What we should pursue once we have a notion of homotopical morphisms?

Question

I know this is a rather vague question but let us give it a try.

Suppose that we have established a notion of homotopy between morphisms in a certain category which is a category of sets endowed with some structure. What would be the standard results from usual homotopy theory that we should pursue?

For example, in my category we have a certain notion of cohomology so it is natural to wonder whether homotopical morphisms induce the same map on cohomology, etc.

I'm not a homotopy theorist so I'm not quite sure how to proceed.

Any references for standart homotopy theory would also be welcome.

Thanks.

Answer 1

Let me just throw you a suggestion that I think is an amazing categorical generalisation (in a way) of the homotopy theory we play with in the category of spaces.

Once we know a few nice classes of maps and a desired class of weak equivalences, once can start to talk about model categories. See texts by Dwyer-Spalinski, Hovey, or the original text by Quillen. Keep in mind the categories of spaces, simplicial sets and chain complexes when reading these abstract definitions.

A beautiful consequence of formal model category theory is things like Whitehead's theorem come for free almost, although constructing model structures is not supposed to be trivial.

The wikipedia article is also reasonably nice to read through,

https://en.wikipedia.org/wiki/Model_category .