posts in this volley, but perhaps this hints at how deep the connection between universal arrows and adjunctions runs.

Our first goal will be to turn the bilinear maps into linear maps. Recall that if and are vector spaces, the set of linear maps from to also forms a vector space under the elementwise operations, i. e.:

With this in mind, we will show that bilinear maps from to an arbitrary vector space correspond bijectively with linear maps from to . First, suppose is a bilinear map. By the definition of a bilinear map, each determines a linear map . Now define a mapping by . To demonstrate that is a linear map, we will show that for all , , and :

For the opposite direction, suppose we begin with a linear map . Define a mapping by . We will similarly show that is linear in each of its arguments:

This gives us a unique way of transforming between bilinear maps and
linear maps into the hom-space. This process, called *currying*,
will also help us to “functorize” our tensor product problem. First,
recall how the covariant hom-functor is usually defined:

The category of vector spaces is rather unique, in that (as we have discussed), is also an object in . In addition to this, for a linear map , the mapping is also a linear map. To see this, let and . Then we have:

This means that in the case of
, we may regard the hom-functor as a functor from
to itself, rather than a functor to
. This is called an *internal hom-functor*. With all
this laid as the backdrop, we will describe the tensor product
as a universal arrow from
to the internal hom-functor
. Let
be such a universal arrow; we will show how
satisfies the universal property of the tensor product.
First, define the bilinear map
by
as previously discussed. Now let
be an arbitrary bilinear map. In order to show that
is the tensor product of
and
, we want to produce a unique linear map
such that the following diagram commutes:

Once again following from the discussion on currying, let be the linear map defined by . From the universal property of , we know that there is a unique linear map such that the following diagram commutes:

The commutativity of this diagram tells us that for all , . Now let and . Using this property, we determine

This means that is the tensor product of and as desired. A similar, but reversed, argument shows that a pair satisfying the universal property for the tensor product induces a universal arrow from to . Then via theorem , we know that the tensor product is unique up to isomorphism. However, it is important to note that we do now know the tensor product actually exists, only that if it does, this uniqueness property holds. While universal arrows are a good framework for proving uniqueness and exploring relations to other categorical concepts (like the internal hom-functor in this example), one must usually return to actual concrete constructions to prove that an object exists which satisfies the universal property. With all that said, I hope this protracted example has helped demonstrate that universal arrows are ubiquitous in mathematics, even though they may sometimes require a little coaxing to reveal themselves. To close out this post, I’d like to prove one final theorem which relates universal arrows to hom-functors. The utility of this theorem will become apparent later on.