### 2-Equivariance and “Weak Pullback”

#### Posted by urs

Here is math question related to strings on orbifolds which I have just submitted to sci.math.research. Replies of all kinds are very welcome.

Consider equivariant principal bundles with connection in functorial language as follows.

Let

be a principal $G$-bundle over a base manifold $M$.

Let there be a finite group $K$ acting freely (for simplicity) on $M$ by diffeomorphisms.

Let $P(M)$ be the groupoid of thin-homotopy classes of paths in $M$.

Let $\mathrm{Trans}(E)$ be the transport groupoid of $E$ with objects the fibers of $E$ (which are $G$-torsors) and morphisms the torsor morphisms between them.

A connection on $E$ is a smooth functor

The action of a group element $k\in K$ on $M$ gives rise to an obvious functor

The pullback under $k$ of the bundle $E$ with connection $\mathrm{trans}$ has the transport functor

given simply by composing $\mathrm{trans}$ with $k$ .

Next consider the quotient $M/K$ and the projection $\pi :M\to M/K$.

Let there be a bundle $E\prime $ with connection $\mathrm{trans}\prime $ on $M/K$. We can pull it back to $M$ by

The bundle with connection
$\mathrm{trans}=\mathrm{trans}\prime \circ \pi $
is *in*variant under $K$. What would we have to do to get something *equi*variant under $K$ instead?

The answer is the following: instead of pulling back $\mathrm{trans}\prime $ globally, we only assume that $\mathrm{trans}\prime $ is locally naturally isomorphic to $\mathrm{trans}$.

My **question** is (at last): what is the general abstract nonsense notion for this idea of “weak pullback of transport functors”?

I’ll make this more precise. The construction I have in mind is the following.

Given $\mathrm{trans}\prime $ on $M/G$, choose a good covering of $M/G$ by open sets $\{{U}_{i}\}$ together with sections ${s}_{i}:{U}_{i}\to M$ such that

Choose on each ${s}_{i}({U}_{i})\subset M$ a transport functor

such that it is naturally isomorphic to $\mathrm{trans}\prime $ restricted to ${U}_{i}$, with the natural isomorphism called ${L}_{i}$ (“L”ift):

By composing the inverse of ${L}_{i}$ with ${L}_{j}$ over ${U}_{i}\cap {U}_{j}$ we get a natural isomorphism upstairs

By construction, ${L}_{j}\circ ({L}_{i}{)}^{-1}$ is associated to an element $k\in K$.

Suppose it is possible to glue all the ${\mathrm{trans}}_{i}$ such that there is a single trans which restricts to them strictly:

If this $\mathrm{trans}$ exists, it will automatically be $K$-*equi*variant in the
following sense:

a) For each $k\in K$ there is a natural isomorphism

b) Moreover, these natural isomorphisms automatically satisfy a triangle ‘coherence law’

saying that the group product is respected.

I am doing the analogous construction for equivariant 2-bundles with 2-transport in order to describe strings on orbifolds and on orientifolds. It turns out to indeed reproduce known constructions when specialized appropriately. Hence it looks like the ‘right’ thing to do.

But I have the feeling that the concept of ‘weak pullback’ of $p$-fucntors, or whatever it should be called, which is used here, is much more general than the application to equivariant $p$-bundles might suggest. Does it have an established name? Can anyone point me to references where this is discussed more generally?

P.S. In case anyone is wondering: I am aware that in applications one is interested in the case where $K$ does *not* act freely. The point of the above is to derive the right notion of (2-)equivariance from the case where it does act freely and then impose that notion on the non-freely acting setup.

## Re: 2-Equivariance and “Weak Pullback”

Urs

There is a vast literature on weak limits (which is what a pullback is, remember) but the best I can do, I think, is refer you to the original master work

“Formal Category Theory: Adjointness for 2-categories” LNM 391

by John W. Gray. On page 217 he defines Cartesian quasi-limits as follows: Let $$F:C\to \mathrm{Cat}$$ be a 2-functor and let $$P:[1,F]\to C$$ (where the bracket thing means comma category) be the canonical projection. The so-called ‘category of sections’ of $[1,F]$ is the pullback in $2\mathrm{Cat}$ of ${P}^{C}$ and ${1}_{C}$. This defines an object $\Gamma (F)$, which is the said quasi-limit.

Of course one needs to sort out the structure of 2cats and comma cats for this to make sense, which Gray does, so its not a ‘trivial’ categorification.