nLab
higher differential geometry

Context

Higher geometry

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 }

          </semantics></math></div>

          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Contents

          Idea

          Higher differential geometry is the incarnation of differential geometry in higher geometry. Hence it is concerned with n-groupoid-versions of smooth spaces for higher nn, where the traditional theory is contained in the case n=0n = 0. For n=1n = 1 these higher structures are Lie groupoids, differentiable stacks, their infinitesimal approximation by Lie algebroids and the generalization to smooth stacks. For higher nn this includes (deloopings of) Lie 2-groups, Lie 3-groups.

          Fully generally, higher differential geometry hence replaces smooth manifolds (and possibly variants such as supermanifolds, formal manifolds, dg-manifolds etc.) by ∞-stacks ((∞,1)-sheaves) on the site of all such. Technically this means that higher differential geometry is the study of an (∞,1)-topos into which standard differential geometry faithfully embeds. This then allows to speak of smooth ∞-groups, Lie ∞-algebroids.

          If the ambient (∞,1)-topos is not 1-localic (for instance over a genuine site of dg-manifolds) then one also speaks of derived differential geometry.

          See at motivation for higher differential geometry for motivation.

          The standard variants of differential geometry have their higher analogs, for instance symplectic geometry generalizes to higher symplectic geometry and prequantum geometry to higher prequantum geometry.

          Formalizations

          One axiomatization is cohesion and differential cohesion.

          Examples

          References

          Exposition is in

          Details are in

          The most classical aspect of higher differential geometry is the theory of orbifolds, Lie groupoids and Lie algebroids and their application in foliation theory. Original reference here include

          • Charles Ehresmann, Catégories topologiques et catégories différentiables Colloque de Géometrie Differentielle Globale (Bruxelles, 1958), 137–150, Centre Belge Rech. Math., Louvain, 1959;

          • Ieke Moerdijk, Dorette Pronk, Orbifolds, sheaves and groupoids, K-theory 12 3-21 (1997) (pdf), Orbifolds as Groupoids: an Introduction (arXiv:math.DG/0203100)

          and standard textbook accounts include

          • Ieke Moerdijk, Janez Mrcun Introduction to Foliations and Lie Groupoids, Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, Cambridge, (2003)

          • Kirill Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge University Press, 2005, xxxviii + 501 pages

            (website)

          • Kirill Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, 124. Cambridge University Press, Cambridge, 1987. xvi+327 pp (MathSciNet)

          For properly appreciating the homotopy theory of Lie groupoids and for passage to more general higher differential geometry it is crucial to understand Lie groupoids as smooth stacks which are geometric: differentiable stacks. Each of the following references provides introduction to this point of view:

          As a warmup for these considerations it may be useful to first look at smooth spaces given by just sheaves on the site of smooth manifolds, see at

          Passing from here to more general smooth groupoids, to smooth 2-groupoids and then eventually to smooth ∞-groupoids involves (∞,1)-topos theory proper, with some tools as discussed at model structure on simplicial presheaves over the site of smooth manifolds, or equivalently just over its dense subsite of Cartesian spaces.

          For motivation for this step see also

          Introductory exposition includes the introductory sections of

          and sections 1.2.4 (geometry of physics – smooth homotopy types) as well as section 1.2.5 (geometry of physics – principal bundles) in the Introduction section of

          This goes on to discuss differential cohomology and the differential cohomology diagram formulated in stable objects in smooth ∞-groupoids (hence in sheaves of spectra on the site of smooth manifolds/Cartesian spaces) in higher differential geometry, see

          Last revised on August 1, 2018 at 10:22:17. See the history of this page for a list of all contributions to it.