The n-Category Café

Here is yet another way to see the same. This is closely analogous the the discussion of (twisted) vector bundles as morphisms into 2-vector transport 2-functors which I gave here.

Let $$GrpSpc$$ be the 2-category whose

- objects are groups

- morphisms $S : G \to H$ are left $G$- right $H$-spaces

- 2-morphisms are maps of $G$-$H$-spaces, commuting with both actions.

This plays the analogous role for principal 1- and 2-bundles as $$\mathrm{Bimodules}$$ does for vector 1- and 2-bundles.

Curvature of a principal 2-bundles with connection is a 2-functor $$\mathrm{curv} : P_2(X) \to \mathrm{GrpSpc}$$ which is such that it trivializes in the sense that there is a morphism $$\mathrm{TRA} : I \to \mathrm{curv}$$ into it.

Here $I : P_2(X) \to \mathrm{GrpSpc}$ is the 2-functor which sends everything to the identity on the trivial group.

The component map of $\mathrm{TRA}$ – a functor on $P_1(X)$ – gives the parallel transport $\mathrm{tra} : P_1(X) \to G\mathrm{Tor}$.

To see this, notice that $\mathrm{curv}$ will send any surface

$$\array{ & \nearrow \searrow^{\gamma} \\ x &\Downarrow^\Sigma& y \\ & \searrow\nearrow_{\gamma^\prime} }$$ to $$\array{ & \nearrow && \searrow^{\mathrm{Aut}(P_x)_{\mathrm{Ad}(\mathrm{tra}(\gamma))}} \\ \mathrm{Aut}(P_x) & & \Downarrow^{\mathrm{tra}(\gamma^{-1}\circ \gamma')}& & \mathrm{Aut}(P_y) \\ & \searrow && \nearrow_{{\mathrm{Aut}(P_x)_{\mathrm{Ad}(\mathrm{tra}(\gamma'))}}} } \,.$$

(Here a group $G$ with an automorphism $\alpha$ in the subscript, $$G_\alpha$$ denotes $G$ as a $G$-$G$ space with the obvious left and the $\alpha$-twisted right action.)

Here $P_x$, $P_y$ are the fibers of a principal $G$-bundle over $X$ and $\mathrm{Aut}(P_x)$ is their automorphism group in the category of $G$-spaces. This is canonically isomorphic to the group $$\mathrm{Ad}P_x := P_x \times_G G$$ which appeared a lot in the above entry.

Notice how the fact that the 2-functor $\mathrm{curv}$ admits a “2-section” $I \to \mathrm{curv}$ forces it to takes values in a sub-2-category of $\mathrm{GrpSpc}$ which in the above entry I denoted $$\mathrm{INN}_{P \times_G P}(\mathrm{Ad}P) \,.$$

This is equivalent to $$\mathrm{INN}(G)$$ and hence, in particular, trivializable.

While $\mathrm{curv}$ is hence trivializable, it is the choice of the trivialization, $\mathrm{TRA}$, which encodes the interesting information:

one and the same curvature 2-functor may be trivialized by several non-equivalent morphisms $\mathrm{TRA}$ and $\mathrm{TRA}'$.

I am emphasizing this point since David Corfield was wondering about this here. The above is one example for the answer which I gave here.