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• Refs:
  • https://arxiv.org/pdf/0806.4160.pdf#page=18
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a stack is a category fibered in groupoids

Fix a category \(\mathsf{T}\), eg \(\mathsf{T} = {\mathsf{Top}}\), then consider $\mathsf{X}\in \mathsf{Cat}_{/ {\mathsf{T}}} $ a category over \(T\) defined by \(\mathsf{X} \xrightarrow{\pi }\mathsf{T}\). Then \(\mathsf{X}\) is fibered in groupoids iff

• Diagrams in \(\mathsf{T}\) lift for every \(x \xrightarrow{f} y\) in the base \(\mathsf{T}\) and every \(X\in \mathsf{X}\) with \(\pi(X) = x\), there is a \(Y\in \mathsf{X}\) and a morphism \(X \xrightarrow{\tilde f} Y\) such that \(\pi(\tilde f) = f\):

Link to Diagram

• Partial triangles lift uniquely:

Link to Diagram

Definition: for a fixed \(x\in \mathsf{T}\), define the fiber as \(\mathsf{X}_x\leq \mathsf{X}\) the subcategory of objects \(X\) with \(\pi(X) = x\). One can check that \(\mathsf{X}_x\in{\mathsf{Grpd}}\).

If \(\mathsf{X}\) satisfies descent, it is a stack:

As a presheaf of groupoids

To prestacks and stacks

See descent data:

Descent condition

# Examples

• \({{\mathbf{B}}G}\to {\mathsf{Top}}\) where \({{\mathbf{B}}G}\) is the groupoid of principal \(G{\hbox{-}}\)bundles \(P\to X\) and the functor is \((P\to X)\mapsto X\). A fiber \({{\mathbf{B}}G}_X\) is the groupoid of \(G{\hbox{-}}\)bundles over the space \(X\).
• For a fixed space \(X\), \({\mathcal{F}}_X \to {\mathsf{Top}}\) where \({\mathcal{F}}_X(T) = {\mathsf{Top}}(T, X)\) is the category of continuous maps \(T\to X\) with the functor \((T\to X)\mapsto T\). The functor \(T\mapsto {\mathcal{F}}_X(T)\) is a sheaf of sets on \({\mathsf{Top}}\).