# Geometric realization of the mapping stack

Some background and notation

Let $Sh_{\infty}(Cartsp)$ be the infinity category of smooth simplicial sheaves on the site of cartesian spaces (convex open subsets of $\mathbb{R}^n$ and smooth maps between them), equipped with the topology of good open covers (contractible finite intersections).

This infinity topos is cohesive and in particular the functor ${\rm disc}:\infty-\mathscr{G}{\rm rpd}\to Sh_{\infty}(Cartsp)$ admits an $\infty$ left adjoint $\Pi$ which preserves finite products. This is the functor I am calling geometric realization.

Question

Let me denote the internal mapping simplicial sheaf, coming from the cartesian closed structure on $Sh_{\infty}(Cartsp)$, by $[M,X]$. Given a smooth compact manifold $M$ and a smooth simplicial sheaf $X$, I would like to compare the geometric realization of $[M,X]$ and the mapping space of the geometric realizations of $M$ and $X$ respectively. Ideally, I would hope for an equivalence $$\Pi[M,X]\simeq Map(\Pi(M),\Pi(X))\;.$$ Is this true? I haven't been able to come up with a counter example and I know this is true if one takes sheaves with values in a stable infinity category (there one can use the fact that arbitrary colimits commute with finite limits along with descent with respect to a finite cover of $M$). Any help here would be greatly appreciated!

• Thanks! The fact that one has this 'singular nerve' representation of $\Pi$ is fantastic. Will this article be appearing on the arXiv soon? – Daniel Grady Jun 13 '17 at 12:11