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.
- unstable homotopy theory
- bordism theory
- cohomology of groups
- rational homotopy theory
- differential topology
- spectral sequences
- 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.
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.