nLab Kan simplicial manifold

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 $\Gamma$ applied to any nonnegatively graded chain complex of abelian Lie groups.

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

5. The nonabelian analogue of $\Gamma$ 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 $\Gamma$ 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.

Related concepts

• simplicial manifold

• higher differential geometry

• internal ∞-groupoid

• smooth ∞-groupoid

References

Early appearances of the concept include

• André Henriques, Integrating $L_\infty$-algebras, Compositio Mathematica, 144, (2008), no. 4, 1017–1045 (arXiv:math/0603563)

• Chenchang Zhu, $n$-groupoids and stacky Lie groupoids, Int Math Res Notices (2009) 2009 (21): 4087-4141. (arXiv:math/0609420)

• Chenchang Zhu, Kan replacement of simplicial manifolds, Lett Math Phys (2009) 90:383–405, arXiv:0812.4150

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)

• Jonathan Pridham, Presenting higher stacks as simplicial schemes, Advances in Mathematics, Volume 238, Pages 184-245 (arXiv:0905.4044)

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

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, section 4.2 of Principal ∞-bundles – Presentations, Journal of Homotopy and Related Structures, March 2014 (arXiv:1207.0249)

• Jesse Wolfson, Descent for $n$-Bundles (arXiv:1308.1113)