Contents

category theory

mapping space

# Contents

## Definition

An internal hom in a (2,1)-topos of stacks or generally in an (∞,1)-topos of ∞-stacks is often called a mapping stack or mapping ∞-stack, in analogy with mapping space.

## Properties

By the general formula for internal homs in toposes, for $X, A \in \mathbf{H}$ two stacks or $\infty$-stacks, their mapping stack assigns to an object $U \in C$ in a site or (∞,1)-site of definition the groupoid or ∞-groupoid given by

$[X, A] \;\colon\; U \mapsto \mathbf{H}(X \times U, A) \,.$

## References

Mapping stacks of orbifolds have been discussed in

• W. Chen, On a notion of maps between orbifolds, I. Function spaces, Commun. Contemp. Math. 8 (2006), no. 5, 569–620.

More general results in the context of differentiable stacks (orbifolds) are in

generalizing the discussion at manifold structure of mapping spaces to stacks.

That the mapping stack out of the circle into a topological stack is again a topological stack is often attributed to

• Ernesto Lupercio, Bernardo Uribe, Loop groupoids, gerbes, and twisted sectors on orbifolds In Orbifolds in mathematics and physics (Madison, WI, 2001), volume 310 of Contemp. Math., pages 163184. Amer. Math. Soc., Providence, RI, 2002.

General mapping stacks of topological stacks are discussed in

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