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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?

  • 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).

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    $\begingroup$ Topos theory is not scary in the least, especially not compared to the higher abstract nonsense you are talking about. But it seems to me you are worried about sinking into generalities – so first you must answer the question, what is it you really want to do? $\endgroup$ – Zhen Lin Nov 28 '13 at 0:46
  • $\begingroup$ @ZhenLin: Since I don't have immediate applications in mind - my answer would be: finally switch from vague handwaving and the "wow this must be cool" impression to actually understanding something and incorporating the basics of their intuition into my own mathematical worldview. And maybe use their insights to understand the relevant bits of mathematical physics. Which is in turn related in some ways to my own research interests. In any case, I don't think that vain curiousity needs to be justified. :) $\endgroup$ – Alexander Shamov Nov 28 '13 at 1:11
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    $\begingroup$ Topos theory is a lamb, and has the benefit of being a field that's pretty well established as category theory goes. You can get a good foothold by just thinking of them as funny set theories. Some of the other stuff you mention haunts me in my sleep... $\endgroup$ – Malice Vidrine Nov 28 '13 at 1:13
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    $\begingroup$ @Lucian: "We do not choose our inclinations. We're born with them." - I totally agree with this. But don't make snap judgements: I actually do feel much more inclined towards the abstract nonsense way of thinking than it is usual among analysts. And that's precisely the reason why I want to learn this. I also don't like the whole idea of dividing mathematics into camps that barely understand each other. $\endgroup$ – Alexander Shamov Nov 28 '13 at 1:23
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    $\begingroup$ @Lucian: Not helpful. AlexanderShamov: Gotcha. I can say you can get a start on topos theory without having to nearly master Mac Lane. And topos theory does play a big part in the branches of abstract nonsense that get at foundational issues, if that's an interest of yours. If topology is more your thing, that stuff's beyond my ken... $\endgroup$ – Malice Vidrine Nov 28 '13 at 2:10
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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.

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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.

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