### Synthetic Differential Geometry and Surface Holonomy

#### Posted by urs

L. Breen and W. Messing in their famous math.AG/0106083 had noted that what is called *synthetic differential geometry* with its use of *combinatorial differential forms* is naturally suited for talking about connections on higher order structures such as gerbes in terms of ‘finite’ morphisms between these structures.

Synthetic differential geometry goes back to category-theoretic ideas by Lawvere and was developed mainly by Anders Kock. There is a textbook

Anders Kock
**Synthetic Differential Geometry**

London Mathematical Society Lecture Notes Series **51**

Cambridge University Press (1981)

as well as a series of more recent papers which discuss things like gauge theory

A. Kock
**Combinatorics of Curvature, and the Bianchi Identity**

Theory and Application of Categories **2** (1996), 69-89

and distribution theory

A. Kock
**Categorical Distribution Theory; Heat Equation**

to appear in Cahiers de Top. et Geom. Diff. Categorique.

preprint available here

from the synthetic point of view.

L. Breen and W. Messing have reformulated and generalized this framework to a scheme-theoretic context in

L. Breen, W. Messing
**Combinatorial Differential Forms**

math.AG/0005087 .

One can safely include synthetic/combinatorial differential geometry in the list of concepts which are very simple and easy to handle in their pedestrian version, but which are powerful and far-reaching enough to admit mind-bogglingly complex generalizations. Breen and Messing discuss the generalized setup. Kock mostly cares about the more pedestrian version.

Combinatorial differential forms make again an appearance in a recent paper by Jussi Kalkkinen which further investigates aspects of Breen&Messing’s work on gerbes with connection:

Jussi Kalkkinen
**Topological Quantum Field Theory on Non-Abelian Gerbes**

hep-th/0510069 .

The paper discusses, motivated by similar construction in physics, how to enlarge the ‘field content’ of local data of a (nonabelian) gerbe by odd-graded ‘ghost’ fields such that odd graded BRST-like nilpotent operators generate the infinitesimal version of gauge transformations on this data.

The approach used is different but not totally unrelated to the construction presented in section 13 of hep-th/0509163.

Incidentally, I have recently been thinking about how to use synthetic differential forms in order neatly relate smooth $p$-holonomy $p$-functors to their associated $p$-forms. This is a kind of technical issue with probably little interest for physically inclined people, but it seems that there is a conceptually nice mechanism at work which relates ‘macroscopic’ $p$-functors to their ‘infinitesimal’ parts.

Some preliminary notes which review material from the theory of smooth (‘diffeological’) spaces as well as synthetic differential geometry and uses them in order to analyse smooth $p$-holonomy $p$-functors can be found here:

Notes: Holonomy on Smooth Path Spaces .

I imagine that much more can be done with synthetic differential geometry in the context of $p$-holonomy, but it should be a first step.

Posted at October 11, 2005 11:57 AM UTC
## Re: Synthetic Differential Geometry and Surface Holonomy

Hi Urs!

It’s been a long time. I’m glad I have one machine that still automatically loads this web site when I launch Mozilla :)

Just glanced at your notes. I think you can guess that I found the section “Combinatorial Differential Forms for Mortals” to be very interesting. Some of those formulas looked very familiar though. That is the language Dimakis and Hoissen like to use when discussing their version of noncommutative geometry. I am still waiting for you to make the connection (pun?) between all this and what we were playing around with :)

Best wishes!

Eric

PS: I hate to be the one to break it to you, but you stopped being mortal a LONG time ago ;)