nLab
topological stack

Definition

Let $\mathrm{Top}$ be the category of compactly generated spaces and continuous maps, equipped with a Grothendieck topology given by usual open covers of topological spaces. This topology is subcanonical. Consider the 2-category $\mathrm{TopStack}$ of 1-stacks of groupoids on $\mathrm{Top}$; by Yoneda $\mathrm{Top}$ is a full subcategory.

By analogy with the case of algebraic stacks one says that a morphism of 1-stacks $f:X\to Y$ in $\mathrm{TopStack}$ is a representable morphism of stacks if for any morphism of 1-stacks $T\to Y$ from a (stack associated to a) topological space $T$ to $Y$ the pullback $T{\times }_{Y}X$ is isomorphic to (a stack associated to) a topological space.

We say that a property $P$ of morphisms is local on the target if satisfaction of this property for a base change of a morphism $f$ along a surjective local homeomorphism implies the property for $f$. Given a property $P$ of morphisms of topological spaces stable under base change along embeddings and local on the target; a representable morphism $f:X\to Y$ of 1-stacks has this property if there exists a topological space $T$ and an epimorphism $T\to Y$ such that the inverse image $T{\times }_{Y}X\to X$ has property $P$.

Following Noohi, we say that

Definition

A 1-stack of groupoids over $\mathrm{Top}$ having a representable epimorphism is a pretopological stack. Any map from a topological space $S$ to a pretopological stack $X$ is representable and the diagonal $X\to X\times X$ is representable as well. The pretopological stack is called topological stack if the chart can be chosen to belong to a class of “local fibrations”; there are axioms which the class of local fibrations have to satisfy; there are several natural choices of this class which modify the variant of topological stacks considered.

References

Articles by Behrang Noohi on this topic:

• Foundations of Topological Stacks I, pdf
• Homotopy types of topological stacks
• Mapping stacks of topological stacks
• K. Behrend, G. Ginot, B. Noohi, P. Xu, String topology for stacks