What is a (the?) good starting point for learning the modern "higher" mathematics? As many of you know, category theorists are currently doing, among other things, a great job in advertising their modern developments. And I must say, this works for me - in particular, I find myself spending more and more time in nLab and the n-category cafe, fully understanding probably tiny bits of what they say there, and feeling that I really should get a grasp of the great picture at some point. My problem is, as for many students from the "analysis camp", almost total lack of background, except maybe some basic algebraic topology and even more basic algebraic geometry.
So the question is - where should I start in order to understand the ideas they refer to?


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*Homotopy theory? This looks like a vast subject, and it's not quite clear for me what aspects of it I should learn. Simplicial methods? Model categories? The latter, for instance, seems pretty abstract and unmotivated until you do some "real" homotopy theory, which looks like a great deal to learn.

*Homological algebra? An even more basic-looking thing, so this might be a good idea to learn it first. Again, to what extent? Is the derived category formalism enough?

*Algebraic geometry - etale cohomology, in particular - at least in order to motivate...

*Topos theory? This is just scary...

*Homotopy type theory? Looks relatively self-contained at first glance, but currently not developed enough to help understanding the "real" stuff. Besides, it drags along logic and type theory as prerequisites, which is even farther away from "the meat"...
Edit: there are related questions here, here and here (see Jacob Lurie's answer, it's really helpful).
 A: So much of my advertisement of topos theory has been done in the comments. I compare topoi to "funny set theories" because they come with their own internal (and structural) form of separation and "power objects", as well as many structures naturally like other common set-theoretic constructions. I actually came to set theory via elementary topos theory, and I found it a pretty natural way to think about set theory. So if "set theory" doesn't make "topoi" sound less daunting, maybe it will work the other way around!
The connections to logic are well known, but it's also the case that many interesting categories can be viewed as presheaf topoi (not the least of which is the category of simplicial sets).
It's not hard to find books explicitly about topoi, but I thought I would mention Steve Awodey's "Category Theory"; it's a readable introduction to category theory generally, but it spends a lot of time on the structure of presheaf categories (which are always topoi), and many of the exercises have a strong topos-theoretic leaning.
A: nLab is an encyclopedia.  Not all of it would be understandable to everybody.
I would vote for algebraic geometry for having a nice mix of regular maths and category theory.  You can couple this with topos theory, since Grothendieck sheaves are an important aspect of it.  MacLane and Moerdijk's book is probably the most accessible book for topos theory.
