## November 7, 2005

### 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

(1)$E\to M$

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

Let $\mathrm{Trans}\left(E\right)$ 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

(2)$\mathrm{trans}:P\left(M\right)\to \mathrm{Trans}\left(E\right)\phantom{\rule{thinmathspace}{0ex}}.$

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

(3)$k:P\left(M\right)\to P\left(M\right)\phantom{\rule{thinmathspace}{0ex}}.$

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

(4)$P\left(M\right)\stackrel{{k}^{*}\mathrm{trans}}{\to }\mathrm{Trans}\left(E\right)=P\left(M\right)\stackrel{k}{\to }P\left(M\right)\stackrel{\mathrm{trans}}{\to }\mathrm{Trans}\left(E\right)$

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

(5)$\mathrm{trans}=\mathrm{trans}\prime \circ \pi \phantom{\rule{thinmathspace}{0ex}}.$

The bundle with connection $\mathrm{trans}=\mathrm{trans}\prime \circ \pi$ is invariant under $K$. What would we have to do to get something equivariant 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 $\left\{{U}_{i}\right\}$ together with sections ${s}_{i}:{U}_{i}\to M$ such that

(6)$M={\cup }_{i}\phantom{\rule{thickmathspace}{0ex}}{s}_{i}\left({U}_{i}\right).$

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

(7)${\mathrm{trans}}_{i}:P\left({s}_{i}\left({U}_{i}\right)\right)\to \mathrm{Trans}\left(E\prime \right)$

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

(8)${L}_{i}:\mathrm{trans}\prime \mid {U}_{i}\to {\mathrm{trans}}_{i}.$

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

(9)${L}_{j}\circ \left({L}_{i}{\right)}^{-1}:{\mathrm{trans}}_{i}\to {\mathrm{trans}}_{j}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\left(\mathrm{on}{U}_{\mathrm{ij}}\right)\phantom{\rule{thinmathspace}{0ex}}.$

By construction, ${L}_{j}\circ \left({L}_{i}{\right)}^{-1}$ 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:

(10)$\mathrm{trans}{\mid }_{{s}_{i}\left({U}_{i}\right)}={\mathrm{trans}}_{i}.$

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

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

(11)$\mathrm{trans}\stackrel{{O}_{k}}{\to }{k}^{*}\mathrm{trans}\phantom{\rule{thinmathspace}{0ex}}.$

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

(12)$\stackrel{{O}_{{k}_{1}}}{\to }\stackrel{{O}_{{k}_{2}}}{\to }=\stackrel{{O}_{{k}_{2}{k}_{1}}}{\to }$

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.

Posted at November 7, 2005 5:41 PM UTC

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### 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 $P:\left[1,F\right]\to C$ (where the bracket thing means comma category) be the canonical projection. The so-called ‘category of sections’ of $\left[1,F\right]$ is the pullback in $2\mathrm{Cat}$ of ${P}^{C}$ and ${1}_{C}$. This defines an object $\Gamma \left(F\right)$, 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.

Posted by: Kea on November 7, 2005 7:59 PM | Permalink | Reply to this

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

Oh dear. Let’s not forget

Basic Concepts of Enriched Category Theory, London Math Soc. Lecture Notes 64 (1982) by M. Kelly

which I think is a reprint of the original LNM of the same name, but I’m not sure.

Posted by: Kea on November 7, 2005 8:17 PM | Permalink | Reply to this

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

Thanks a lot. So you say is what I am looking for is called a quasi-limit?

Maybe my use of ‘pullback’ is a red herring, I wonder. Actually in what I wrote I don’t really use a pullback construction explicitly. I note that composition of transport functors with diffeos induces pullbacks on the bundles involved - but implicitly.

Hm, I gotta run now. Have to learn a little about sigma models on stacks before going to bed in order to prepare for Eric Sharpe, who is visiting us tomorrow.

Posted by: Urs on November 7, 2005 8:25 PM | Permalink | Reply to this

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

Actually, Kelly’s book talks about indexed limits, which I think were later called weighted limits. Its one of my dreams to one day understand all these concepts but at the rate I’m going that’s going to take a LONG time!

Posted by: Kea on November 7, 2005 9:24 PM | Permalink | Reply to this

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

A pity we cannot draw diagrams here. Kelly points out (page 119) that Kan extensions could replace indexed limits as the ‘proper’ index notion. My favourite illustration of this is the following: recall the triangle defining transposes and exponential objects in a Cartesian closed category. This diagram involves a total of 3 arrows and 4 objects. Now draw the (co)Kan diagram with 4 arrows and fill the spaces in with 3 2-arrows. This is precisely (if one chooses the functors properly) the categorified triangle!

Posted by: Kea on November 7, 2005 10:17 PM | Permalink | Reply to this

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

Of course I might mention that the word ‘Cartesian’ in the above is like a sledge hammer being used to open a can of worms. Once one has 2-categories floating about and one wishes to consider their products, the universal one is NOT Cartesian, but rather the Gray product.

Posted by: Kea on November 7, 2005 10:59 PM | Permalink | Reply to this

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

Hi Marni,

many thanks indeed for all these comments. I do appreciate it.

I feel, however, a little lost! :-)

Maybe with your assistance I’ll be able to see clearer how what you have in mind applies to my problem.

Maybe I reformulate it as follows, more abstractly.

Say I have a functor $S\stackrel{F}{\to }T$. Also assume I have another functor $S\prime \stackrel{p}{\to }S$.

In the example that I talked about it happened to be the case that the mere composition of these functors

(1)$S\prime \stackrel{p}{\to }S\stackrel{F}{\to }T$

induced a pullback on some of the data defining these functors. This lead me, maybe unwisely, to mention the term pullback, even though there is no pullback cone involved here. Not directly at least.

I was interested furthermore in the situation where there are invertible automorphisms $S\prime \stackrel{k}{\to }S\prime$ acting on $S\prime$ such that

(2)$S\prime \stackrel{k}{\to }S\prime \stackrel{p}{\to }S=S\prime \stackrel{p}{\to }S\phantom{\rule{thinmathspace}{0ex}}.$

This implies that the gadget that I, maybe unwisely, called the pullback of $F$, namely

(3)$S\prime \stackrel{p}{\to }S\stackrel{F}{\to }T$

is also invariant under composition with these automorphisms.

I am interested in something weaker than that. I want to cook up from $F$ a functor $\Phi \left(F\right)$

(4)$S\prime \stackrel{\Phi \left(F\right)}{\to }T$

which is not plain invariant, but something I dared to call equivariant (which is possibly also not the best terminology, even though it amounts, in the special example I considered, to standard terminology.)

Namely I want there to be a natural isomorphism ${O}_{k}$

(5)$\begin{array}{c}S\prime \stackrel{k}{\to }S\prime \stackrel{\Phi \left(F\right)}{\to }T\\ ↓{O}_{k}\\ S\prime \stackrel{\Phi \left(F\right)}{\to }T\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

I indicated a way how to construct such a $\Phi \left(F\right)$ in the special example that I am interested in. What I am trying to find out is if that construction is known in more general terms.

Right now I cannot say if what you hinted at is indeed secretly an approach to an answer to that question. If it is, you need to help me see how! :-)

Posted by: Urs on November 8, 2005 9:38 PM | Permalink | Reply to this

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

Hi Urs

Sorry. I guess I got a little carried away with the weak pullback idea. I’m just hoping that you’ll start putting all that creative energy into true tricategorical constructions rather than String specific ones.

Anyway, putting side the equivariance question for a second: assuming you are really dealing with pseudofunctors then the diagram for ${O}_{k}$ lives in 2cat proper and is thus a pseudonatural transformation, which is automatically invertible.

A definition of 2-equivariance? This seems like a good question for a mathematician! But as far as I can tell it MUST involve limits. The usual EGxM/G (homotopically same as M/G for free action) is defined using pullbacks. I googled on this and did actually come up with some stuff about G-sheaves and Kan extensions but I’m afraid it all looked rather mysterious to me. I’ll keep thinking about it.

Posted by: Kea on November 8, 2005 11:29 PM | Permalink | Reply to this

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

assuming you are really dealing with pseudofunctors then the diagram for ${O}_{k}$ lives in 2cat

Wait, in what I wrote about equivariance of bundles, the ${O}_{k}$ are natural transformations between parallel transport 1-functors. So the triangle diagram satisfied by them lives in the functor category, which, for the case I discussed, is a 1-category.

But what I am of course really interested in is the categorification of that. This yields transport 2-functors and the ${O}_{k}$ are then pseudonatural transformations between these satisfying a ‘solid’ triangle in the corresponding 2-functor 2-category. The triangle is filled with a modification of pseudonat. transformations, which itself then makes a tetrahedron diagram 2-commute.

which is automatically invertible

True. In fact all natural or pseudonatural transformations or modifications thereof in my setup are invertible, since the transport (1-, 2-) functors they act between all take values in groupoids.

A definition of 2-equivariance? This seems like a good question for a mathematician!

Yes, maybe. With all due modesty I am claiming to have a good notion of 2-equivariance. It is ‘good’ in the sense that it is the right thing to handle gerbes with connection and curving on orbifolds and orientifolds. It’s the rather obvious generalization of the equivariance condition in the way I stated it in the above entry for bundles.

With that 2-equivariance definition in hand, I was now wondering if maybe I was reinventing the wheel. If maybe this is just a special case of some well known general abstract nonsense.

The usual $EG×M/G$ (homotopically same as $M/G$ for free action) is defined using pullbacks.

Hm, I am not sure I see what you are talking about. I assume $EG$ is supposed to denote the universal $G$-bundle. Now what precisely is defined using pullbacks? Sorry.

Posted by: Urs on November 9, 2005 7:55 PM | Permalink | Reply to this

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

You can find the Kelly’s Enriched book
which you mention from
London Math Soc Lec Note Series
retyped recently at the tac archive
as the link number 8 on the page

http://www.tac.mta.ca/tac/reprints/index.html

Posted by: Zoran Skoda on November 24, 2005 9:44 PM | Permalink | Reply to this

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