The Stacks project

Definition 94.12.3. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. The $2$-category of algebraic stacks over $S$ is the sub $2$-category of the $2$-category of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ (see Categories, Definition 4.35.6) defined as follows:

  1. Its objects are those categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ which are algebraic stacks over $S$.

  2. Its $1$-morphisms $f : \mathcal{X} \to \mathcal{Y}$ are any functors of categories over $(\mathit{Sch}/S)_{fppf}$, as in Categories, Definition 4.32.1.

  3. Its $2$-morphisms are transformations between functors over $(\mathit{Sch}/S)_{fppf}$, as in Categories, Definition 4.32.1.

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