higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth 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
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
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.
One axiomatization is cohesion and differential cohesion.
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 Mrčun 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:
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
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
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
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