# Deeper studies in Category Theory: suggestions and references.

I discovered Category Theory one year and a half ago and I got addicted. I studied some of the basic concepts of this branch (limits, adjunctions etc.), including some sightseeing into abelian categories and cohomology theory and some look at sheaves on sites for my undergraduate thesis. I also plan in the immediate future to try and get confident with triangulated and derived categories, if I can bear the subject.

I'd now like to go into deeper studies in CT, so I'd appreciate to have some references (books or on-line sources) and suggestions, especially about what topics should I cover (and discover), also in view of possible research outcomes. For the moment, I'd like to stick to arguments which are as near as possible to "pure" Category Theory, without studying it through its applications in different fields of mathematics (even though this do match my interests and is somehow unavoidable, I think, for a reasonable understanding of CT itself).

For sake of concreteness, I can say I feel rather attracted from Higher Category Theory (even if I don't know much about it nor about how much background it is needed) and Categorical Algebra, but I'm open to any other (maybe more approachable or more fundamental) suggestion for further studying.

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Since you mentioned categorical algebra, you can go through all volumes of Borceux's handbook. That should keep you occupied for a while.

Higher category theory is receiving quite some attention lately. The nLab is full of discussions, ideas, and information. You can roam there for hours and learn quite a lot by following the references.

There is also the recent project on homotopy type theory if you are interested in foundations other than topos theory.

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+1 for nLab! I love that website –  Avitus Nov 26 '13 at 9:21

The Gelfand Manin Methods of Homological Algebra contains much information on derived categories and related constructions, with much detail. Forget about the first chapter (apart from the basic definitions on category theory) and go directly to the category of chain complexes. At the end of the book you have a short introduction to triangulated categories.

For this topic I suggest Neeman's Triangulated Ctegories, though. The first 2 chapters contain what it should be known about triangulated categories and subcategories. It is a very well written book, with a rather strange-baroque font.

Do not forget to have a look at Weibel's An Introduction to Homological Algebra: it is nice, concise but full of examples.

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why the downvote? –  Avitus Nov 26 '13 at 10:39

There's always Lurie's "Higher Topos Theory", which I believe is on the arXiv as well as in print. Between this and the nLab I'm pretty sure you could keep yourself busy with $\infty$-categories for quite some time.

I've just started to get into this book myself, and it's delightful. Though you will probably find it helpful to have some algebraic topology under your belt to really get into it. (I do not, and am finding it rather challenging as a result.)

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The arXiv version is out-of-date. The latest version can be found on Lurie's website. –  Zhen Lin Nov 26 '13 at 10:21
Cheers, good to know! –  Malice Vidrine Nov 26 '13 at 10:33

I'm surprised nobody mentioned topos theory! Please, have a look to Moerdijk-Mac Lane "Sheaves in Geometry and Logic", I think it's kind of Pandora's box of pure Category Theory (like almost everything written by Moerdijk, obviously: maybe a good step further should be the amazingly short, amazingly deep book "Classifying spaces and classifying topoi").

I think that after this, what you want to do strongly depends on which side of the story you like to expand: geometry? algebraic topology? logic? Interplay between the three?

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Are there really people who study category theory and don't study topos theory? Madness. –  Malice Vidrine Nov 26 '13 at 22:16
Plenty! There's an equally vast branch of categorical algebra (i.e. category-theoretic abstract algebra), after all... –  Zhen Lin Nov 26 '13 at 22:28
Lots of people do algebraic geometry without knowing what a Grothendieck topos is (obviously minor subtleties about general (=elementary) toposes would be too much for them). This is madness. –  tetrapharmakon Nov 27 '13 at 8:24

As Ittay pointed out, Borceux's books are pretty good.

You might also want to check out Professor Vakil's Algebraic Geometry notes (although they're "Algebraic Geometry" notes, it should be noted that most advanced books on Algebraic Geometry have a lot devotion to categorical machinery before getting to classification of curves, which you might find interesting).

Also, Professor Lurie has a lot of notes on his website on higher category theory and connections to topology.

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