Applications of Category Theory in Abstract Algebra

Question

Almost every text on Category theory uses categories such as Ab, Grp, and so on as examples to work with but can category theoretic methods actually help us understand the structures better? In particular, does Category Theory aid us in proving some significant abstract algebraic results that are otherwise tedious to prove? Furthermore, is there any structural insight about abstract algebraic objects that is not readily apparent from algebra itself but becomes crystal clear with a category theoretic approach? I do not have a specific format for the answer in mind. As category theory is quite general, the answer may also include ways to say use results from other branches such as analysis or topology, so as to prove an abstract algebraic result which was tedious to prove using an algebraic approach. In this light, any insight about the relationship between category theory and abstract algebra is welcome.

Answer 1

Here is the simplest interesting example I can think of:

Exercise: Let $n, m \in \mathbb{N}$. Prove that if the free groups $F_n$ and $F_m$ are isomorphic then $n = m$.

You could imagine giving this to a student who was only given the explicit combinatorial construction of the free groups in terms of reduced words and them really struggling to analyze explicitly what an isomorphism between two free groups looks like, it would be a mess. But if you instead have the universal property of the free groups it is completely obvious; this universal property says if $G$ is any group we have a natural bijection

$$\text{Hom}(F_n, G) \cong G^n$$

and now the answer falls out immediately from taking $G$ to be any nonzero finite group, say $G = C_2$.

In fact the universal property can show more generally that if $X, Y$ are two sets and the free groups $F_X$ and $F_Y$ are isomorphic then $|X| = |Y|$. The above counting argument is no longer available (actually I don't know if it's true either way whether $|2^X| = |2^Y|$ implies $|X| = |Y|$) but instead we can argue as follows: by the universal property, the abelianization of $F_X$ is the free abelian group on $X$, namely the direct sum $\bigoplus_{x \in X} \mathbb{Z}$ of $|X|$ copies of $\mathbb{Z}$. And by a second application of the universal property, tensoring with $\mathbb{Q}$ produces the free $\mathbb{Q}$-vector space on $X$, namely the direct sum $\bigoplus_{x \in X} \mathbb{Q}$. This is a vector space of dimension $|X|$ and now we can appeal to the uniqueness of dimension.

Speaking of tensor products, that's the next area in abstract algebra where I think universal properties really provide a lot of clarity and elegance. Proving that $V \otimes W \cong W \otimes V$ or that $U \otimes (V \otimes W) \cong (U \otimes V) \otimes W$ without the universal property is tedious and with the universal property it's very easy. Then there's the tensor-hom adjunction which is very useful and fundamentally categorical, we learn that tensor products preserve colimits, we can define extension of scalars and talk about complexification, etc. etc.

As a student the first place I really used category theory for myself was thinking in terms of left and right adjoints (free constructions etc.) and organizing information about what functors preserved what limits and colimits using adjunction properties; I found this very helpful. E.g. does abelianization preserve limits or colimits? It's a left adjoint, so colimits. This means it preserves cokernels and hence it preserves epimorphisms, but that we shouldn't necessarily expect it to preserve kernels or monomorphisms (and it doesn't).

Answer 2

Furthermore, is there any structural insight about abstract algebraic objects that is not readily apparent from algebra itself but becomes crystal clear with a category theoretic approach?

I don't know if this counts, but without category theory, I'm not entirely sure how one sees which properties of mod-$R$ are shared by mod-$M_n(R)$ without using category theory except by ad-hoc methods. An answer to that (and more) is given by the theory of Morita equivalence. Since many ring-theoretic conditions are characterized by conditions on their category of modules, it means we have gained insight about rings and their matrix rings, I suppose.

can category theoretic methods actually help us understand the structures better?

I think the process itself of considering structures as a category often helps us understand the structures better.

Firstly, a lot of the constructions and tools can be re-expressed as phenomena in category theory observed elsewhere, and that itself is simplifying. It's especially helpful across disciplines.

Secondly, it enables us to relate categories to each other to solve problems. It does this even without directly saying something about a category. For example, category theory doesn't really tell us which polynomials are solvable, but the category of groups via category theory informs us which polynomials are solvable.

Answer 3

There are alot of good reasons to state things categorically.

1. Universal Properties
   Category theory is made to give a precise meaning to universal properties. I don't think it is possible to circumvent stating and using universal properties in algebra nowadays (tensor products, (co)products, pushouts and pullbacks, exact sequences, quotients etc. come to mind). Even if you are not interested in jumping between mathematical worlds and want to do "just algebra", category theory allows you to quickly decide or at least make conjectures on how these universal properties interact with each other (think for example the compatibilities of tensor products with quotient constructions...).

2. Generalization to other Settings
   Did you know you can try to do algebra in many different settings? Phrasing ordinary algebra in category theoretic terms allows you to quickly setup algebraic structures in other settings. For example groups with a compatible structure of a smooth manifold, so called Lie groups, are easily defined as group objects in the category $\mathsf{SmMan}$ of smooth manifolds. Of cause you can also define them in an ad-hoc way, but phrasing things categorically allows you to quickly transfer known results about groups into this new setting.

3. Jumping Mathematical Worlds
   Functoriality allows you to pass from one mathematical theory to another in a very clean way. The most basic result in category theory, which is incredibly effective, is that a functor preserves isomorphisms. For example forgetting the manifold structure of a Lie-group turns out to be a functor $\mathsf{LieGrp} \rightarrow \mathsf{Grp}$. Thus isomorphic Lie-groups have in particular isomorphic underlying groups. Arguably in most examples this is easy to verify by hand too. But it is a no-brainer knowing functoriality and furthermore we now can give a precise meaning to the questions like "what kind of structure is preserved if we turn $X$ into $Y$?" by making it a question about the functor.

4. Free Constructions
   As combination of 1. and 3. it is worthwhile to talk about free functors, that is left adjoint functors. To mention one example: It is a rather easy thing to show abstractly that free functors compose. Once you know that the forgetful functors $\mathsf{Grp} \rightarrow \mathsf{Set}$, $\mathsf{Ab}\rightarrow\mathsf{Set}$ and $\mathsf{Ab}\rightarrow\mathsf{Grp}$ all have left adjoint functors given by constructing the free group $F_X$ and free abelian group $\Bbb Z^{\oplus X}$ on a set $X$, respectively quotienting by the commutator subgroup $G/[G,G]$, it is a formality to show that $F_X/[F_X,F_X] \cong \Bbb Z^{\oplus X}$. I think this is less painful than explicitly constructing this isomorphism from generators and relations.

5. Commutative Diagrams
   They allow to graphically encapsule equations, which at the very least helps (me) to memorize things more easily. Arguably, you can still draw the same diagrams without referring to category theory. But using the functoriality mentioned in 2. allows you to justify equations by inferring them from some other mathematical context.

6. More Reasons

I don't think you can really expect to prove deep theorems about some specific mathematical theory by using plain category theory. Category theory is meant to be general (in the sense of 2. and 3.), so you can expect categorical proofs to work out in several examples. Of cause you can put more and more adjectives on your category just to be able to prove your theorem, but at some point the generality of category theory might become a burden rather than a helpful tool. The upshot is that category theory allows you separate formal nonsense from the essence of your theory (at least to a certain extent).

Answer 4

A quick example from back when I was studying about quantales. Rosenthal, 1990, Quantales and their applications.

Let $P$ be a quantale, i.e, $P$ is a complete lattice equipped with an associative binary operation $\cdot$ that satisfies the identities $$ p \cdot \left(\bigvee p_\alpha \right) = \bigvee p\cdot p_\alpha \quad\mbox{and}\quad \left(\bigvee p_\alpha\right ) \cdot p = \bigvee p_\alpha\cdot a. $$ So, we have induced join-preserving maps $p\cdot -:P\to P$ and $-\cdot p:P\to P$. Thus, by the adjoint functor theorem, we have morphisms $p\rightarrow _r -:P\to P$ and $p\rightarrow _\ell -:P\to P$ that satisfy $$ p\cdot q \leqslant s \Leftrightarrow q\leqslant p\rightarrow _rs \quad\mbox{and}\quad q\cdot p\leqslant s \Leftrightarrow q\leqslant p\rightarrow _\ell s. $$

I tried to figure out explicitly what $p\rightarrow _{r/\ell}$ might look like for sufficiently complicated examples, but couldn't succeed. Yet the adjoint functor theorem predicts these maps exist, which is just mindboggling.

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Often when one runs into some kind of Galois connections, the adjoint morphism is not explicitly given but its existence is justified via the adjoint functor theorem.