I'm a graduate student interested in categorical algebra. I want to understand Grothendieck categories and in particular the Gabriel-Popescu theorem. See Wikipedia page.
I found a lot of litterature on the subject, like the Tohoku paper, the Freyd's book, the thesis of Gabriel, etc... But all those references are not adressed to a "student public". They are often too old to be pleasant to read or to specific to be approchable for beginners. I would like to find a book where all definitions are detailed, examples are given, equivalences of definitions are explained, and theorems like the Gabriel-Popescu are developped. My question is then : does there exists a pedagogic reference on Grothendieck categories? It can be a syllabus. I also speak French.
You may try to have a look to the chapters about "Abelian categories" and "Grothendieck categories" in Stenström's excellent book "Rings of Quotients, An Introduction to Methods of Ring Theory".
In those chapters you will find very detailed proofs for the Gabriel-Popescu Theorem and for several basic facts about Grothendieck categories (existence of products, existence of an injective cogenerator, existences of injective envelopes, ...).