The n-Category Café

Assume Getler = Getzler ;-)
but any chance of a defintion of a Getzler cat?

Thanks for catching the typo. It wasn’t the only one, but typos in names are (even more) unpleasant.

Yes, as Todd says, by a “Getzler-category of fibrant objects” I here mean his slight variation of the set of axioms of a Kenneth-Brown-category-of-fibrant-objects. As described roughly starting in the section A notion of category of fibrant objects.

Is the terminology (Serre) cofibration standard in some subculture?

Nice notes. I hope John Huerta learns this stuff and applies it to his Lie super-2-algebras and Lie super-3-algebras.

I hope John Huerta learns this stuff and applies it to his Lie super-2-algebras and Lie super-3-algebras.

I talked with John H. about this a bit in Oberwolfach.

On he one hand it is very straightforward to generalize the setup to any notion of geometry, in particular to supergeometry. For instance in the main running example, replace Cartesian probe spaces $\mathbb{R}^n$ by super-Cartesian probe spaces $\mathbb{R}^{p|q}$. This is in fact essentially the same step that gives synthetic differential $\infty$-geometry only that in one case the nilpotent elements of the function rings on test spaces skew-commute, in the other they commute. One can also do both and consider synthetic super $\infty$-geometry and $\infty$-Lie super-groupoids and $\infty$-Lie super-algebroids.

Setting all this up is quite straightforward. Two summers ago while at the Hausdorff center in Bonn, I tried with my office mate back then to warm up for the evident notion of integration of $\infty$-Lie superalgebras to $\infty$-Lie supergroups by getting our hands dirty with working out the basic examples, such as the integration of the super-translation super Lie algebras using the abstract integration procedure. This was supposed to warm us up for the integration of the supergravity super Lie 3-algebra and its cousins. But it turned out that we found the integration of just the simple super-translation super Lie algebras in terms of the abstract procedure quite tedious. (Of course writing out the integration formally is a tautology, what is tedious is cutting it down to something equivalent but recognizable.) Possibly we were being dense, but it caused me to not further follow up on that.

And it also caused me to step back a bit and wonder if this is going in the right direction. With everybody doing derived algebraic geometry, concentrating on supergeometry feels a bit like pre-school: everybody knows how to count over the natural numbers, but you only know 1 and 2. ;-)

It is striking how many aspects of supergeometry that people are fond of are really secretly aspects of derived geometry. In lots of cases the $\mathbb {Z}_2$-grading of supergeometry is naturally refined to an $\mathbb{N}$-grading.

Not in all cases of course. But maybe those where it is not naturally refined this way, maybe it is secretly refined this way?

Be that as it may: doing differential cohomology in a super/derived $(\infty,1)$-topos should be a straightforward application of the general theory, though with some interesting but possibly tedious aspects.