nLab mapping stack

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

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

Related concepts

• mapping ∞-stack

References

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

• David Roberts, Raymond Vozzo, The Smooth Hom-Stack of an Orbifold, In : Wood D., de Gier J., Praeger C., Tao T. (eds) 2016 MATRIX Annals. MATRIX Book Series, vol 1. Springer, Cham (2018) (arXiv:1610.05904, doi:10.1007/978-3-319-72299-3_3)

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

• Behrang Noohi, Mapping stacks of topological stacks, Journal für die reine und angewandte Mathematik, Volume 2010, Issue 646 (arXiv:0809.2373, doi:10.1515/crelle.2010.067)