synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
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.
Diffeological groups were intoduced in
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.
Textbook account:
Last revised on June 16, 2021 at 12:19:11. See the history of this page for a list of all contributions to it.