The Stacks project

112.4 Related references on foundations of stacks

• Vistoli: Notes on Grothendieck topologies, fibered categories and descent theory [vistoli_fga]

  Contains useful facts on fibered categories, stacks and descent theory in the fpqc topology as well as rigorous proofs.

• Knutson: Algebraic Spaces [Kn]

  This book, which evolved from his PhD thesis under Michael Artin, contains the foundations of the theory of algebraic spaces. The book [LM-B] frequently references this text. See also Artin's papers on algebraic spaces: [Artin-Algebraic-Approximation], [ArtinI], [Artin-Implicit-Function], [ArtinII], [Artin-Construction-Techniques], [Artin-Algebraic-Spaces], [Artin-Theorem-Representability], and [ArtinVersal]

• Grothendieck et al, Théorie des Topos et Cohomologie Étale des Schémas I, II, III also known as SGA4 [SGA4]

  Volume 1 contains many general facts on universes, sites and fibered categories. The word “champ” (French for “stack”) appears in Deligne's Exposé XVIII.

• Jean Giraud: Cohomologie non abélienne [giraud]

  The book discusses fibered categories, stacks, torsors and gerbes over general sites but does not discuss algebraic stacks. For instance, if $G$ is a sheaf of abelian groups on $X$, then in the same way $H^1(X, G)$ can be identified with $G$-torsors, $H^2(X, G)$ can be identified with an appropriately defined set of $G$-gerbes. When $G$ is not abelian, then $H^2(X, G)$ is defined as the set of $G$-gerbes.

• Kelly and Street: Review of the elements of 2-categories [kelly-street]

  The category of stacks form a 2-category although a simple type of 2-category where are 2-morphisms are invertible. This is a reference on general 2-categories. I have never used this so I cannot say how useful it is. Also note that [stacks-project] contains some basics on 2-categories.