Formal Lie groupoids
The notion of smooth groupoid is the first generalization of the notion of smooth space to higher differential geometry. A smooth groupoid is a stack on the site CartSp. This is equivalently an 1-truncated smooth infinity-groupoid.
This concepts contains notably the deloopings of Lie groups and more generally of diffeological groups, and the concept of Lie groupoid.
A smooth stack or smooth groupoid is a stack on the site SmoothMfd of smooth manifolds or equivalently (and often more conveniently) on its dense subsite CartSp of just Cartesian spaces and smooth functions between them, equipped with the standard coverage of good open covers.
for the (2,1)-category of stacks on this site, equivalently the result of taking groupoid-valued presheaves and then universally turning local (as seen by the coverage) equivalences of groupoids into global equivalence in an (infinity,1)-category.
By generalizing here groupoids to general Kan complexes and equivalences of groupoids to homotopy equivalences of Kan complexes, one obtains smooth ∞-stacks or smooth ∞-groupoids, which we write
We often write Smooth∞Grpd for short.
By the (∞,1)-Yoneda lemma there is a sequence of faithful inclusions
SmoothMfd SmoothGrpd Smooth∞Grpd.
This induces a corresponding inclusion of simplicial objects and hence of groupoid objects
For a groupoid object we write
for its (∞,1)-colimiting cocone, hence (without subscript decoration) denotes the realization of as a single object in .
By the Giraud-Rezk-Lurie axioms of the (∞,1)-topos this morphism is a 1-epimorphism – an atlas – and its construction establishes is an equivalence of (∞,1)-categories , hence morphisms in are equivalently diagrams in of the form
This is evidently more constrained that just morphisms
by themselves. The latter are the “generalized” or Morita morphisms between the groupoid objects , . These can be modeled as --bibundles.
Every Lie groupoid presents a smooth groupoids. Those of this form are also called differentiable stacks.
A 0-truncated smooth groupoid is equivalently a smooth space.
For a smooth group, its delooping is a smooth groupoid, the moduli stack of smooth -principal bundles.
Discussion of diffeological groups (such as diffeomorphism groups and quantomorphism groups) goes back to
- Jean-Marie Souriau, Groupes différentiels, in Differential Geometrical Methods in Mathematical Physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math. 836, Springer, Berlin, (1980), pp. 91–128. (MathScinet)
Discussion of smooth stacks as target spaces for sigma-model quantum field theories is in
Discussion of geometric Langlands duality in terms of 2d sigma-models on stacks (moduli stacks of Higgs bundles over a given algebraic curve) is in