# nLab higher differential geometry

Contents

## Theorems

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

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

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 $n$, 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

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