nLab
Kan 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.

Examples

  1. Any ordinary manifold, interpreted as a constant simplicial object.

  2. The nerve of a Lie groupoid. In particular, the delooping of any Lie group, which represents principal bundles with this Lie group as a structure group.

  3. The Dold–Kan functor Γ\Gamma applied to any nonnegatively graded chain complex of abelian Lie groups.

  4. In particular, applying Γ\Gamma to the chain complex U(1)[n]\mathrm{U}(1)[n], we get the Kan simplicial manifold representing bundle (n1)(n-1)-gerbes.

  5. The nonabelian analogue of Γ\Gamma applied to any crossed module whose two constituent groups are Lie groups and the involved homomorphisms and actions are smooth.

  6. The nonabelian analogue of Γ\Gamma applied to any hypercrossed complex whose constituent groupoids are Lie groupoids and the involved homomorphisms and actions are smooth.

  7. As a special case of the previous example, any simplicial Lie group? is a Kan simplicial manifold.

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 December 24, 2020 at 15:40:13. See the history of this page for a list of all contributions to it.