I've been doing a lot of reading on homotopy theory. I'm very drawn to this subject as it seems to unify a lot of topology under simple principles. The problem seems to be that the deeper I go the more confused I get since the number of possible formalisms grows exponentially. The more i read the harder it gets to seperate the fundamental principles from the formal manipulations.
For example: I know the basics of cofiber and fiber sequences and why they work concretely but I still have no clear idea of how to organize them into a clear picture that would work the same in the homological setting.
Here are things i've been trying to understand axiomatically in terms of first principles and so far have been unsuccesful:
- General notion of a derived functor between categories with weak equivalences.
- Homotopy (co-)limits - cofibrant and fibrant replacements (Which as i understand are a special case of 1).
- Stable homotopy category.
From the reading i've done on nlab it seems a lot of homotopy theory can be expressed neatly in terms of $(\infty ,1)$-categories. For me it's a pretty good argument to learn that formalism.
1. Should I learn $(\infty ,1)$-category theory?
2. If not, is there a way to gain a formal unified undertanding of homotopy theory which feels less like walking around in a dark room and more like climbing a mountain?