nLab higher differential geometry

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 $n$, where the traditional theory is contained in the case $n = 0$. For $n = 1$ these higher structures are Lie groupoids, differentiable stacks, their infinitesimal approximation by Lie algebroids and the generalization to smooth stacks. For higher $n$ 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

• Smooth∞Grpd, SuperFormalSmooth∞Grpd.

• Lie 2-groupoid

• double Lie algebroid, Lie algebroid-groupoid, double Lie groupoid

Related concepts

• higher structure

• higher Lie theory

• higher prequantum geometry

• higher complex analytic geometry

• L-infinity algebras in physics

References

Exposition is in

• Urs Schreiber, Higher Structures in Mathematics and Physics, talk at Oberwolfach Workshop 1651a, 2016 Dec. 18-23

Details:

• geometry of physics, categories and toposes, …

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)

Textbook accounts:

• Ieke Moerdijk, Janez Mrcun Introduction to Foliations and Lie Groupoids, Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, Cambridge, (2003) (doi:10.1017/CBO9780511615450)

• 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:

• Jochen Heinloth, Some notes on differentiable stacks (pdf)

• Kai Behrend, Ping Xu, Differentiable Stacks and Gerbes (arXiv:0605.5694).

• Metzler, Topological and smooth stacks (arXiv:math/0306176)

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

• Urs Schreiber, geometry of physics – smooth spaces

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

• Urs Schreiber, twisted smooth cohomology in string theory, lectures at ESI program on quantum fields and K-theory, 2012

Introductory exposition includes the introductory sections of

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech Cocycles for Differential characteristic Classes, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 (arXiv:1011.4735)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, A higher stacky perspective on Chern-Simons theory, in Damien Calaque et al. (eds.) Mathematical Aspects of Quantum Field Theories Mathematical Physics Studies, Springer 2014 (arXiv:1301.2580)

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

• Urs Schreiber, differential cohomology in a cohesive topos (arXiv:1310.7930)

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

• Ulrich Bunke, Thomas Nikolaus, Michael Völkl, Differential cohomology theories as sheaves of spectra (arXiv:1311.3188)

On derived differential geometry and higher Lie theory cast in (differential) cohesive $(\infty,1)$-topos theory:

• Luigi Alfonsi, Charles A. S. Young, Towards non-perturbative BV-theory via derived differential cohesive geometry [arXiv:2307.15106]

exposition

• Luigi Alfonsi: Derived $n$-plectic geometry: towards non-perturbative BV-BFV quantisation and M-theory, talk at M-Theory and Mathematics (Jan 2023) [slides: pdf, video: YT]

Review:

• Urs Schreiber, Higher Topos Theory in Physics [arXiv:2311.11026]

• Luigi Alfonsi, Higher geometry in physics, in: Encyclopedia of Mathematical Physics 2nd ed [arXiv:2312.07308]