# Contents

## Definition

A (smooth) simplicial manifold is a simplicial object in the category Diff of smooth manifolds.

Under the canonical inclusion Diff $\hookrightarrow$ Top this is in particular a simplicial topological space.

## Interpretation in terms of higher category theory

The role of simplicial manifolds in most of the literature is best understood from the nPOV, by thinking of $[\Delta^{op}, Diff]$ as a subcategory $[\Delta^{op}, Diff] \hookrightarrow [\Delta^{op}, PSh(Diff)]$ of that of simplicial presheaves on Diff. Using the local model structure on simplicial presheaves which presentable (∞,1)-category the (∞,1)-topos of ∞-stacks, one is entitled to think of any simplicial manifold as representing a Lie ∞-groupoid.

In fact, more is true: as discussed there in detail, in the projective local model structure on simplicial presheaves on Diff there is a cofibrant resolution functor with values in simplicial manifolds. (Here we allow Diff to be closed under coproducts indexed by sets; that is, we do not require our manifolds to be connected or second-countable, although we may require their components to be second-countable). Therefore every ∞-stack on Diff may be presented by a simplicial manifold.

Revised on August 24, 2010 17:46:29 by Urs Schreiber (131.211.36.96)