Category Theory usage in Algebraic Topology

Question

First my question:

How much category theory should someone studying algebraic topology generally know?

Motivation: I am taking my first graduate course in algebraic topology next semester, and, up to this point, I have never taken the time to learn any category theory. I've read that category theory helps one to understand the underlying structure of the subject and that it was developed by those studying algebraic topology. Since I do not know the exact content which will be covered in this course, I am trying to find out what amount of category theory someone studying algebraic topology should generally know.

My university has a very general outline for what the course could include, so, to narrow the question a bit, I will give the list of possible topics for the course.

Possible Topics:

• unstable homotopy theory
• spectra
• bordism theory
• cohomology of groups
• localization
• rational homotopy theory
• differential topology
• spectral sequences
• K-theory
• model categories

All in all, I am well overdue to learn the language of categories, so this question is really about how much category theory one needs in day to day life in the field.

Update

I emailed the professor teaching the course and he said he hopes to cover the following (though maybe it is too much):

• homotopy, homotopy equivalences, mapping cones, mapping cylinders
• fibrations and cofibrations, and homotopy groups, and long exact homotopy sequences.
• classifing spaces of groups.
• Freudenthal theorem, the Hurewicz and the Whitehead theorem.
• Eilenberg-MacLane spaces and Postnikov towers.
• homology and cohomology theories defined by spectra.

Answer 1

The list of possible topics that you provide vary in their categorical demands from the relatively light (e.g. differential topology) to the rather heavy (e.g. spectra, model categories). So a better answer might be possible if you know more about the focus of the course.

My personal bias about category theory and topology, however, is that you should mostly just learn what you need along the way. The language of categories and homological algebra was largely invented by topologists and geometers who had a specific need in mind, and in my opinion it is most illuminating to learn an abstraction at the same time as the things to be abstracted. For example, the axioms which define a model category would probably look like complete nonsense if you try to just stare at them, but they seem natural and meaningful when you consider the model structure on the category of, say, simplicial sets in topology.

So if you're thinking about just buying a book on categories and spending a month reading it, I think your time could be better spent in other ways. It would be a little bit like buying a book on set theory before taking a course on real analysis - the language of sets is certainly important and relevant, but you can probably pick it up as you go. Many topology books are written with a similar attitude toward categories.

All that said, if you have a particular reason to worry about this (for instance if you're worried about the person teaching the course) or if you're the sort of person who enjoys pushing around diagrams for its own sake (some people do) then here are a few suggestions. Category theory often enters into topology as a way to organize all of the homological algebra involved, so it might not hurt to brush up on that. Perhaps you've already been exposed to the language of exact sequences and chain complexes; if not then that would be a good place to start (though it will be very dry without any motivation). Group cohomology is an important subject in its own right, and it might help you learn some more of the language in a reasonably familiar setting. Alternatively, you might pick a specific result or tool in category theory - like the adjoint functor theorem or the Yoneda lemma - and try to understand the proof and some applications.

Answer 2

I agree with Paul Siegel's very nice answer, and would just like to add one thing that's a little too long for a comment.

Depending on what direction you take, algebraic topology can become practically synonymous with higher category theory. This can come in multiple ways. First, the category of topological spaces has spaces as its objects, continuous maps as its morphisms, homotopies as its 2-morphisms, homotopies between homotopies as its 3-morphisms, etc. Stated perhaps a little too cavalierly, the point of model categories is essentially to set up a general framework for studying higher categories that may (or may not) look like that of spaces. But then, higher categories themselves are also a lot like spaces. In this analogy, functors are like continuous maps, natural transformations are like homotopies, etc. (The fact that there are these two totally distinct ways that spaces and categories interact really blindsided me the first time I saw it.)

Anyways, the point is that if you pursue algebraic topology seriously, you may eventually have to get very friendly with category theory and be okay with using ridiculous and scary phrases like "homotopy left Kan extension" and such. It seems that usage of higher category theory in algebraic topology is very much on the rise, so it's possible that in twenty years, algebraic topologists will have no choice but to become well-versed in all this stuff. (I'm certainly not. Not yet, at least.) Just a heads-up.

Answer 3

The bottlenecks seem to be K-theory and Model categories. I can't think of any categorial tools you need for any of the other topics that aren't a proper subset of those used in these two. Then it depends, like everything in mathematics, on how deep you want to go. My advice would be to take a book on those topics and just try to read it. You will see right ahead what categorial language you need. Most topology books specifically highlight any categorial reasoning and many even include a appendix on category theory.