### Rational Homotopy Theory in an (oo,1)-Topos

#### Posted by Urs Schreiber

In the previous entry John described higher gauge theory with the declared intent of not emphasizing its higher category theory.

The other extreme of this has its own charms: try to describe higher gauge theory *entirely* using formal category theory on an ambient $\infty$-topos. I have been entertaining myself with searching for this intrinsic $\infty$-topos-theoretic formulation. The present state of my understanding is summarized on this page:

This lists structures that are available on purely formal grounds in an $(\infty,1)$-topos: its shape, its cohomology, its homotopy, its rational homotopy, its differential cohomology.

Here *differential cohomology in an $(\infty,1)$-topos* is just another word for *higher gauge theory* .

In the last days maybe I was able to fill what used to be a gap in the abstract story that I was trying to tell. It helped to read the remarkable recent article

- David Ben-Zvi, David Nadler,
*Loop spaces and connections*

and its emphasis of the issue discussed in the remarkable

- Bertrand Toën,
*Champs affine*

This provides a nice point of view on *rational homotopy theory* from $\infty$-topos theory. Some aspects of this I had understood before. For instance the left adjoint $\infty$-functor that produces the global function dg-algebra on an $\infty$-stack discussed there is what I used to call the Chevalley-Eilenberg algebra of an $\infty$-stack. But after reading this again now I obtained a clear simple abstract picture that I did not quite have before.

On the page

there is first a section that describes this abstract picture. Then there is a section that recalls key definitions and results from Bertrand Toën’s article from this perspective.

Posted at March 2, 2010 12:26 PM UTC
## Re: Rational Homotopy Theory in an (oo,1)-Topos

Where is the rational? How does a reflexive embedding give us a characteristic 0 field? or is this just the usual overuse of a single word?

Alternatively, why isn’t this just

Homotopy theory in an (oo,1)-topos?