synthetic differential geometry
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Examples
-Lie groupoids
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Like a Lie groupoid is an internal groupoid in the category Diff of smooth manifolds, a diffeological groupoid is more generally an internal groupoid in the larger category of diffeological spaces.
If we regard Lie groupoids as special cases of stacks on Diff (smooth stacks), then diffeological groupoids are a little more general special cases.
The path groupoid of a smooth manifold (and indeed of a diffeological space) is a diffeological groupoid.
A pointed connected diffeological groupoid is a diffeological group, generalizing the notion of Lie group.
Original discussion of diffeological groups:
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. (doi:10.1007/BFb0089728, mr:607688)
Jean-Marie Souriau, Groupes différentiels et physique mathématique, In: Denardo G., Ghirardi G., Weber T. (eds.) Group Theoretical Methods in Physics. Lecture Notes in Physics, vol 201. Springer 1984 (doi:10.1007/BFb0016198)
motivated by the examples appearing in geometric quantization, such as the (Hamiltonian) diffeomorphism group and its quantomorphism group extension.
The definition of diffeological groupoids:
Historical survey:
On Morita equivalence of diffeological groupoids:
Nesta van der Schaaf: Diffeology, Groupoids & Morita Equivalence, MSc thesis, Radboud University (2020) [pdf]
Nesta van der Schaaf: Diffeological Morita Equivalence, Cahiers de Topologie et Géométrie Différentielle Catégoriques LXII 2 (2021) 177-238 [arXiv:2007.09901, pdf]
On Lie theory for diffeological groupoids:
As a special case of smooth groupoids (and smooth infinity-groupoids):
Hisham Sati, Urs Schreiber, Equivariant Principal -Bundles, Cambridge University Press (2025) [arXiv:2112.13654]
Last revised on October 31, 2025 at 08:17:36. See the history of this page for a list of all contributions to it.