hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
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.
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
Mapping stacks of orbifolds have been discussed in
Weimin Chen, On a notion of maps between orbifolds, I. Function spaces, Commun. Contemp. Math. 8 (2006), no. 5, 569–620 (doi:10.1142/S0219199706002246)
André Haefliger, On the space of morphisms between étale groupoids, In: P. Robert Kotiuga (ed.) A Celebration of the Mathematical Legacy of Raoul Bott (arXiv:0707.4673, ams:crmp-50)
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
General mapping stacks of topological stacks are discussed in
Last revised on July 10, 2020 at 10:41:32. See the history of this page for a list of all contributions to it.