The n-Category Café

There are several ways to say what a principal fiber bundle $P$ over a space $X$ for a structure group $G$ is.

An elegant way it to let $\mathbf{G} = X \times G$ be the trivial bundle of groups, and require that the the morphism

which in terms of ordinary elements reads

be an isomorphism.

When stated this way, this is straightforwardly generalized to the case where $\mathbf{G}$ is more generally a groupoid, with object space $G_0$. We then equip $P$ with a morphism

saying, heuristically, which fibers of $P$ are acted on by morphisms starting at which object in $\mathbf{G}$. The action of the category $\mathbf{G}$ on $P$ is then defined in just the same way as the action of $\mathbf{G}$ on itself (by composition of morphisms) - only that $P$ behaves like consisting of morphisms that just have a target, but no source.

So then a principal groupoid bundle is a bundle $P$ equipped with an action by $\mathbf{G}$ and such that the natural map

is an isomorphism.

What I call a “groupoid bundle” here is what Igor (and many others) call a $\mathbf{G}$-torsor.

It is well known that these $\mathbf{G}$-torsors are classified by nonabelian cocycles

The goal is to categorify everything in sight.

So instead of an ordinary groupoid, we now let $\mathbf{G}$ be a weak 2-category (a bicategory) all of whose morphisms are equivalences: this is a weak 2-groupoid, or bigroupoid.

Everything generalizes accordingly. See this excerpt of Igor’s thesis for some details.

Igor shows how to extract from a local trivialization of a torsor for a bigroupoid a cocycle with values in that bigroupoid. These cocycles are not triangles as above, but 2-commuting tetrahedra labelled in $\mathbf{G}$.

For bigroupoids with just a single object, this problem was also addressed by Toby Bartels.

Igor’s construction works with everything taking place internal to a fixed given topos $\mathbf{E}$. Like for instance that of sheaves over smooth manifolds, for the case of smooth 2-bundles. The admissable notion of “total space” of a 2-bundle (reconstructed from a 2-cocycle, say) depends on the choice of this topos.

There’d be more to say. But so much for now.