∞-Lie theory (higher geometry)
Formal Lie groupoids
A smooth -groupoid is an ∞-groupoid equipped with cohesion in the form of smooth structure. Examples include smooth manifolds, Lie groups, Lie groupoids and generally Lie infinity-groupoids, but also for instance moduli spaces of differential forms, moduli stacks of principal connections and generally of cocycles in differential cohomology.
The (∞,1)-topos of all smooth -groupoids is a cohesive (∞,1)-topos. It realizes a higher geometry version of differential geometry.
Many properties of smooth -groupoids are inherited from the underlying Euclidean-topological ∞-groupoids. See ETop∞Grpd for more.
There is a refinement of smooth -groupoids to synthetic differential ∞-groupoids. See SynthDiff∞Grpd for more on that.
For any open cover of a paracompact manifold also is paracompact. Hence we may find a differentiably good open cover . This is then a refinement of the original open cover of .
We discuss the relation of to other cohesive (∞,1)-toposes.
The cohesive (∞,1)-topos ETop∞Grpd of Euclidean-topological ∞-groupoids has as site of definition CartSp. There is a canonical forgetful functor
The functor extends to an essential (∞,1)-geometric morphism
such that the (∞,1)-Yoneda embedding is factored through the induced inclusion SmoothMfd Mfd as
Using the observation that preserves coverings and pullbacks along morphism in covering families, the proof follows precisely the steps of the proof of this proposition.
(Both of these are special cases of a general statement about morphisms of (∞,1)-sites, which should eventually be stated in full generality somewhere).
The essential global section (∞,1)-geometric morphism of factors through that of ETop∞Grpd
Observe that CartSp is (the syntactic category of) a Lawvere theory: the algebraic theory of smooth algebras (-rings). Write for the category of its algebras. Let be the full subcategory on the infinitesimally thickened points.
Let CartSp be the full subcategory on the objects of the form with and . Write
for the canonical inclusion.
This follows as a special case of this proposition after observing that is an infinitesimal neighbourhood site of in the sense defined there.
In SynthDiff∞Grpd we have ∞-Lie algebras and ∞-Lie algebroids as actual infinitesimal objects. See there for more details.
The (1,1)-topos on the 0-truncated smooth -groupoids is
the sheaf topos on SmthMfd/CartSp discussed at smooth space.
The concrete objects in there
are precisely the diffeological spaces.
Structures in the cohesive -topos
We discuss the general abstract structures in a cohesive (∞,1)-topos realized in .
This section is at
Smooth -groupoids and related cohesive structures play a central role in the discussion at
For standard references on differential geometry and Lie groupoids see there.
The -topos is discussed in section 3.3 of
A discussion of smooth -groupoids as -sheaves on and the presentaton of the -Chern-Weil homomorphism on these is in
For references on Chern-Weil theory in Smooth∞Grpd and connection on a smooth principal ∞-bundle, see there.
The results on differentiable Lie group cohomology used above are in
- P. Blanc, Cohomologie différentiable et changement de groupes Astérisque, vol. 124-125 (1985), pp. 113-130.
- Graeme Segal, Cohomology of topological groups , Symposia Mathematica, Vol IV (1970) (1986?) p. 377
A review is in section 4 of
Classification of topological principal 2-bundles is discussed in
and the generalization to classification of smooth principal 2-bundles is in
Further discussion of the shape modality on smooth -groupoids is in