A (smooth) simplicial manifold is a simplicial object in the category Diff of smooth manifolds.
Under the canonical inclusion Diff Top this is in particular a simplicial topological space.
The role of simplicial manifolds in most of the literature is best understood from the nPOV, by thinking of as a subcategory 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.
For more information, see the article Kan simplicial manifold.
Last revised on December 24, 2020 at 19:39:52. See the history of this page for a list of all contributions to it.