nLab
Kan-fibrant simplicial manifold

Contents

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Internal (,1)(\infty,1)-Categories

Contents

Idea

A Kan-fibrant simplicial manifold is a simplicial manifold (for instance simplicial topological manifold or simplicial smooth manifold) which satisfies a suitable analog of the Kan complex-condition on a simplicial set. (Typically, to get an interesting theory, Kan-fibrancy on simplicial manifolds is imposed in a suitable local sense, meaning that horns have continuous/smooth fillers in a open neighbourhoods of all points, but possibly not globally.)

Motivated by the standard way (see at homotopy hypothesis) in which bare Kan complexes (hence Kan-fibrant simplicial sets) present geometrically discrete ∞-groupoids and given that the nerve of a Lie groupoid is an example of a locally Kan-fibrant simplicial manifold (a 1-truncated one), such Kan-fibrant simplicial manifolds are often referred to Lie infinity-groupoids (or Lie n-groupoids for finite truncation) (Zhu 06).

With such a suitably local definition, there should be the structure of a homotopical category on Kan-fibrant simplicial manifolds which embeds homotopically full and faithful into a local model structure on simplicial presheaves over a suitable site of manifolds, hence such that this inclusion presents and full sub-(∞,1)-category of the (∞,1)-sheaf (∞,1)-topos over manifolds (“smooth ∞-groupoids”).

Some care is needed in correctly interpreting the “Lie” condition in view of the homotopy theory. For instance every ∞-stack on the category of smooth manifolds (“smooth ∞-groupoid”) is presented by a simplicial manifold, just not in general by a suitably Kan-fibrant simplicial manifold (NSS 12, section 2.2).

A homotopy-correct characterization of the sub-\infty-category presented by the Kan-fibrant simplicial objects as that of geometric ∞-stacks modeled on manifolds is in (Pridham 09) (see around p. 17 for the differential geometric version).

Kan-fibrant simplicial manifolds have received particular attention as the result of Lie integration of L-∞ algebroids. See at Lie integration for more on that.

References

Early appearances of the concept include

Characterization of the homotopy theory of Kan-fibrant simplicial manifolds as geometric ∞-stacks modeled on smooth manifolds is in (see aroung p. 17 for the differential geometric version)

Discussion of principal ∞-bundles in Smooth∞Grpd =Sh (SmoothMfd)= Sh_\infty(SmoothMfd) which are represented by locally Kan-fibrant simplicial manifolds is in

Last revised on March 8, 2016 at 13:46:00. See the history of this page for a list of all contributions to it.