### Principal ∞-Bundles

#### Posted by Urs Schreiber

I have been speaking about *principal ∞-bundles* in an $\infty$-topos for a while now, in print starting around the article

- Hisham Sati, Urs Schreiber, Jim Stasheff,
*Twisted differential String- and Fivebrane structures*.

Last year Thomas Nikolaus and myself visited Danny Stevenson in Glasgow. We labored day and night for two weeks with the aim to compile an article with a comprehensive account of the matter. As it goes with these glorious plans, they don’t fit the intended time schedule, and so we find ourselves still working on the article these days, always almost finished. At least Danny has meanwhile put a major piece of his on the arXiv, which we are making use of:

Danny Stevenson,

*Classifying theory for simplicial parametrized groups*David Roberts, Danny Stevenson,

*Simplicial principal bundles in parametrized spaces*.

See below the fold for an abstract of what we are after now.

While we are not done yet with typing, next week I will speak about the subject (to an audience of topologists). I decided to prepare something like a first-order talk script, possibly usable as a pdf-handout, which gives a concise overview of what it’s all about, what the main theorems are and what the impact on applications is. I still have a bit of time to fine-tune this, but since some feedback can be useful for this, I am hereby posting these notes:

*Principal $\infty$-bundles – Theory and applications*, pdf-handout notes.

I’d be interested in hearing whatever comments you might have.

There is a little table on page 5, which indicates something that I am fond of, and which I have been talking about here on the $n$Café in various guises every now and then. It tabulates examples of $\infty$-bundles of smooth moduli $\infty$-stacks and indicates what happens when you interpret these as universal associated coefficient bundles for nonabelian cohomology. Due to size limitations of a “handout”, this table is a small piece of a more extensive table, which is discussed in section 4.4

- Urs Schreiber,
*differential cohomology in a cohesive topos*.

Along the lines of such a “table of twists” I will probably also give the lectures at ESI in a few weeks.

The theory of G-principal bundles makes sense in any (∞,1)-topos, such as that of topological or of smooth ∞-groupoids. It provides a geometric model for structured higher nonabelian cohomology. For suitable group objects $G$ these $G$-principal ∞-bundles reproduce the theory of ordinary principal bundles, of principal 2-bundles, of gerbes and 2-gerbes, of bundle gerbes and bundle 2-gerbes and generalizes them to higher analogs of arbitrary degree.

We discuss the general abstract theory of principal ∞-bundles, observe that it is induced directly by the ∞-Giraud axioms in an (∞,1)-topos and show the equivalence to the intrinsic nonabelian cocycles, hence their classification by nonabelian cohomology. We discuss that for every extension of ∞-groups there is a corresponding notion of ∞-bundles twisted by a principal ∞-bundle and that these *twisted ∞-bundles* are classified by the corresponding twisted cohomology.

We give explicit presentations of this theory by model structures of simplicial presheaves and notaby by weakly principal bundles in categories of locally fibrant simplicial sheaves. In the smooth context we discuss in some detail presentations by submersive locally fibrant simplicial smooth manifolds: *Lie ∞-groupoids*.

Finally we discuss a wealth of examples and applications.

## Re: Principal ∞-Bundles

This is nice! Let me ask a very simple question from page 2:

What is a point of the site $SmthMfd$? (Why) does it have enough of them?