The n-Category Café

In the first part of his talk Danny Stevenson gave a nice introduction to how the abelian sheaf cohomology and the nonabelian cohomology of a topological space $X$ are all unified into the intrinsic cohomology of the $(\infty,1)$-topos $\mathbf{H} = Sh_{(\infty,1)}(X)$ (or its hypercompletion) over that space, as described at $n$Lab: cohomology.

He has been working for some time now on results that show how this intrinsic cohomology of $\mathbf{H}$ relates to that back in $\infty Grpd \simeq Top$. Not sure if he will make it to that in his talk, but roughly the statement is that for $\mathbf{B}A \in \mathbf{H}$ the $(\infty,1)$-sheaf induced by a suitably well behaved topological $\infty$-group $A$, and for $\mathcal{B}|A| \in Top$ its geometric realization, the corresponding cohomologies computed in $\mathbf{H}$ coincide with those computed in $Top$ in that $H(X,A) := \pi_0 \mathbf{H}(X= *, \mathbf{B}A) \simeq \pi_0 Top(X,\mathcal{B}|A|)$.

(This is not in the talk, though, I see now, hopefully I am allowed to say it here anyways.)

This generalizes the corresponding reesult by Baez-Stevenson and Baas, Kro et al. which we discussed here at some point, and which is the restriction of the above to the case that $A$ is a topological 2-group, in which case the cohomology in question classifies $A$-principal 2-bundles on $X$, or equivalently generalized $A$-gerbes on $X$.

Danny approaches this by representing $A$ as a simplicial topological group and looking at its Moore complex.

From Michael Murray’s talk – Bispaces and Bibundles:

• Bibundles are not a new idea. They certainly goes back to work of Breen on bitorsors in 1990.
• Also discussed by Aschieri, Cantini, and Jurco in 2005.
• However when you look for examples there are not as many of them as there are principal bundles.
• Our aim is to address this existence question.
• It turns out that we need to use crossed modules instead of just Lie groups $G$.
• While my coworkers [David Roberts and Danny Stevenson] are keen crossed module and 2-group people I resisted this at first.
• Let me take you through the reasons for adding this extra complexity.

Hmm, shades of the old Klein 2-geometry there.

Uli Bunke gives a survey of differential cohomology in general and his work with Thomas Schick on models for differential K-theory in particular.

Notice by the way that recently he gave a precise formulation of Killingbacks’ old worldsheet Green-Schwarz anomaly cancellation of the heterotic string using differential K-theory. See $n$Lab: quantum anomaly – spinning particles and super-branes for the reference.

Christian Bär made the following statement in his talk.

Remember the formula for the path integral of the non-relativistic quantum particle coupled to gravity and electromagnetism as given in standard physics textbooks, as a limit over finite-dimensional multiple integrals over spaces of piecewise straight (geodesic) paths supported on points $(\gamma : \{1, 2, \cdots, N\} \to X)$.

For the special simple case of trivial gravitational and electromagnetic background field physics textbooks give this as

And this has a straightforward generalizaton to non-trivial background fields. (I was going to point to $n$Lab: path integral, but this entry is still badly underdeveloped…)

Christian Bär’s statement was (that’s at least how I understood the message of his talk): for imaginary time he has a proof that this limit over $N$ does indeed exist and does indeed yield the expected value.

Happy Birthday, Alan!! It is many years since I have seen Alan, but it is still difficult to believe that he has reached this age.