nLab mapping stack




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,AHX, A \in \mathbf{H} two stacks or \infty-stacks, their mapping stack assigns to an object UCU \in C in a site or (∞,1)-site of definition the groupoid or ∞-groupoid given by

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


Mapping stacks of orbifolds have been discussed in

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

Last revised on July 10, 2020 at 14:41:32. See the history of this page for a list of all contributions to it.