The String Coffee Table

In these notes I describe how (weak) principal $p$-bundles with $p$-connection over categorically trivial base $p$-spaces $M$ (and hence also (weakened) ($p-1$)-bundle-gerbes with connection, curving, etc.) are completely encoded in a single $p$-functor

from the lifted path $p$-groupoid $P_p^C(U)$ (where $U\to M$ is a good cover) to the weak structure $p$-group(oid) $G_p$. This will be called the global holonomy $p$-functor.

It is discussed how all the cocylce relations describing a $p$-bundle with $p$-connection transparently follow from the functoriality of ${\mathrm{hol}}_p$ and how the $p$-gauge transformations of the $p$-bundle with $p$-connection come from natural transformations between different such global $p$-holonomy functors.

This should be called the integral picture and is discussed in section 2. By going to a ‘differential version’ of the ${\mathrm{hol}}_p$-functor one should arrive at something that should be called the differential picture of a $p$-bundle with $p$-connection, where the above $p$­-groupoids are replaced by $p$-algebroids and where the functor between them becomes an algebroid morphism

from some sort of path $p$-algebroid to the $p$-algebroid ${𝔤}_p$ coming from the structure $p$-group(oid) $G_p$.

I do not know yet how to do this differentiation globally (if possible at all), but locally (i.e. on a given patch $U_i$ of the good cover $U$) this should essentially be what Thomas Strobl and collaborators have been studying, motivated by studies of the Poisson $\sigma$-model and other topological field theories.

In section 3 I essentially review aspects this approach, trying to put it in context with the integral picture. I discuss how the consistency conditions known from the integral picture (like the vanishing of the fake curvature) arise in the differential picture and how the notion of gauge transformations (locally) in both pictures coincide.

At the currently, certainly incomplete, level of understanding, one can easily see certain things in one picture but not, or not so easily, in the other. For instance in the intergal picture the global issues related to transitions from one patch to another are very transparent, while they remain to be fully understood in the differential picture. On the other hand, due to the intricacies of weak $p$-group(oid)s it is hard to translate the general diagrams that we discuss into formulas for local data when the structure group is weak and/or really a groupoid. But the analogous generalization, namely from strict $p$-algebras to (not weak but) semistrict $p$-algebroids, is straightforwardly done in the differential picture.