Learning higher differential geometry

Question

I have read parts of the motivation on nlab and all the posts on MO I could find on the subject, and by now there are a few questions on my mind. If they trivial for someone who understands the theory, I apologise beforehand. My questions are:

In Higher Differential Geometry, are we able to ...

1. ... generalise metrics to smooth spaces?
2. ... generalise connections, e.g. LC and Chern connections to smooth spaces?
3. ... explicitly calculate classical geometric structures (metrics, geodesics, connections, curvature, cohomology, etc.) for some interesting smooth spaces, and are we always able to do so if we are able to do so in classical geometry? Do the calculations get easier or more clear conceptually?

And finally: what is an achievable, if not royal, road to learning higher differential geometry for someone who has no prior knowledge in higher category theory/higher topos theory (maybe a reading list which is as concise as possible)?

Thank you very much for your time to read my questions.

Answer 1

You seem to be asking about my $n$Lab entries on higher differential geometry and related. I have now added there a

• commented References section

with some pointers. See there for an extended and fully hyperlinked version of the following. Also, maybe you find useful some of the talk slides and lecture notes that I keep here.

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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) , 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

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

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 geometry: 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

• geometry of physics – smooth spaces

Passing from here to more general smooth groupoids, to smooth groupoids and then eventually to smooth ∞-groupoids involves (∞,1)-topos theory proper, with some tools such as the model structure on simplicial presheaves over the site of smooth manifolds (or equivalently just that 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 of the differential cohomology hexagon 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)