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:
- I think seeing examples in the wild will reinforce my understanding of Categories.
- 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.