nLab simplicial manifold

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.

Constructions on simplicial manifolds

• The simplicial de Rham complex generalizes the de Rham complex from ordinary manifolds to simplicial manifolds.

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 presents 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.

Kan simplicial manifolds

For more information, see the article Kan simplicial manifold.

Related concepts

• Lie groupoid

• ∞-stack

• Kan simplicial manifold