Deeper studies in Category Theory: suggestions and references.

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.


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.)


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?


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.