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 groupoid presentation [Y ₁ ⇉ Y ₀] such that Y ₀ and Y ₁ are compact topological spaces. If Y ₀ and Y ₁ are only locally compact, we show that Map(𝒴, 𝒳) is a paratopological stack. In particular, it has a classifying space (hence, a natural weak homotopy type). We also show 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 a 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.