The Stacks project

79.1 Introduction

This chapter is devoted to advanced topics on groupoids in algebraic spaces. Even though the results are stated in terms of groupoids in algebraic spaces, the reader should keep in mind the $2$-cartesian diagram
\begin{equation} \label{spaces-more-groupoids-equation-quotient-stack} \vcenter { \xymatrix{ R \ar[r] \ar[d] & U \ar[d] \\ U \ar[r] & [U/R] } } \end{equation}

where $[U/R]$ is the quotient stack, see Groupoids in Spaces, Remark 78.20.4. Many of the results are motivated by thinking about this diagram. See for example the beautiful paper [K-M] by Keel and Mori.

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