nLab
diffeological groupoid

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

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          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          \infty-Lie theory

          ∞-Lie theory (higher geometry)

          Background

          Smooth structure

          Higher groupoids

          Lie theory

          ∞-Lie groupoids

          ∞-Lie algebroids

          Formal Lie groupoids

          Cohomology

          Homotopy

          Examples

          \infty-Lie groupoids

          \infty-Lie groups

          \infty-Lie algebroids

          \infty-Lie algebras

          Contents

          Idea

          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.

          Examples

          • The path groupoid of a smooth manifold (and indeed of a diffeological space) XX is a diffeological groupoid.

          • A pointed connected diffeological groupoid is a diffeological group, generalizing the notion of Lie group.

          References

          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. (MathScinet)

          motivated by the examples appearing in geometric quantization, such as the (Hamiltonian) diffeomorphism group and its quantomorphism group extension.

          Last revised on August 2, 2018 at 03:01:48. See the history of this page for a list of all contributions to it.