Mapping stacks of topological stacks

TL;DR

This paper establishes conditions under which the mapping stack of topological stacks remains a topological or paratopological stack, providing a framework for understanding their homotopy types and examples like loop stacks of classifying stacks.

Contribution

It proves that mapping stacks of topological stacks are topological or paratopological under certain conditions and introduces an invariance theorem for their homotopy types.

Findings

Mapping stacks of topological stacks are topological if Y admits a compact groupoid presentation.

Mapping stacks are paratopological if Y admits a locally compact groupoid presentation.

The weak homotopy type of mapping stacks is invariant under replacement of X by its classifying space.

Abstract

We prove that the mapping stack Map(Y,X) of topological stacks X and Y is again a topological stack if Y admits a compact groupoid presentation. If Y admits a locally compact groupoid presentation, we show that Map(Y,X) is a paratopological stack. In particular, it has a classifying space (hence, a natural weak homotopy type). We prove an invariance theorem which shows that the weak homotopy type of the mapping stack Map(Y,X) does not change if we replace X by its classifying space, provided that Y is paracompact topological space. As an example, we describe the loop stack of the classifying stack BG of a topological group G in terms of twisted loop groups of G.