Category-Theory notions and examples in Algebraic Topology

Question

I never really dived in to category theory, you see. I'm familiar with core concepts at a basic level, but I really lack context and depth. Sadly, it is unlikely that I'll have the time to relearn it thoroughly soon.

Lately, I've began reading Hatcher's "Algebraic Topology". I'll be taking a course on the subject, following this reference soon. It obvious, even to myself, that algebraic topology is littered with category-theoretic notions. I was thinking to myself:

"Hey, that's an awesome way to see some category theory in context!

I can see two advantages:

1. I think seeing examples in the wild will reinforce my understanding of Categories.
2. Abstracting the Algebraic-Topology notions will allow me with a new perspective of the topic at hand.

Unfortunately, Hatcher doesn't put a lot of emphasis on the categorical notions (at least in the first 70 pages or so). I'm not skilled enough to abstract-ify and distill the concepts myself.

Would have been great if I had a list of important examples, with their underlying category-theory notions highlighted. I'm looking for answers like:

"See this concept here? This is a natural transformation of between the functors $F,G$ from the category Top to the category Foo. As common with natural transformations, it leads to corollaries such as ..."

References are welcome as well.

Many thanks!

Answer 1

If you are looking to learn category theory in the context of algebraic topology, in my humble opinion, Hatcher's book is not your best bet! It takes more of a geometric approach to the subject.

I would suggest Tammo tom Dieck's Algebraic Topology book, which has more of a categorical viewpoint. If you find homological algebra to your liking, then you have Weibel's classic Introduction to Homological Algebra, and Gelfand & Manin's Methods of Homological Algebra. For more advanced topics, take a look at Peter May's A Concise Course in Algebraic Topology and its sequel.

As for example, here is my favorite one : the $Ext$ term appearing in universal coefficient theorem for cohomology is an instance of a very general phenomena called a derived functors. A left exact functor (on a nice category) can be "derived" so that you have a long exact sequence. As a concrete example, suppose you have a short exact sequence $0 \to A \to B \to C \to 0$ of $R$-modules, for some commutative ring $R$. Then, for any $R$-module $M$, the functor $\hom(\_, M)$ is left exact, i.e, we get the exact sequence $$0 \to \hom(C,M) \to \hom(B,M) \to \hom(A,M).$$ In order to extend this to the right, we need the $Ext$-functors. We have the long exact sequence $$0 \to \hom(C,M) \to \hom(B,M) \to \hom(A,M)\to Ext^1(C,M) \to Ext^1(B,M) \to Ext^1(A, M) \to \cdots \to Ext^n(C,M) \to Ext^n(B,M) \to Ext^n(A, M) \to \cdots.$$ Turns out, if $R$ is a PID, then all higher $Ext^{n \ge 2}$ vanish identically. This leads to the universeal coefficient theorem. Similarly, the $Tor$ is the left derived functor for the functor $\_ \otimes M$ of taking tensor with $M$.

Answer 2

Functors in algebraic topology are used to distinguish topological spaces (i.e., $X$ and $Y$ can't be homeomorphic because, for some functor $F$ the images $F(X)$ and $F(Y)$ are'nt isomorphic) or to show that a subspace $X\subseteq Y$ is not a retract, i.e., the inclusion $i:X\to Y$ does not have a left inverse, in many cases by finding a functor $F$ such that $F(Y)$ is trivial but $F(X)$ is not (in the sense that $F(i):F(X)\to F(Y)$ cannot have a left inverse). That the circle $S^1$ is not a retract of the disc $D^2$ can be shown with the fundamental group functor and that $S^n$ is not a retract of $B^{n+1}$ needs the singular homology functor $H_n$.

Answer 3

May's Concise Course in Algebraic Topology provides a great bridge between Algebraic Topology and Category Theory in my opinion. It definitely serves more like a reference book than one you can read outright (especially without experience in either subject), which is actually stated in the book itself.

For more advanced concepts like enrichment (like as in 2-categories or abelian categories) and model categories for homotopy theory, you may want to refer to Riehl's Categorical Homotopy Theory. Sources like Loregian's co/end calculus also provide similar explanations and displays some graphical calculus.

Finally, any Category Theory book will most definitely contain topological examples.