∞-Lie theory (higher geometry)
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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see also algebraic topology
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category object in an (∞,1)-category, groupoid object
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.
Any ordinary manifold, interpreted as a constant simplicial object.
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.
The Dold–Kan functor $\Gamma$ applied to any nonnegatively graded chain complex of abelian Lie groups.
In particular, applying $\Gamma$ to the chain complex $\mathrm{U}(1)[n]$, we get the Kan simplicial manifold representing bundle $(n-1)$-gerbes.
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.
The nonabelian analogue of $\Gamma$ applied to any hypercrossed complex whose constituent groupoids are Lie groupoids and the involved homomorphisms and actions are smooth.
As a special case of the previous example, any simplicial Lie group? is a Kan simplicial manifold.
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)
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)
Last revised on December 24, 2020 at 15:40:13. See the history of this page for a list of all contributions to it.