The Stacks Project

Tag 03B4

102.4. Related references on foundations of stacks

    The code snippet corresponding to this tag is a part of the file guide.tex and is located in lines 115–166 (see updates for more information).

    \section{Related references on foundations of stacks}
    \item Vistoli:
    \emph{Notes on Grothendieck topologies, fibered categories and descent theory}
    Contains useful facts on fibered categories, stacks and descent theory in the
    fpqc topology as well as rigorous proofs.
    \item Knutson: \emph{Algebraic Spaces} \cite{Kn}
    This book, which evolved from his PhD thesis under Michael Artin,
    contains the foundations of the theory of algebraic spaces. The book
    \cite{LM-B} frequently references this text. See also Artin's papers on
    algebraic spaces: \cite{Artin-Algebraic-Approximation},
    \cite{ArtinI}, \cite{Artin-Implicit-Function},
    \cite{ArtinII}, \cite{Artin-Construction-Techniques},
    \cite{Artin-Algebraic-Spaces}, \cite{Artin-Theorem-Representability}, and
    \item Grothendieck et al, \emph{Th\'eorie des Topos et Cohomologie \'Etale des
    Sch\'emas I, II, III} also known as SGA4 \cite{SGA4}
    Volume 1 contains many general facts on universes, sites and fibered
    categories. The word ``champ'' (French for ``stack'') appears in
    Deligne's Expos\'e XVIII.
    \item Jean Giraud: \emph{Cohomologie non ab\'elienne} \cite{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.
    \item Kelly and Street: \emph{Review of the elements of 2-categories}
    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 \cite{stacks-project} contains some basics on 2-categories.

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