The n-Category Café

Principal ∞-Bundles – general theory and presentations

Posted by Urs Schreiber

A few weeks back I had mentioned that with Thomas Nikolaus and Danny Stevenson we are busy writing up some notes on bundles in higher geometry. Now we feel that we are closing in on a fairly stable version, and so we thought it may be about time to share what we have and ask for feedback.

As it goes, this project has become a collection of three articles now. They are subdivided as

Principal ∞-bundles

1. General theory (pdf)

2. Presentations (pdf)

3. Applications (not yet out)

The idea is that the first proceeds in full abstraction, using just the axioms of $\infty$-topos theory (so roughly, up to the technical caveats discussed here at length: using just the axioms of homotopy type theory), while the second discusses models of the axioms by presentations in categories of simplicial (pre)sheaves.

More explicitly, in the first one we discuss how, in a given $\infty$-topos, principal $\infty$-bundles are equivalent to cocycles in the $\infty$-topos, how fiber $\infty$-bundles are associated to principal $\infty$-bundles and how their sections are cocycles in twisted cohomology classifying twisted principal $\infty$-bundles and extensions of structure $\infty$-groups. We close by identifying the universal twisting bundles/local coefficient bundles with $\infty$-gerbes and discuss how this reproduces various notions of $n$-gerbes.

In the second one we show how principal $\infty$-bundles are equivalent to cocycles in simplicial hyper-Čech-cohomology, and we prove a strictification result: in a 1-localic $\infty$-topos the space of principal $\infty$-bundles over any $\infty$-stack is modeled by ordinary simplicial bundles with an ordinary action of a simplicial group, the only weakening being that the principality condition holds only up to local weak equivalence. We discuss what this looks like for discrete geometry and for smooth geometry.

For a tad more detail see the abstracts (General Theory abstract, Presentations abstract). And for full details, including references (see page 4 of part 2 for a discussion of the literature) etc. see – of course – the writeups themselves: 1. General theory, 2. Presentations.

All comments would be welcome!