Reference request for Category Theory 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.
 A: 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:


*

*categorical logic (from a math perspective): Sheaves in geometry and logic by Mac Lane and Moerdijk and Sketches of an elephant by Johnstone, and also most of Lawvere's papers,

*categorical logic (from a CS perspective): Categorical logic and type theory by Jacobs,

*algebraic geometry: all SGAs are written with heavy category theory, Algebraic geometry by Hartshorne, and I should mention Stack Project,

*sheaf theory: Sheaves in geometry and logic by Maclane and Moerdijk and Categories and Sheaves by Shapira and Kashiwara

*abelian categories and homological agebra: Grothendieck's Tohoku original paper Sur quelques points d'algèbre homologique is very readable if you read french, and An Introduction to Homological Algebra by Weibel seems to be quite a hit, and Categories and Sheaves by Shapira and Kashiwara

*homotopy theory: Homotopical algebra by Quillen started it all but is not that easy to read, Model categories by Hovey might be an easier entry point, and Grothendieck's tapuscript Pursuing stacks is worth at least browsing,

*higher category theory: Higher Topos Theory by Lurie, Higher operads, higher categories by Leinster

*Also, try not to forget to take a look at Categories for the working mathematician by Mac Lane from times to times, there is always something in it that we thought was not there.


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...
A: 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.
