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Although this question has been asked before, I want to ask a kind of more specific/different request concerning category theory (perhaps a bit more precise). I have gone through Emily's Riehl book ''Category theory in context'' and I can say, I understand it in a very good level. However this a basic book in category theory, which is a huge area of research hence quite broad, that covers up the main theory that almost everyone needs to know in algebra nowadays. So I was wondering if there is a "natural" extension of that book going towards the knowledge you need, to deal with the advanced stuff popping out category theory, like algebraic geometry, categorical logic, homotopy theory, sheaf theory etc. In other words, a book that comprises the appropriate categorical language/knowledge to go through deeper theories and results in mathematics.

DISCLAIMER:

As I have been mentioned above, I do know the vastness of category theory and what I am asking is probably a little bit absurd, however there could probably be a textbook treating a second step in category theory, so that's why am asking.

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As you say, category theory and its applications are vast, and a book going deeper in category theory will necessarily narrow the field it covers (otherwise, it is called the nlab).

Now depending on which field you want to go deeper in, there is some textbooks that are recognized as references. Here a non exhaustive list of those I'm thinking about:

Personally, I learned a lot from Sheaves in geometry and logic when I had only a basic categorical luggage. It is very readable, it is quasi self contained (especially every basic notion is recalled as some point) and it goes quite far inside topos theory.

Some of the books I cited before have a free preprint (or final) version. The other are famous enough that you can find a way to have a look at them...

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  • $\begingroup$ There are also these books aimed at certain applications: Category Theory for the Sciences, by David Spivak. mitpress.mit.edu/books/category-theory-sciences Basic Category Theory for Computer Scientists, by Benjamin Pierce. mitpress.mit.edu/books/… Category Theory for Computing Science by Michael Barr and Charles Wells. tac.mta.ca/tac/reprints/articles/22/tr22.pdf $\endgroup$ – SixWingedSeraph Jun 14 '17 at 14:01
  • $\begingroup$ @SixWingedSeraph Thanks for these addings, I didn't thought about the more applied stuff. $\endgroup$ – Pece Jun 14 '17 at 15:43
  • $\begingroup$ @KevinCarlson Remark I put Lurie in "higher category theory", not in "homotopy theory" (I assume people goes through some homotopical algebra before higher stuff in general but I might be wrong). That being said, the first chapters of HTT are not that hard to read iirc, Lurie makes a pretty good job conveying intuitions. And HTT's appendices are a gold mine of usual results in homotopical algebra. (Of course, it is also a matter of taste.) $\endgroup$ – Pece Jun 14 '17 at 17:34
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To add to the other suggestions: I think Riehl's second book, Categorical Homotopy Theory, is the best manageable introduction to higher category theory and homotopical algebra. It also contains a brief introduction on enriched category theory, but you should really read Kelly's book on the latter subject.

2-category theory (and low dimensional category theory in general) is an important area lacking a monograph, but Steve Lack's "A 2-Categories Companion" was designed to approximate this role: it's more a guide to the literature in an area which I'd argue is in the middle of a second flowering.

Finally, Borceux had a three-volume magnum opus surveying all of category theory at the time of writing, which may be the best way to get an idea of what all is a part of the field.

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  • $\begingroup$ I'd be very interested to know more specifically what you have in mind when you speak of "a second flowering". $\endgroup$ – Alexander Campbell Jun 14 '17 at 23:11
  • $\begingroup$ @AlexanderCampbell Well, you would know better than me, and I'm perfectly prepared to be entirely off base about this, but I was thinking of work like arxiv.org/abs/1301.3191, arxiv.org/abs/1104.2111, arxiv.org/abs/math/0607646, and comp.mq.edu.au/~rgarner/Papers/CT07.pdf, which it seems to me picks up after a bit of a slowdown ca. 1985-2000. $\endgroup$ – Kevin Carlson Jun 15 '17 at 0:40
  • $\begingroup$ I should also mention the program of Riehl and Verity to base $\infty$-category theory on 2-category theory as a very notable area of application. $\endgroup$ – Kevin Carlson Jun 15 '17 at 0:45

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