I am reading about infinity categories.
My source is http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf
My aim is to think categorically so that all the constructions I deal with are natural (till now homotopy colimit\limit, the perspective on derived functors is very nice. Also I liked the construction of model category letting you tell when two maps are homotopic. Also simplicial sets are a good way to capture degeneracies). I am trying to study Spectra as a concrete example.
Also I would like to be able to read later about derived algebraic geometry.
However, a lot of the book is very dull to me; proving what the fibrations\cofibrations are in the model structure on simplicial sheaves. I am okay with just being told the answer; it's frustrating to go through 100's of pages proving it.
My questions are:
- What are the fun important parts (by the aims I described) that I should read about?
- Am I doing harm to myself by accepting technical facts like who the fibrations\cofibrations are?
Rumors I've heard from smart friends are - compact objects, some adjoint theorems
I am obviously no expert, but to close this question; I found that instead of sludging through books, the first lectures of the following were very helpful to me-
https://www.youtube.com/channel/UCk1WaD2LYVvNiRPteCBeSsA
(Seeing that a limit in infinity categories is the standard homotopy limit, and working out why showed me how infinity categories keep track of the homotopy as well).