# The Stacks Project

## Tag 03B4

### 102.4. 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.

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}
\label{section-related}

\begin{itemize}
\item Vistoli:
\emph{Notes on Grothendieck topologies, fibered categories and descent theory}
\cite{vistoli_fga}
\begin{quote}
Contains useful facts on fibered categories, stacks and descent theory in the
fpqc topology as well as rigorous proofs.
\end{quote}
\item Knutson: \emph{Algebraic Spaces} \cite{Kn}
\begin{quote}
This book, which evolved from his PhD thesis under Michael Artin,
contains the foundations of the theory of algebraic spaces. The book
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
\cite{ArtinVersal}
\end{quote}
\item Grothendieck et al, \emph{Th\'eorie des Topos et Cohomologie \'Etale des
Sch\'emas I, II, III} also known as SGA4 \cite{SGA4}
\begin{quote}
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.
\end{quote}
\item Jean Giraud: \emph{Cohomologie non ab\'elienne} \cite{giraud}
\begin{quote}
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.
\end{quote}
\item Kelly and Street: \emph{Review of the elements of 2-categories}
\cite{kelly-street}
\begin{quote}
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.
\end{quote}
\end{itemize}

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