+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Category theory +-- {: .hide} [[!include category theory - contents]] =-- #### Mapping space +--{: .hide} [[!include mapping space - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## 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](https://doi.org/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](https://arxiv.org/abs/0707.4673), [ams:crmp-50](https://bookstore.ams.org/crmp-50)) More general results in the context of [[differentiable stacks]] ([[orbifolds]]) are in * {#RobertsVozzo18} [[David Michael 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](https://arxiv.org/abs/1610.05904), [doi:10.1007/978-3-319-72299-3_3](https://doi.org/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 * {#Noohi08} [[Behrang Noohi]], _Mapping stacks of topological stacks_, Journal für die reine und angewandte Mathematik, Volume 2010, Issue 646 ([arXiv:0809.2373](http://arxiv.org/abs/0809.2373), [doi:10.1515/crelle.2010.067](https://doi.org/10.1515/crelle.2010.067)) [[!redirects mapping stacks]] [[!redirects Hom stack]] [[!redirects Hom stacks]] [[!redirects hom stack]] [[!redirects hom stacks]] [[!redirects Hom-stack]] [[!redirects Hom-stacks]] [[!redirects hom-stack]] [[!redirects hom-stacks]]