December 8, 2006

Local Transition of Transport, Anafunctors and Descent of n-Functors

Posted by Urs Schreiber

In discussion here we discovered that we need to harmonize our use of the terms in the title of this post. Here is my take at what is going on. Please feel free to add your comment and correct me where I need correction.

Notation is an issue here. I will take the liberty of using my own ideosyncratic notation in the following, the one I have been using all along here in discussions.

So, in my notation, the issue is the following:

we have a domain $n$-category

(1)$P_n(X)$

and a codomain

(2)$T$

and we want to talk about $n$-functors

(3)$\mathrm{tra} : P_n(X) \to T$

without actually doing so. Instead, we want to assume we can handle $\mathrm{tra}$ “locally”. This means we assume a morphism

(4)$p : P_n(U) \to P_n(X)$

and assume we know what $\mathrm{tra}$ pulled back along $p$ is.

Or not even quite that. But consider this for a moment, and I will shortly discuss how to forget $\mathrm{tra}$ alltogether, without actually losing information.

So, in general, the thing we have now locally, on $P_n(U)$, might not even take values in $T$ anymore. Instead, there might be a morphism

(5)$i : T' \to T$

of codomains and our local functor takes values in $T'$:

(6)$\mathrm{tra}_U : P_n(U) \to T' \,.$

If you like, you can consider $T' = T$ and $i = \mathrm{Id}_T$ in all of the following. But I claim that you will want to allow non-identity $i$ eventually.

Okay, so saying that $\mathrm{tra}$ locally looks like something taking values in $T'$ is saying that there is an equivalence

(7)$\array{ P_n(U) &\stackrel{p}{\to}& P_n(X) \\ \mathrm{tra}_U\downarrow\;\; &\sim \Downarrow t& \;\; \downarrow \mathrm{tra} \\ T' &\stackrel{i}{\to}& T } \,.$

This equivalence is a local trivialization of our $n$-functor. More precisely, in order to emphasize the assumptions that went into thi, I say it is a $p$-local $i$-trivialization.

(And I am claiming that, more generally, we may want to weaken the equivalence $t$ to a mere special ambidextrous adjunction. All we ever do in this setup can be done with just assuming special ambijunctions here. I talk about applications where this relaxed notion of local trivializaiton is important in On 2d QFT: From Arrows to Disks.)

Okay fine. But we may find ourselves in a situation where the globally defined $\mathrm{tra}$ is just not available as a concrete object, but where we sort of reconstruct it from just knowing its local version $\mathrm{tra}_U$ together with some data encoded in the trivialization morphism $t$.

As I discuss in detail in Transport, Trivialization, Transition one can show that given a $p$-local $i$-trivialization as above one can derive the following data.

Let $(P_n(U))^{[n]}$ be the $(n-1)$-fold pullback of $P_n(U)$ along itself. If $U \to X$ is a cover of $X$, this simply amounts to going to $(n-1)$-fold intersections of patches.

The data we get is this:

On double intersections an equivalence

(8)$g : p_1^*\mathrm{tra}_U \to p_2^*\mathrm{tra}_U \,.$

on triple intersections an equivalence

(9)$f : p_{23}^* g \circ p_{12}^* g \to p_{13}^*$

(10)$\array{ p_2^* \mathrm{tra}_U &\stackrel{p_{23}^* g}{\to}& p_3^*\mathrm{tra}_U \\ p_{12}^*g \uparrow\;\;\; &\nearrow& \;\; \downarrow p_{34}^* g \\ p_1^*\mathrm{tra}_U &\stackrel{p_{14}^*g}{\to}& p_4^* \mathrm{tra}_U } \;\;\; \simeq \;\;\; \array{ p_2^* \mathrm{tra}_U &\stackrel{p_{23}^* g}{\to}& p_3^*\mathrm{tra}_U \\ p_{12}^*g \uparrow\;\;\; &\searrow& \;\; \downarrow p_{34}^* g \\ p_1^*\mathrm{tra}_U &\stackrel{p_{14}^*g}{\to}& p_4^* \mathrm{tra}_U }$

and so on.

(Here in the last diagram the triangles are supposed to be filled by the corresponding $f$s.)

This, I call the transition data of my original $n$-functor $\mathrm{tra}$. Ross Street would call it the descent data, I think. Compare pages 2 and 3 of

Ross Street
Descent Theory

with definition 5 in TraTriTra, if you like.

Notice how the globally defined $\mathrm{tra}$ does not appear anywhere anymore. Only the local $\mathrm{tra}_U$ does and lots of morphisms between its pullbacks, all of which can be obtained from the local trivialization $t$, if one was given.

But, and that’s the point, even if no $t$ was given to begin with, and no globally defined $\mathrm{tra}$, we can consider transition data as above. In fact, and that’s why all this is a good idea, under suitable circumstances we may reconstruct from the transition data the globally defined $\mathrm{tra}$.

If so, we have demonstrated that our $p$-locally $i$-trivializable $n$-functors form an $n$-stack. An $n$-stack of $Q$s on $X$ means, in words, that $Q$s on $X$ are the same as glued $Q$s on $U$, where $U \to X$ is a cover.

More technically, we have an $n$-category of descent data

- its objects are $n$-simplices colored with transition morphisms as indicated above

- its morphisms are the “obvious” morphisms of $n$-simplices. There are a couple of ways to see the same obvious structure here. For $n=2$ I spell it out in TraTriTra. And it’s the same as what Ross Street uses in his theory of higher descent.

Before listing a couple of examples, I will now try to say what an anafunctor is and how the above, for $n=1$ relates to anafunctors.

Anafunctors were defined in

M. Makkai
Avoiding the axiom of choice in general category theory
Journal of Pure and Applied Algebra, Volume 108, Number 2, 22 April 1996, pp. 109-173(65)
.ps from M. Makkai’s site
anafun1.pdf (title and contents)
anafun2.pdf (introduction)
anafun3.pdf (main text)

Definition(M. Makkai) For $A$ and $B$ two categories, an anafunctor

(11)$F : A \to B$

from $A$ to $B$ is a span

(12)$\array{ |F| &\stackrel{F_1}{\to}& B \\ F_0 \downarrow \;\; \\ A }$

with $F_0$ surjective on objects and morphisms and such that every morphism in $A$ has at most one pre-image with given source and target object.

So, in words, instead of going directly from $A$ to $B$ we wirst kind of resolve $A$ in terms of $|F|$ and then go from $|F|$ to $B$.

I will now try to indicate how from any object in the category of $p$-local $i$-transition data for a 1-functor we obtain an anafunctor in a canonical fashion.

In fact, I did already discuss this in On $n$-transport: Universal Transition, albeit for the case $n=2$ (which of course includes the case $n=1$ we are restricting to right now.)

Namely, from a given cover

(13)$p : P_1(U) \to P_1(X)$

we can form the category

(14)$P_1(U^\bullet)$

of paths in the transition groupoid. Its morphism are generated from those in $P_1(U)$ together with a unique morphism between any two objects in $P_1(U)$ with the same projection, divided out by an obvious equivalence relation.

This comes equipped canonically with a projection

(15)$\array{ P_1(U^\bullet) \\ \downarrow \\ P_1(X) }$

and, if you think about it, the relations mentioned above are precisely such that they ensure that every morphism in $P_1(X)$ has a unique lift with specified endpoints.

(I think I now understand that this is exactly what the pullback diagram (109) in

Toby Bartels
Higher gauge theory I: 2-Bundles
math.CT/0410328

achieves, too.)

Then, any $p$-local transition data (descent data) $(\mathrm{tra}_U,g)$ canonically defines a functor

(16)$(\mathrm{tra}_U,g) : P_1(U^\bullet) \to T \,.$

In conclusion, from a given $p$-local transition data (descent data) we construct the anafunctor

(17)$(\mathrm{tra}_U,g) : P_1(X) \to T$

given by the span

(18)$\array{ P_1(U^\bullet) &\stackrel{(\mathrm{tra}_U,g)}{\to}& T \\ \downarrow \\ P_1(X) } \,.$

I’d think that, conversely, every anafunctor can be interepreted as an object in the category of $p$-local transition data of a 1-functor for given $p$.

Examples:

The most natural examples are obtained by letting $X$ be a smooth space, letting $U \to X$ be a surjective submersion and letting $P_n(X)$ be some notion of smooth $n$-paths in $X$, letting $p : P_n(U) \to P_n(X)$ be the obvious induced morphisms on $n$-paths and letting everything in sight be smooth.

Then all that remains to vary is the morphism $i : T' \to T$. Just choosing different such $i$ we re-obtained an entire zoo of well-known structures.

For $G$ any ordinary Lie group and for

(19)$i = \mathrm{Id}_{\Sigma(G)}$

the category of $p$-local $i$-transition data is canonically isomorphic to the category of $G$-cocycles describing locally trivialized $G$-bundles with connection on $X$. That’s an easy exercise.

For $G_2$ a strict Lie-2-group and for

(20)$i = \mathrm{Id}_{\Sigma(G_2)}$

the category of $p$-local $i$-transition data is canonically isomorphic to the category of $G_2$-cocycles describing locally trivialized $G_2$-2-bundles with “fake flat” connection. This is the example discussed in John’s paper with me #. A greatly simplified proof is given here.

There is a rather obvious way generalizing this from a structure 2-group to a structure 2-groupoid. The resulting cocycle data is that discussed by Igor Bakovic # (he does not discuss connections, though).

The fake-flatness constraint is lifted as follows:

For $G_2$ a strict Lie-2-group and $\mathrm{INN}(G_2)$ its 3-group of inner automorphisms and for

(21)$i = \mathrm{Id}_{\Sigma(\mathrm{INN}(G_2))}$

the category of $p$-local $i$-transition data is canonically isomorphic to the category of $G_2$-cocycles describing locally trivialied $G_2$-2-bundles with arbitrary connection, reproducing the data found by Breen-Messing. The proof is given here.

Just a special case of this, but worth mentioning is this:

For $G_n = \Sigma^n(U(1))$ the $n$-fold suspension of $U(1)$ and

(22)$i = \mathrm{Id}_{\Sigma^n(U(1))}$

the category of $p$-local $i$-transition data is canonically isomorphic to the category of Deligne cocycles. I prove this for up to $n=2$ here and I claim that it is clear that the statement holds for all $n$ (but I haven’t written down a proof for that).

In all these examples we had $i = \mathrm{Id}$ the identity. So all these examples should correspond to anafunctors.

But, I claim, for $n \gt 1$ it is useful to have nontrivial $i$. Identity $i$ will always give us cocycles for a “full local trivialization”. But often we want to do local pre-trivializations.

For instance: locally trivializing a gerbe yields as transition bundle also known as a bundle gerbe. Only if we, in turn, also locally trivialize this transition bundle do we get back to the Deligne cocycles mentioned above.

So, for

(23)$i : \Sigma(\Sigma(\mathbb{C}^\times)) \stackrel{\subset}{\to} \Sigma(1d\mathrm{Vect}_\mathbb{C})$

the canonical inclusion, the 2-category of $p$-local $i$-transition data is canonically isomorphic to the 2-category of line bundle gerbes with connection. The proof is here.

In fact, we should be thinking in terms of the chain of inclusions

(24)$\Sigma(\Sigma(\mathbb{C}^\times)) \stackrel{\subset}{\to} \Sigma(1d\mathrm{Vect}_\mathbb{C}) \stackrel{\subset}{\to} \mathrm{Bim}(\mathrm{Vect}_\mathbb{C})$

which manifestly makes line bundle gerbes the transition data of rank-1 line-2-bundles. This aspect is discussed here.

This has an obvious generalization. For $H$ any ordinary group, for $H\mathrm{BiTor}$ the monoidal category of $H$-bitorsors, and for the canonical inclusion

(25)$i : \Sigma(\mathrm{AUT}(H)) \stackrel{\subset}{\to} \Sigma(H\mathrm{BiTor})$

the 2-category of $p$-local $i$-transition data is canonically isomorphic to the 2-category of principal (nonabelian ) bibundle gerbes with connection as defined (on objects) by Aschieri-Jurčo. Again, the connection is “fake flat” as long as we work just with 2-groups. The proof is here.

Generally, if we take $i$ to be a representation of an $n$-group

(26)$i : \Sigma(G_n) \to n\mathrm{Vect}$

we get the local data of an associated $n$-bundle.

The discussion of line bundle gerbes above can be understood as an example for that for the canonical representation of $\Sigma(U(1))$ on $\mathrm{Bim}(\mathrm{Vect}) \subset 2\mathrm{Vect}$. This is explained here.

For 3-bundles and 2-gerbes one would consider even longer chains of inclusions of codomains, corresponding to more steps of local trivializations: a 2-gerbe has a transition 1-gerbe, which has a transition bundle, which has a transition function.

A chain of inclusions for instance relevant for 1-dimensional vector 3-bundles I have discussed here.

All these examples assume that the transition morphisms are equivalences. If we instead consider transition data (anafunctors) and require the transitions to be just special ambidextrous adjunctions, we pass from the world of classical transport ($n$-bundles with connection) to that of quantum transport (evolution/propagation in $n$-dimensional quantum field theory).

For instance, the inclusion

(27)$i = \mathrm{Id}_{\Sigma(\mathrm{Vect})}$

and demanding transitions to be just special ambijunctions yields the local data that is the Fukuma-Hosono-Kawai state sum model of 2-dimensional topological field theory. This I discussed here.

2-dimensional rational conformal field theory is, essentially, just topological field theory internalized in a modular tensor category $C$ more sophisticated than $\mathrm{Vect}$. Accordingly, transition data with respect to

(28)$i : \Sigma(C) \stackrel{\subset}{\to} \mathrm{Bim}(C) \stackrel{\subset}{\to} \mathrm{TwBim}(C)$

can be understood as underlying the decoration prescription (“state sum model”) used in the FRS description of rational conformal field theory. This I talk about here and here.

So much for now. I’d be content with addressing all these examples of descent data for $n$-functors as $n$-anafunctors. Maybe “generalized” $n$-anafunctors. Please let me know if that would be reasonable use of terminology.

Further notes on this topic, discussed in the comment section below, are these:

A note on the peculiarity encountered in composing morphisms of anafunctors. Compare with Toby’s remarks below.

A note on how anafunctors and functors with transition data are equivalent, followed by a proposal for defining 2-anafunctors in terms of transition data of 2-functors. Essentially a reformulation of what I had written about transitions and the universal transition before, now with an eye on the language of anafunctors.

Posted at December 8, 2006 5:45 AM UTC

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Re: Local Transition of Transport, Anafunctors and Descent of n-Functors

In his article on anafunctors, M. Makkai presents almost everything in terms of two equivalent definitions. The one exception is the composition of morphisms of anafunctors, which is not presented in the otherwise more elegant definition in terms of spans.

I would like to understand composition of their morphisms in the more elegant span form.

I understand that Toby Bartels writes down exactly that, even internally, in his thesis. But for my own benefit, I want to see the relevant structure stripped off the complexity introduced by writing down everything internalized.

In the little time I have left today, I came up with this.

I’d be grateful if Toby, or anyone else, could quickly check if the composition of morphisms of anafunctors that I write down is correct and coincides with the standard definition.

Thanks!

Posted by: urs on December 8, 2006 4:52 PM | Permalink | Reply to this

Re: Local Transition of Transport, Anafunctors and Descent of n-Functors

I’m missing something here - probably an unspoken assumption. In the example $A=P_1(X)$ we have a functor with values
in Cat. Definition 3 seems to depend crucially on X and sigma seems to be a
________ from X to |G|.

Posted by: jim stasheff on December 9, 2006 3:01 AM | Permalink | Reply to this

Re: Local Transition of Transport, Anafunctors and Descent of n-Functors

I’m missing something here […]

Oops, thanks. I wrote $X$ where it should have written $A$.

Posted by: urs on December 9, 2006 12:54 PM | Permalink | Reply to this

Re: Local Transition of Transport, Anafunctors and Descent of n-Functors

Yeah, that’s right. (There’s a typo at the end of the linear diagram on the last page, but that’s obvious.)

Notice that neither I nor Makkai would allow ourselves to discuss σ (hence nor t). This is because you need the Axiom of Choice to prove that a (strict) functor σ exists, and even then it won’t be smooth. (Or to put it another way, it exists in the strict 2category of 2spaces and 2functors only if the axiom of choice holds in the category of spaces.) That may be one reason why Makkai doesn’t include this version of the definition.

Now, we can include everything in the diagram at the top of the last page except for the map t. Then we could argue that, because of the properties of the functor from |F| ×X |G| ×X |H| to |F| ×X |H| is a surjection (or more generally cover) on objects and morphisms, there is still a natural transformation between the functors that we want (which, as you point out, no longer have |G| in them). Indeed, this is precisely what the sentence directly after diagram (118) in my thesis (version 3) is about.

I can’t read Makkai right now, since I don’t have a PostScript reader on this machine (and its hard drive is full to the bursting point), so I can’t give an intelligent reason as to why Makkai didn’t make the same argument. But I’m sure that he knew how to do it.

Posted by: Toby Bartels on December 10, 2006 1:24 AM | Permalink | Reply to this

morphisms of anafunctors

Many thanks indeed to Jim Stasheff and Toby Bartels for pointing out typos! I think I have corrected them now, the corrected file is here.

Toby wrote:

Notice that neither I nor Makkai would allow ourselves to discuss $\sigma$ (hence nor $t$). This is because you need the Axiom of Choice […]

That may be one reason why Makkai doesn’t include this version of the definition.

Okay, I understand that. Or at least the idea. Since the diagrams I drew were supposed to be just a translation of what Makkai does on p.15 , when he says

[…] with any $t \in |G|X;$

I am not sure how choosing that $t$ does not involve the axiom of choice. I guess the answer is implicit somewhere in your (Toby’s) diagram (118). Indeed, as you write

Indeed, this is precisely what the sentence directly after diagram (118) in my thesis (version 3) is about.

Posted by: urs on December 11, 2006 10:54 AM | Permalink | Reply to this

Re: morphisms of anafunctors

I am not sure how choosing that t does not involve the axiom of choice.

But t is just a single element of a set (known to be occupied). You don’t need choice to choose a singleton!

The diagram in 3.(iii) is making the same point (in its own way) as your diagram in Morphisms of Anafunctors. But it’s doing so point by point, with no need for a global choice function.

Here is the general situation: Let A, B, and C be sets, let f be a surjection from A to B, and let g be a function from A to C. Suppose that one wants to define a function h from B to C such that the triangle commutes. One might (using Choice) pick a section f′ of f, then let h be the composite of f′ and g. Under certain circumstances, one can prove that this composite is independent of the section chosen. This is what you do.

Alternatively, one might look at each element x of B and consider how g(y) varies as y varies over the fibre of f over x. Under certain circumstances, one can prove g(y) is independent of y, so long as f(y) is fixed. Then one may define h(x) to be g(y) for any y over x. This is what Makkai does; it does not require Choice, but it makes explicit reference to elements.

Finally, one can consider the kernel of f, which is a subset of the Cartesian square of A. (Membership-theoretically, it consists of pairs (y, z) such that f(y) = f(z). But it is defined by a universal property; category theorists call its inclusion map ‘kernel pair’.) There are two projection maps from this kernel to A, and under certain circumstances, one can prove that their composites with g are equal. In that case, it is a theorem of set theory that a unique function h exists as desired. And this theorem does not require Choice; in fact, it holds in any category, as long as f is a regular epimorphism and has a kernel pair. This is what I do.

Posted by: Toby Bartels on December 11, 2006 10:30 PM | Permalink | Reply to this

Re: morphisms of anafunctors

In that case, it is a theorem of set theory that a unique function h exists as desired.

That’s very good. Have you ever thought about lifting the corresponding construction in your thesis from 1- to 2-categories?

Posted by: urs on December 12, 2006 10:15 AM | Permalink | Reply to this

Re: Local Transition of Transport, Anafunctors and Descent of n-Functors

This looks reasonable. As far as checking for errors, I’ve only looked at the references to anafunctors; but on the level of rough intuition, I like all of it.

I do want to stress one point, however: Anafunctors should be seen as the standard notion of morphism between categories (or at least internal categories). You can get away with using (strict) functors only if the Axiom of Choice holds in your context category. (In particular, Mac Lane and Eilenberg were able to get away with this in their original definition of functor, because they used Choice in the category of sets.) Thus Makkai wrote his paper originally because he wanted to allow for the possibility that Choice fails in the category of sets, and I used anafunctors because I was working in the category of smooth manifolds, where Choice definitely fails. (The precise statement of Choice for this purpose, incidentally, is that every cover has a section; you need to equip the category with a notion of what a cover is to even discuss what an anafunctor is. In the category of sets, a cover is a surjection; in the category of smooth manifolds, it’s probably best to let a cover be a surjective submersion, although my thesis required a surjective local diffeomorphism, which fortunately yields an equivalent 2category.)

So, when considering whether some descent data define a morphism between categories, it should be enough that they define an anafunctor.

Posted by: Toby Bartels on December 10, 2006 1:41 AM | Permalink | Reply to this

Re: Local Transition of Transport, Anafunctors and Descent of n-Functors

By the way, if you’re interested in Makkai’s notion of saturated anafunctor, I’ve fairly well convinced myself that these are (in the contenxt of Lie groupoids) Hilsum–Skandalis morphisms, which are fairly well established. So the Lie groupoid theorists are probably doing things correctly.

Posted by: Toby Bartels on December 10, 2006 1:49 AM | Permalink | Reply to this

Re: Local Transition of Transport, Anafunctors and Descent of n-Functors

I’d be interested in any *conceptual* comments on span’ in anafunctor
in comparison to roof’ as used in e.g. inverting equivalences in alg top/
homotopy theory.

Posted by: jim stasheff on December 10, 2006 7:25 PM | Permalink | Reply to this

Re: Local Transition of Transport, Anafunctors and Descent of n-Functors

I’d be interested in any conceptual comments on ‘span’ in anafunctor in comparison to ‘roof’ as used in e.g. inverting equivalences in alg top/ homotopy theory.

As far as I know, ‘span’ is just a general term in (elementary) category theory for a pair of morphisms with common domain. I don’t know what ‘roof’ means.

I would argue that the proper notion of morphism between spaces considered up to homotopy is not a (continuous) map, nor a homotopy-equivalence class of such (a useless step anyway if you’re going to use ωgroupoids), but a span whose first leg induces isomorphisms on all homotopy groups. Is this relevant?

Posted by: Toby Bartels on December 11, 2006 3:35 AM | Permalink | Reply to this

spans and roofs

I’d be interested in any conceptual comments on ‘span’ in anafunctor in comparison to ‘roof’ as used in e.g. inverting equivalences in alg top/ homotopy theory.

As far as I know, ‘span’ is just a general term in (elementary) category theory for a pair of morphisms with common domain. I don’t know what ‘roof’ means.

I guess Jim Stasheff is referring to the use of “roof” as common in the theory of localization (many-object version of localization of a ring).

See for instance p. 19-20 of this.

So, a “roof” is a “span” in a certain context.

Posted by: urs on December 11, 2006 1:19 PM | Permalink | Reply to this

2-anafunctors?

So is there any literature on 2-anafunctors?

Given all that I wrote above about how anafunctors are like descent data of functors I would think I know what an 2-anafunctor should be. But does anyone talk about 2-anafunctors?

One reason why I was interested in the composition of morphisms of anafunctors that we dicussed above is that for 2-anafunctors the analog problem is an old one in the theory of bundle gerbes:

What is a morphism between bundle gerbes that live on different surjective submersions $Y$?

There are proposed solutions to that, as in Danny Stevenson’s thesis, but it would be nice to have a general understanding.

Namely, as I have tried to indicate, a bundle gerbe is nothing but what should be called a 2-anafunctor. Morphisms of bundle gerbes correspond to (pseudo-)natural transformations of these 2-anafunctors, and the subtlety concerning morphsims of gerbes on different $Y$ correspond precisely to the subtlety we discussed above.

Posted by: urs on December 11, 2006 11:09 AM | Permalink | Reply to this

Re: 2-anafunctors?

[A] bundle gerbe is nothing but what should be called a 2-anafunctor.

Under certain circumstances, a bundle is nothing what should be called an anafunctor. Specifically, a principal Gbundle over B is nothing but what should be called an anafunctor from the (categorially) discrete groupoid of B to the one-object groupoid G. More generally, an anafunctor (at least is properly saturated?) from H to G defines an Hprincipal (H,G)bibundle (that is a Hilsum–Skandalis morphism).

Since a bundle gerbe is nothing but what should be called a 2bundle, you seem to be right on track!

Posted by: Toby Bartels on December 11, 2006 10:38 PM | Permalink | Reply to this

Re: 2-anafunctors?

[A] bundle gerbe is nothing but what should be called a 2-anafunctor.

Under certain circumstances, a bundle is nothing what should be called an anafunctor.

There is a slight difference in these two statements. If by “is” we mean anything stronger than “is equivalent to” then we should say that

an anafunctor (with values in a group) is the transition function of a principal bundle (possibly with connection).

Same for bundle gerbes. A bundle gerbe is not a 2-bundle (at least not on base space) - it is the “transition 2-function” of a 2-bundle.

Sorry for being nitpicky about this, but this is essentially the point which got our discussion here started.

The category of anafunctors from $P_1(X)$ to $\Sigma(G)$ is canonically isomorphic to the category of “differential $G$-1-cocycles” on $X$. And it is (only, and non-canonically) equivalent to the category of $G$-bundles (in the sense of total spaces $B$ with projection $B \to X$, etc.) with connection on $X$.

Of course in the end we don’t need to care about anything beyond equivalence, and probably that’s what you are doing already. But I think for the moment, where we are concerned with the peculiarities of the details of anafunctors, it pays to point out that some equivalences in the game here are actually isomorphisms, while others are not.

More generally, an anafunctor (at least is properly saturated?) from $H$ to $G$ defines an $H$principal $(H,G)$bibundle (that is a Hilsum-Skandalis morphism).

Interesting! Let me see if I understand that:

To me, the cocylce of an $(H,G)$-bibundle on $X$ for $H$ and $G$ groups with $G$ acting on $H$, forming a crossed module of groups, is an anfunctor

(1)$g : \mathrm{Disc}(X) \to (H \to G)$

from the discrete category of $X$ to $(H \to G)$, where $(H\to G)$ denotes the 2-group corresponding to the above crossed module, regarded here as a mere groupoid (i.e. not using the monoidal structure on that).

In other words, this is the cocycle of an $H$-bundle with the special property that its image under the map $t : H \to G$ is a trivial $G$-cocycle (with fixed trivialization).

Is this roughly what you have in mind when you say “$H$-principal $(H,G)$-bibundle”?

a Hilsum-Skandalis morphism

I had to look this up. Found one definition in Henriques & Metzelder: Presentation of noneffective orbifolds on p. 7:

Definition 2.6. Let $G$ and $H$ be smooth groupoids. A (Hilsum-Skandalis) morphism from $G$ to $H$ consists of a manifold $P$, maps $s_P : P \to G_0$, $t_P : P \to H_0$, a left action of $G$ on $P$ with the base map $s_P$ , a right action of $H$ on $P$ with base map $t_P$ , such that:

(1) $s_P$ is $H$-invariant, $t_P$ is $G$-invariant;

(2) the actions of $G$ and $H$ on $P$ are compatible: given $p \in P$, $(gp)h = g(ph)$;

(3) $s_P : P \to G_0$, as an $H$-bundle with base map $t_P$ , is principal. We will often abuse notation and denote such a morphism simply by $P$.

Posted by: urs on December 12, 2006 10:02 AM | Permalink | Reply to this

Re: 2-anafunctors?

Henriques & Metzelder

(Notice that (1) and (2) make it a bibundle, while (3) makes it Hprincipal.)

Posted by: Toby Bartels on December 14, 2006 1:51 AM | Permalink | Reply to this

Re: 2-anafunctors?

Henriques & Metzelder

(Notice that (1) and (2) make it a bibundle, while (3) makes it $H$principal.)

Okay. So morally, a Hilsum-Skandalis morphism is to groupoids like a bimodule is to algebras. It is a groupoid bi-torsor.

The only subtlety is that where for a bimodule we just demand compatible left and right action, for a bitorsor we in addition want to demand one of the two actions to be principal. (They may both happen to be principal if $H = G$, but need not.)

So then there should be a more or less obvious composition of HS-morphisms by a coequalizer construction, and a 2-morphism between two parallel HS morphisms should be any groupoid bundle homomorphism compatible with the left and right actions.

Very nice. This is now getting close to a question I meant to ask: what is the relation between anafunctors and profunctors?

In his introduction Makkai mentions that anafunctor alludes to profunctor by the analogy with mitosis, where we have, among others, an anaphase and prophase.

I haven’t really understood this analogy at all so far!

But given that you claim that a saturated anafunctor is, in the context of Lie groups, a Hilsum-Skandalis morphism, and that it seems as if such a morphism is nothing but a bitorsor, hence something like a functor on $H \times G^\mathrm{op}$, it’s now beginning to look as if

saturated anafunctors are profunctors

Posted by: urs on December 14, 2006 11:21 AM | Permalink | Reply to this

Re: 2-anafunctors?

But given that you claim that a saturated anafunctor is, in the context of Lie groups, a Hilsum-Skandalis morphism.

I’m not sure that this is really true. But something like it is.

Posted by: Toby Bartels on December 14, 2006 11:22 PM | Permalink | Reply to this

Re: 2-anafunctors?

what is the relation between anafunctors and profunctors?

On the one hand, only saturated anafunctors can be described this way as profunctors. On the other hand, only representable profunctors define anafunctors. However, up to equivalence, it’s fair to say that anafunctors are the same as representable profunctors. (That’s because every anafunctor is naturally ismorphic to a saturated one, and this respects composition, given an equivalence of bicategories.)

The idea that H-S morphisms correspond to saturated anafunctors is therefore akin to the idea that representability of a functor corresponds to principality of a bundle. I’m not sure if that’s true, but something like it should be.

I haven’t really understood this analogy at all so far!

As far as I can tell, it’s not meant to be anything more precise than that “pro” and “ana” can appear together to describe different variations on the same theme.

Posted by: Toby Bartels on December 14, 2006 11:39 PM | Permalink | Reply to this

Re: 2-anafunctors?

I have a proposal for what a $n$-anafunctor should be. Explicit for $n=2$ with a rather straightforward generalization to higher $n$.

My proposal is that an $n$-anafunctor is the unique morphism obtained from an $n$-functor with transition data under the construciton of what I call the “universal transition” # – a certain pushout.

For this to make sense, it should reduce to Makkai’s definition of anafunctors for $n=1$. And it does. I have written up a short note on that here:

Comments are very welcome. In essence, I am just restating what I wrote elsewhere concerning “transition of $n$-transport”, but now trying to make direct contact with the anafunctor language.

Posted by: urs on December 12, 2006 2:40 PM | Permalink | Reply to this

Re: 2-anafunctors?

On Anafunctors and Transitions.

Posted by: Toby Bartels on December 14, 2006 1:52 AM | Permalink | Reply to this

Re: 2-anafunctors?

On Anafunctors and Transitions.

I have now corrected some funny blunders in labelling some of the diagrams. Sorry for that. But I guess you got the idea nevertheless.

Posted by: urs on December 14, 2006 2:12 PM | Permalink | Reply to this

forgetting structure, remembering choice

I hate myself for writing the fourth comment in a row here - but remember that it could be worse: I could be writing a new entry instead! ;-)

Anyway: now that we have made some progress in understanding each other’s terminology, I would like to hear in more detail Toby’s and/or John’s opinion on the way I did approach this issue.

It seems that I did implicitly use anafunctors all along, addressing them (originally, in my thesis) as functors on the “Čech path groupoid” and (later) as functors on paths in the transition groupoid of a cover.

But I always considered these functors as possibly being obtained from having “locally trivialized” a globally defined functor, i.e. one that was literally defined on the domain, not just on a surjection of the domain.

Of course, as we have all emphasized various times, that “globally defined” thing is usually not itself continuous/smooth in a direct sense.

In the language of anafunctors, I think the language that I adopted amounts to the following, and I’d be interested in anyone’s comment on that strategy:

While ambient categories like $\mathrm{Top}$ or $C^\infty$ don’t have sections for every epimorphism (“the axiom of choice fails”), there is a forgetful functor to $\mathrm{Set}$ and - if we allow ourselves to use the axiom of choice on bare sets, which I am willing to do - after applying that functor the sections do exist.

So, for instance, for $A$ some domain and $|F| \to A$ some surjection, all in our ambient category, we don’t in general have any section

(1)$s : A \to |F|$

but we do as soon as we forget the extra structure (e.g. the topology, or the smooth structure).

This allows to consider functors on $A$, not internal to our ambient category. Instead of them being continuous/smooth by way of being internal to $\mathrm{Top}$ or to $C^\infty$, it makes sense (I think) to equip them with a smooth structure by specifying that and how their pullback to $|F|$ is equivalent (internal just to $\mathrm{Set}$) to a functor (on $|F|$) which is continuous/smooth in the sense that it lives internal to $\mathrm{Top}$/$C^\infty$.

It seems to me that this amplifies a property of anafunctors which is present, but which is only implicit in the way Makkai defines them and even less obvious in the way Toby defined them (so implicit, in fact, that I completely missed it until recently): and that’s the relation to descent data.

As the name suggests, this is precisely about descending down along the surjection

(2)$\array{ |F| \\ \downarrow \\ A } \,,$

expressing the idea that the entity upstairs is like something downstairs having been pulled up.

That’s important, isn’t it? For instance this is, to my mind, the very justification for the precise definition of anafunctor that Makkai gives:

one might ask why exactly we demand an anafunctor to be a span with the extra property that the surjection of the left leg has unique lifts with specified source and targets.

This is the crucial point. I expect this becomes quite amplified once one tries to talk about $n$-anafunctors: one will have to figure out what this condition really means.

And, I think, what it really means is that there is a certain descent property around. For $n=1$ this just happens to be encoded in codiscrete categories.

Posted by: urs on December 11, 2006 5:38 PM | Permalink | Reply to this

Re: forgetting structure, remembering choice

Urs wrote:

… even less obvious in the way Toby defined them…

Okay, Toby — you’ve got to fix this while preparing your thesis for publication. Talk about descent!

Posted by: John Baez on December 28, 2006 8:06 AM | Permalink | Reply to this
Read the post Local Transition of Transport, Anafunctors and Descent of n-Functors
Weblog: The n-Category Café
Excerpt: Concepts and examples of what would be called transition data or descent data for n-functors.
Tracked: December 12, 2006 3:00 PM

Michael Makkai’s text on anafunctors

On some systems the postscript files provided on Michael Makkai’s site don’t display properly. Since I was asked, I have now converted the PS files to PDFs, which do seem to be more well-behaved:

anafun1.pdf (title and contents)

anafun2.pdf (introduction)

anafun3.pdf (main text)

Since the original files are freely available, I guess it is okay that I provide the above links.

Posted by: urs on December 14, 2006 12:48 PM | Permalink | Reply to this

Re: Michael Makkai’s text on anafunctors

Since the original files are freely available, I guess it is okay that I provide the above links.

I’ve brought this discussion to Makkai’s attention already by email, so perhaps he will say something if he objects.

Posted by: Toby Bartels on December 14, 2006 11:21 PM | Permalink | Reply to this

Re: Michael Makkai’s text on anafunctors

As Makkai has pointed out to me by email, there is already a brief discussion of 2-anafunctors in his paper.

Posted by: Toby Bartels on December 25, 2006 1:14 PM | Permalink | Reply to this

Re: Michael Makkai’s text on anafunctors

As Makkai has pointed out to me by email, there is already a brief discussion of 2-anafunctors in his paper.

I assume you are referring to what is called morphisms of anabicategories and anafunctors of anabicategories in the paper, the definition of which begins on p. 55 (part 4 of chapter 3).

This definition is given in the componentwise style in which also the first version of the definition of anafunctors themselves was presented. For the latter, there is the equivalent but more concise formulation in terms of spans. Is an equivalent definition of morphisms of anabicategories known which is analogous to the definition of anafunctors in terms of spans?

Posted by: urs on December 27, 2006 10:21 PM | Permalink | Reply to this

Re: Michael Makkai’s text on anafunctors

Is an equivalent definition of morphisms of anabicategories known which is analogous to the definition of anafunctors in terms of spans?

I’m sure that there is one, but if it’s not in Makkai’s paper, then it probably hasn’t been written down, perhaps never even fully worked out. So you should probably just develop your own idea and then compare it with Makkai’s componentwise definition afterwards.

Perhaps I should have quoted Makkai’s email more fully. I can’t get at it right now, but he said something like ‘The discussion was very brief and not very explicit.’. So hardly a reason to stop working on your own approach!

Posted by: Toby Bartels on December 28, 2006 5:49 AM | Permalink | Reply to this

Re: Michael Makkai’s text on anafunctors

but if it’s not in Makkai’s paper

At least I haven’t seen it.

So you should probably just develop your own idea and then compare it with Makkai’s componentwise definition afterwards.

I am sort of on vacation currently, but I thought about this a little. I’d be surprised if the “descent of 2-functors” that I talked about would not give rise to a morphism of anabicategories as defined by Makkai. (The converse statement, if true, might be harder to see.)

Essentially, it amounts to noticing that the descent data of a 2-functor, while not ensuring that any two lifts with fixed enpoints of a given 1-morphism are equal, do precisely ensure that any two lifts are connected by a unique 2-isomorphism.

This should exactly mean that a 2-functor on these lifts restricts to a 1-anafunctor on the respective Hom-categories. And that, in turn, is the main aspect of M. Makkai’s definition of “anafunctor of anabicategoies”.

So hardly a reason to stop working on your own approach!

Okay, good! :-)

But I’d be very interested in learning more about what other poeple do and did in this direction.

Posted by: urs on December 28, 2006 2:28 PM | Permalink | Reply to this

Re: Michael Makkai’s text on anafunctors

Urs wrote:

In this paper by Eugenia Cheng an equivalence between “ordinary” bicategories and opetopic bicategories is shown.

Doesn’t this indicate that opetopic bicategories are rather not like anabicategories?

No. Anafunctors are just like functors, given the axiom of choice. So, anabicategories are just like bicategories, given the axiom of choice. And, similarly, opetopic bicategories are just like bicategories, given the axiom of choice.

You’re interested in applications ‘internal to’ categories like $Top$ or $C^\infty$, where the axiom of choice fails. In this situation, I guess that opetopic bicategories will still be just like anabicategories. But, they’ll be different from ordinary bicategories.

You already know the following stuff, but in case anyone out there is falling behind:

A functor $f: X \to Y$ assigns to each object of $X$ an object of $Y$. An anafunctor $f: X \to Y$ assigns to each object of $X$ an object of $Y$ up to canonical isomorphism.

Anafunctors show up all over the place in category theory: for example, the ‘product’ operation in a category with products is an anafunctor, since the product of objects is defined up to canonical isomorphism.

We can convert an anafunctor to a functor by choosing a specific object $f(x) \in Y$ for each object of $x \in X$. The choice doesn’t matter much, because $f(x)$ is defined up to canonical isomorphism. So, we can get different functors from different choices, but they’re all naturally isomorphic.

However, to actually do the choice we typically need the axiom of choice!

In contexts where our categories have not a set of objects but (for example) a space of objects, the axiom of choice fails — so anafunctors become significantly more general than functors.

Posted by: John Baez on December 28, 2006 8:44 PM | Permalink | Reply to this

Re: Michael Makkai’s text on anafunctors

I’m not sure what an ‘anabicategory’ is. But, I’ll guess that it’s like a bicategory, but where the composition functor:

$\circ : hom(x_1,x_2) \times hom(x_2,x_3) \to hom(x_1,x_3)$

has been replaced by an anafunctor.

If so, it should be fairly similar to what James Dolan and I call simply a ‘2-category’ in our opetopic approach to n-categories. The main difference is that instead of treating binary composition and identities

$i: 1 \to hom(x_1,x_1)$

as fundamental, we treat all $k$-ary compositions

$\circ_k : hom(x_1,x_2) \times \cdots \times hom(x_{k-1},x_k) \to hom(x_1,x_k)$

on an equal footing.

I similarly suspect that an ‘anabifunctor’ may be a lot like what James and I called a ‘virtual functor between 2-categories’.

This is part of a much bigger story you probably don’t want to hear. Briefly, what we did is take Makkai’s anafunctor idea together with the theory of operads and use them to develop a general ‘opetopic’ definition of $n$-category. Then, Makkai took our ideas and developed them into $n$-categorical foundation of mathematics! But, for technical reasons, he used the word ‘multitope’ instead of ‘opetope’.

Most of this work will not be useful to you in the short run, because it focuses on general $n$ instead of $n = 2$.

But, here’s something that focuses on the $n = 2$ case:

Tom Leinster may also have written some useful stuff about the $n = 2$ case.

(You’ll meet Eugenia and Tom the week after next!)

Posted by: John Baez on December 28, 2006 7:38 AM | Permalink | Reply to this

Re: Michael Makkai’s text on anafunctors

I’m not sure what an ‘anabicategory’ is. But, I’ll guess that it’s like a bicategory, but where the composition functor […] has been replaced by an anafunctor.

Yes, exacxtly.

I similarly suspect that an “anabifunctor” […]

The main idea is, rather naturally, that a “morphism of anabicategories” is like a morphism of bicatgeories, but with functors on the Hom-categories replaced by anafunctors.

BTW, Makkai uses the term “anabifunctor” for something else, namely for a functor on a product category $A \times X$ that restricts to an anafunctor in each of its two arguments. The ana-version of a 2-functor he calls a “morphism/anafunctor of anabicategories”.

Just one instance of the general problem with “bi”. What is a “bitorsor”? A 2-sided torsor or a torsor for a 2-group? The former.

But then what is a “bigroupoid bitorsor”? Now this is both: the groupoid is categorified, hence is its torsor, which in addition is 2-sided.

The usage of the prefix “bi” here leads to ambiguous terminology.

[…] like what James and I called a ‘virtual functor between 2-categories’.

Oh, interesting. So: in your and my work on 2-connections we work out the descent data for a 2-functor with values in a 2-group.

This defines a 2-functor on 2-paths in the transition 2-groupoid of the chosen cover.

Does this define a “virtual functor between 2-categories”?

As I have mentioned in my last reply to Toby # it seems like this does define a “anafunctor of anabicategories”.

Eugenia Cheng, Opetopic Bicategories

Thanks for that! I’ll take a look at it.

Posted by: urs on December 28, 2006 2:52 PM | Permalink | Reply to this

Re: Michael Makkai’s text on anafunctors

Eugenia Cheng, Opetopic Bicategories

Thanks for that! I’ll take a look at it.

I’d need more time to fully absorb this. But one quick question:

in this paper by Eugenia Cheng an equivalence between “ordinary” bicateories and opetopic bicategories is shown.

Doesn’t this indicate that opetopic bicategories are rather not like anabicategories?

Posted by: urs on December 28, 2006 4:37 PM | Permalink | Reply to this

Re: Michael Makkai’s text on anafunctors

Then, Makkai took our ideas and developed them into n-categorical foundation of mathematics!

Makkai has (by email) specifically directed me to The multitopic omega-category of all multitopic omega-categories; corrected.

Posted by: Toby Bartels on December 28, 2006 10:28 PM | Permalink | Reply to this

Re: Michael Makkai’s text on anafunctors

Urs wrote:

So: in your and my work on 2-connections we work out the descent data for a 2-functor with values in a 2-group.

This defines a 2-functor on 2-paths in the transition 2-groupoid of the chosen cover.

Does this define a “virtual functor between 2-categories”?

I haven’t worked this out carefully, but in April of this year, at the very end of my third Unni Namboodiri Lecture at Chicago, I wrote something like this:

Ultimately we expect to find:

For any smooth 2-group $G$, principal $G$-2-bundles with 2-connection over a smooth space $B$ are classified by smooth 2-anafunctors from the path 2-groupoid of $B$ to $G$.

And why stop at 2? The basic principle of Galois theory keeps growing…

So: yes — I believe that modulo variations in terminology, I was saying just what you’re saying now!

If anyone doesn’t know what this stuff has to do with Galois theory, they should read all the lectures — there’s actually a very big beautiful story here.

Posted by: John Baez on December 28, 2006 9:00 PM | Permalink | Reply to this

Re: Michael Makkai’s text on anafunctors

So: yes — I believe that modulo variations in terminology, I was saying just what you’re saying now!

Okay, good.

In this case, for a change, my question was really aimed at the details:

do I understand correctly that you do expect that whatever a 2-anafunctor is in the end, it should be smething involving a surjection of the domain 2-category which involves some kind of descent condition, like I tried to indicate above?

I vaguely recall that somewhere, quite a while ago, you said something like how anafunctors are just another way to talk about stacks.

I think I understand why and how this is true. Did anyone write up this observation in any detail?

This would mean that 2-anafunctors should correspondingly be related to 2-stacks. And this then brings in those descent conditions (2-commuting tetrahedra, really), which are just the usual descent data encountered in the definition of 2-stacks.

I believe I am able to draw considerable parts of this picture. But it would be good to know which parts exactly have been drawn before.

Posted by: urs on December 28, 2006 10:04 PM | Permalink | Reply to this

Re: Michael Makkai’s text on anafunctors

Urs wrote:

I vaguely recall that somewhere, quite a while ago, you said something like how anafunctors are just another way to talk about stacks.

I think I understand why and how this is true. Did anyone write up this observation in any detail?

All I know is that Toby Bartels has rigorously related gerbes to smooth anafunctors. As you know, gerbes are a special case of stacks.

More precisely:

One big result in Toby’s thesis is that the 2-category of nonabelian $G$-gerbes over a space $B$ is equivalent to the 2-category of principal $AUT(G)$-2-bundles over $B$, where $AUT(G)$ is the automorphism 2-group of the Lie group $G$. More generally, for 2-group G, he defines a principal G-2-bundle over $B$ to be a smooth anafunctor

$P \to B$

equipped with some extra stuff.

(He uses the term ‘2-map’ instead of the lengthy ‘smooth anafunctor’, but he refers to Makkai’s work on anafunctors. Personally I prefer ‘map’ to ‘2-map’, since ‘2-maps’ sound like 2-morphisms, but we’re talking about morphisms.)

Toby has been chillin’ in Nebraska. He recently visited Riverside, and we talked about future plans. He plans to add extra material to his thesis to cover the case when $B$ is a full-fledged 2-space instead of just a space… in order to please you, he said!

I think Toby’s thesis could use more expository prose. So, if anyone out there (Urs, Bruce?) has attempted to read Toby’s thesis and found that certain sections would be easier to understand if he added more explanations, please let Toby or me know what sort of explanations you would have liked. A comment on this blog would do the job nicely.

After it’s polished up, we’ll try to get Toby’s thesis published. It’ll be called ‘Higher Gauge Theory I: 2-Bundles’.

Our own paper on 2-connections will then turn into ‘Higher Gauge Theory II: 2-Connections’. We really should modify it a little to make the role of anafunctors a bit clearer. Because we wrote some parts of it very long ago, we mistakenly say the right kind of map between smooth 2-spaces is a smooth functor, rather than a smooth anafunctor.

Posted by: John Baez on December 30, 2006 6:27 PM | Permalink | Reply to this

2-bundles

Urs wrote:

I vaguely recall that somewhere, quite a while ago, you said something like how anafunctors are just another way to talk about stacks.

I think I understand why and how this is true. Did anyone write up this observation in any detail?

All I know is that Toby Bartels has rigorously related gerbes to smooth anafunctors.

Ah, okay. That’s good. But maybe I was thinking here of something else you might have meant.

I was thinking of the observation that an anafunctor can equivalently be regarded as a certain object in a category of descent data. This is one sense in which they are related to stacks.

As you all know, a stack of something is a weak presheaf of something such that each something on X comes from something on a cover of X equipped with gluing data.

If “something” here is functor, then anafunctor is like “something on the cover with gluing data”.

if anyone out there (Urs, Bruce?) has attempted to read Toby’s thesis

I had read the first version in detail and found it ro be very clear indeed.

At the later version I looked only more briefly. As we had discussed, I would have found a clarifying sentence in the section on anafunctors helpful. But now it is clear to me.

I am officially on vacation currently, and have to use a a very slow internet connection at the moment, which makes every step on the web difficult, so I haven’t checked right now if there is a new version which I had not seen before.

But last time I checked the proof of this result:

[…] that the 2-category of nonabelian $G$-gerbes over a space $B$ is equivalent to the 2-category of principal $\mathrm{AUT}(G)$-2-bundles over $B$,

was not contained yet.

There is a point about this which to my mind would deserve a little clarification. I believe I have asked the corresponding question a few times before. Please remind me if I just happen to have forgotten the answer.

The question is this: I have seen that Toby shows that every 2-bundle gives rise to a nonabelian 2-cocycle which also classifies a gerbe.

The more subtle step is always that going the other way around: to show that from every such 2-cocycle we can build the total 2-space of a 2-bundle that gives rise to this cocycle when locally trivialized.

This step I have not seen discussed. (But, as I said, possibly it is now in the latest version which I have not seen yet.)

I am expecting that what makes this converse step work is precisely the fact that Toby demands the projection

(1)$P \to B$

to be “just” an anafunctor. This probably allows us to take $P$ to simply be

(2)$U \times G \,,$

probably with a couple of isomorphisms thrown in, where $U$ is the cover of $B$ and $G$ the 2-group.

Anyway, last time I looked at the text I did not see a discussion of this reconstruction of the total 2-space from the cocycle data. So this would be a point I would like to see discussed more.

add extra material to his thesis to cover the case when $X$ is a full-fledged 2-space instead of just a space… in order to please you, he said!

Oh, I’d be very interested indeed.

It is good that we talk about this, because meanwhile I have developed some ideas of my own that address the point which originally made me want to consider full-fledged 2-spaces. It would be very interesting to compare notes.

Maybe just briefly: originally I had the feeling that if we go from bundles to 2-bundles, we should, for good measure, also pass from points to strings.

The base space of a bundle plays the role of the configuration space of the particle charged under that bundle.

Hence, I was originally thinking, the base space of a 2-bundle should be the configuration space of the something like a string coupled to that 2-bundle. This suggested that the base space of a 2-bundle should be a full-fledged 2-space.

But I didn’t make much progress with understanding 2-bundles on full 2-spaces. So I’d be very interested in what Toby can say about this!

In particular, at some point I began adopting a point of view which didn’t mention base spaces and total spaces explicitly, but tried to encode everything in 2-functors on 2-categories of 2-paths. But in the end all these descriptions should be different aspects of the same structure.

The relation between the base space of a 2-bundle and the configuration space of a string now looks, from my point of view, like this:

for me, a 2-bundle with connection is entirely a 2-functor from a domain $P_2$ of “2-paths” to some suitable codomain.

If I want to couple a “string” to this, I first say how this string looks like “internally” by defining a small “parameter space” category. For the open string for instance the category with 2-objects that looks like

(3)$\mathrm{par} = \{a\to b\}$

is a suitable model.

Then the configuration space of this string is simply the 2-functor 2-category

(4)$[\mathrm{par},P_2]$

(or rather a sub-2-category of that obtained by retaining only those morphisms that relate “gauge equivalent configurations”, but never mind at the moment).

Now, just by abstract nonsense we can “transgress” the 2-bundle with connection to this configuration space.

I talked about that at several places. For instance here.

I’d be very interested in understanding if and how from the data described this way we can reconstruct a total 2-space of a 2-bundle on a full-fledged 2-space. That would be very interesting.

Our own paper […]

We really should modify it a little to make the role of anafunctors a bit clearer.

There are a few things that should be made clearer. I was hoping that maybe in Toronto I could hand you a couple of notes and ask you to see if they could be merged into a joint something.

Concerning Toby’s work, we should also not miss to find some time to talk with Igor Bakovic in Toronto (it looks as if, against all odds, he will mange to come, which is great). He is almost finished with a thesis on 2-groupoid 2-bundles, which contains quite a few things that should be nicely complementary to what Toby did.

Posted by: urs on December 30, 2006 8:19 PM | Permalink | Reply to this

Re: 2-bundles

I haven’t checked right now if there is a new version which I had not seen before.

The latest version, version 3, was published on June 26. I’m sure that you’ve seen it.

But last time I checked the proof of this result [Theorem 3] was not contained yet.

It’s supposed to be there, in the preceding discussion, but I know that you need more details. Actually, it’s really Theorem 2 that needs the details; I hope that you’re happy with the equivalence between gerbes and (what I call) 2-transitions and only want further details on the equivalence between 2-transitions and 2-bundles. This theorem is also proved in its preceding discussion, and the bit that needs the details is Proposition 22 (which amounts to the essential surjectivity of the obvious forgetful functor from 2-bundles to 2-transitions). The details are needed in the first paragraph of the proof, where I remark that every equivalence 2-relation involving a 2-cover has a 2-quotient. Actually, this remark is true by definition; see the axioms of 2-covers in Section 2.1.5. So where I really need details is Section 2.2.5, where I explain why the 2-covers defined there satisfy those axioms. To be even more precise, I need details in the penultimate paragraph, but really that whole section could use more details. That section is, more than anything else, what I need to work on.

That said, since you (Urs) didn’t trace the problem back to Section 2.2.5, I presumably need more exposition elsewhere; I solicit your advice on that.

The more subtle step is always tat going the other way around: to show that from every such 2-cocycle we can build the total 2-space of a 2-bundle that gives rise to this cocycle when locally trivialized.

This is essentially Proposition 22.

I am expecting that what makes this converse step work is precisely the fact that Toby demands the projection

(1)$P \to B$

to be “just” an anafunctor. This probably allows us to take P to simply be

(2)$U \times G ,$

probably with a couple of isomorphisms thrown in, where U is the cover of B and G the 2-group.

Yes, that is precisely what P is!

Notice that, since I don’t restrict to the principal case in most of my thesis, I say E instead of P and F instead of G (in some contexts, when it is the fibre that is wanted). Now look at the first sentence in the proof of Proposition 22 (in Section 2.5.4) to see that E is indeed constructed as a 2-quotient of F × U. (In general, in forming quotients of categories, one not only throws in isomorphisms, as you said, but also identifies parallel morphisms. But it seems to me that the latter won’t happen in this case.)

Posted by: Toby Bartels on December 30, 2006 10:28 PM | Permalink | Reply to this

Re: 2-bundles

This is essentially Proposition 22.

Ah, thanks. I was looking for a statement like this prop. 22, but didn’t see it. Sorry, I should have read the text more carefully.

Now that I am finally aware of it, I very much like the clear proof of prop. 22.

The powerful fact which makes this work is that

every equivalence 2-relation involving a 2-cover has a 2-quotient

(very top of p. 68).

It seems to me that we can appreciate the power of this fact by noticing that this achieves essentially what Wirth proved in his mapping cylinder theorem for homotopy fibrations.

What precisely is the status of this statement?

It appears first at the beginning of section 2.1.5 on p. 34. There you call it an axiom for 2-covers.

It is all very much analogous to the case for 1-covers. Though there the corresponding statement is a property derived from an independent definition of 1-cover. (I haven’t looked at the text [Ele1] that you refer to.)

Anyway: it seems that you define a 2-cover to be something that makes, among other things, the statement

every equivalence 2-relation involving a 2-cover has a 2-quotient

true. Is that so?

When I tried to understand this reconstruction business on my own a while ago, I was trying to understand it first in a concrete example: given a 2-cocycle descibing an abelian gerbe, i.e. something like

(1)$f_{ijk}f_{ikl} = f_{ijl}f_{jkl}$

with $f_{\cdots}$ taking values in $U(1)$, what is the total 2-space of a 2-bundle that has such a 2-transition?

I understand that your proof of prop. 22 is non-constructive, in that it only asserts that such a total 2-spaces does exist.

But would you know how to explicitly construct it?

Finally: can you help me see where exactly in the proof of prop. 22 the properties of anafunctors play a role?

Maybe I have this question more generally: which of your constructions would break down if you were to use internal functors instead of internal anafunctors?

A)

You point out an “analogy” between $n$-groups and transitions in principal $n$-bundles (for $n=1$ and $n=2$).

I think of this analogy this way: the data of a $G$-$2$-transition can be understood as defining a $\Sigma(G)$-enrichment of the codiscrete category obtained from the cover.

(For $G$ having only identity morphisms this statement then also applies to $n=1$.)

B)

You discuss pullbacks in detail. Unless I am missing something, the quotients you discuss are certain pushouts. So for instance (13) is like the opposite of (3).

I would find a brief indication helpful concerning how the categoriefied versions of these pullback and pushouts are both instances of a certain notion of 2-(co)-limit. I seem to recall that there are various slightly different flavors of 2-(co)-limits.

C)

There seems to be a typo in the very first sentence of section 2.5.4. The first “$G$-2-bundle” should probably be a “$G$-bundle”.

D)

Above diagram (12) there seems to be a typo: the codomain of that new map should read $X$ instead of $R^{[0]}$.

Analogously in the line above the categorified version (86).

Posted by: urs on December 31, 2006 12:29 PM | Permalink | Reply to this

Re: 2-bundles

given a 2-cocycle descibing an abelian gerbe, […] what is the total 2-space of a 2-bundle that has such a 2-transition?

I should mention that I have an idea how that total 2-space should look like explicitly. I describe this here.

I would be interested in understanding such concrete examples in terms of the general formalism.

Posted by: urs on January 1, 2007 12:04 PM | Permalink | Reply to this

Re: 2-bundles

Sorry, Urs; I missed these comments the first time, travelling across a frozen landscape, ringing in the New Year in a motel with a picture of a sleeping alien out front. (Warning: Links may not be strictly relevant.)

It [that “every equivalence 2-relation involving a 2-cover has a 2-quotient”] appears first at the beginning of section 2.1.5 on p. 34. There you call it an axiom for 2-covers.

Right, because in this portion of the paper (as in essentially all of part 2 except for section 2.2), I’m working in as general a 2-category of 2-spaces as possible, not necessarily the specific 2-category of smooth 2-spaces (whatever even that is precisely). So the 2-category of 2-spaces is simply required to be equipped with a notion of 2-cover satisfying these conditions.

It is all very much analogous to the case for 1-covers. Though there the corresponding statement is a property derived from an independent definition of 1-cover.

Actually, in the corresponding section 1.1.5, these are also axioms. Here (as in essentially all of part 1 except for section 1.2), I’m working in as general a category of spaces as possible, not necessarily the specific category of smooth spaces (whatever even that is precisely). So the category of spaces is simply required to be equipped with a notion of cover satisfying these conditions, again as axioms. (Perhaps this is a place that I could improve the wording.)

(I haven’t looked at the text [Ele1] that you refer to.)

(There’s no reason that you should, except in as much as everybody ought to read that book. I referenced it only because these axioms appear there, although the context is —as far as I know— quite different.)

So these parallel sections, 1.1.5 and 2.1.5, need to be complemented by something that explains what (2)-covers are in the specific (2)-category of (2)-spaces. Looking at this category specifically is the province of sections 1.2 and 2.2 (which, unlike as in the rest of the paper, do not correspond to one another except at the grossest level). For (2)-covers, the relevant sections are 1.2.2 and 2.2.5.

So really, there are two things (at least) that I need to add to the paper: * Fuller discussion (in section 1.2.2) of why surjective submersions in the category of manifolds satisfy the axioms in section 1.1.5 (on the basis of established facts about manifolds); * Fuller discussion (in section 2.2.5) of why 2-covers as there defined satisfy the axioms in section 2.1.5 (on the assumption that 2-spaces are categories internal to a category equipped with a notion of cover that satisfies the axioms in section 1.1.5).

But would you know how to explicitly construct it?

In principle, yes! In practice, my thesis is largely an abstract exercise in matters that (even where well established and by no means new!) I have little experience with. (It didn’t start out that way, but it ended up that way.) This is why, for example, I haven’t been able to follow Gel’fand’s advice to include the simplest nontrivial example (which advice would really improve the paper, I think). While my proofs should tell one how to make explicit constructions, I get lost when I try.

So what might really help, Urs, is if you and I could go over some specific examples —probably whichever ones you want best to understand— and work through everything. This would not only help you understand my thesis better, and not only help me explain it better, but also help me understand its practical content better (since otherwise it is only so much abstract nonsense). In other words, it would help me understand better what you and John are going to do with it!

So I would very much like to work this out in Toronto, if you have time!

I have also noted the following very minor random comments:

Thanks. I’ve fixed the typos (in the unpublicised version whose URL you should know), and I’ll try to write the requested explanations over the next few months.

Posted by: Toby Bartels on January 7, 2007 2:12 AM | Permalink | Reply to this

Re: 2-bundles

Hi Toby,

thanks for the explanation. I get the point now.

So I would very much like to work this out in Toronto, if you have time!

At least, I’d be very interested indeed in talking about this. Whether we will find enough time I can’t tell, with so many other things of interest going on. But we should really try.

Posted by: urs on January 7, 2007 5:11 PM | Permalink | Reply to this

Re: 2-bundles

Bruce Bartlett might want to notice proposition 13 (p. 41) in Toby’s thesis, since this is what we talked about recently #.

Like an $G$-bundle with connection on $X$ is characterized by an anafunctor from paths in $X$ to $\Sigma(G)$, a $G$-bundle without connection is characterized by an anafunctor from constant paths in $X$ to $\Sigma(G)$. This last part is, paraphrased, what Toby’s prop 13 asserts.

Posted by: urs on December 31, 2006 1:08 PM | Permalink | Reply to this

Re: 2-bundles

Before the fireworks start in a few hours, I want to briefly sketch how the problem of reconstructing a total structure from its local transition data looks like from the point of view where $n$-bundles with connection are encoded entirely in terms of the parallel transport functors.

I’ll describe the case $n=1$ of ordinary bundles, where the solution is familiar.

First the usual notation. Skip the next paragraph if you have seen enough of this.

Let $X$ be a base space, let

(1)$U \to X$

be a cover, let

(2)$P_1(X)$

be the groupoid of paths in $X$ and let $G$ be a Lie group and

(3)$\Sigma(G)$

the category with a single object and $G$ worth of morphisms. Finally, let

(4)$P_1(U^\bullet) \to P_1(X)$

be the 2-cover induced from $P_1(X)$ as Toby describes it. We may realize $P_1(U^\bullet)$ as the groupoid generated from $P_1(U)$ and $U^{[2]}$ modulo the “obvious” relations which ensure, if you want to put it this way, that every morphism in $P_1(X)$ has at most one lift to $P_1(U^\bullet)$ with given source and target.

Okay, now what I want to say:

A differential $G$-cocycle on $X$ describing the transition data of a $G$-bundle with connection on $X$ is an anafunctor

(5)$\array{ P_1(U^\bullet) &\stackrel{(\mathrm{tra}_U,g)}{\to}& \Sigma(G) \\ \downarrow \\ P_1(X) } \,.$

For the following we want to consider the “graph” of this functor (in the sense of graphs of functions). We can equivalently think of this as an anafunctor with codomain $\Sigma(G)$ replaced by

(6)$P_1(U^\bullet) \times \Sigma(G)$

by writing

(7)$\array{ P_1(U^\bullet) &\stackrel{\mathrm{Id}\times(\mathrm{tra}_U,g)}{\to}& P_1(U^\bullet) \times \Sigma(G) \\ \downarrow \\ P_1(X) } \,.$

Now, it would be nice to have an arrow-theoretic way to construct from this data the corresponding “total space structure”. By this I mean the smooth functor on $P_1(X)$ with values in the smooth transport groupoid of the $G$-bundle obtained via reconstruction from the given cocycle data.

We know that from the 1-cocycle itself ($g_{ij}g_{jk}=g_{ik}$) contained in the above data we can reconstruct a $G$-bundle

(8)$P \to X$

and from the remaining cocycle data ($A_i = g_{ij}A_j g_{ij}^{-1} + g_{ij}d g_{ij}^{-1})$) we get a connection

(9)$\nabla$

on $P$.

We also know (see the first part of this for a reminder) that the parallel transport of this connection is a smooth functor

(10)$\mathrm{tra}_\nabla : P_1(X) \to P \times_G P$

from paths to the smooth transport groupoid

(11)$P \times_G P \,,$

whose objects are the points of $X$ and whose morphisms are the torsor morphisms between the fibers of $P$ over the respective points of $X$.

Inserting this functor into the above diagram we get

(12)$\array{ P_1(U^\bullet) &\stackrel{ \mathrm{Id}\times (\mathrm{tra}_U,g)}{\to}& P_1(U^\bullet) \times \Sigma(G) \\ \downarrow \\ P_1(X) &\stackrel{\mathrm{tra}_\nabla}{\to}& P \times_G P } \,.$

This should be trying to tell us that the transport groupoid $P \times_G P$ (the analog of the total space of the $G$-bundle $P$ in the world of parallel transport) is a pushout of the span given by our original anafunctor.

In fact, there is an obvious surjection

(13)$\array{ P_1(U^\bullet) \times \Sigma(G) \\ \downarrow \\ P \times_G P }$

(encoding essentially the morphism $\tilde j$ in Toby’s diagram (26)) which makes this diagram commute:

(14)$\array{ P_1(U^\bullet) &\stackrel{ \mathrm{Id}\times(\mathrm{tra}_U,g) }{\to}& P_1(U^\bullet) \times \Sigma(G) \\ \downarrow && \downarrow \\ P_1(X) &\stackrel{\mathrm{tra}_\nabla}{\to}& P \times_G P } \,.$

And I think that, furthermore, this is the universal pushout cone of the span coming from our original anafunctor.

(You should not trust me on that but check it yourself. If you think I am wrong, please let me know.)

All this suggests an obvious generalization to 2-bundles.

Here, we do know that the right notion of differential 2-cocycle is a 2-anafunctor

(15)$\array{ P_2(U^\bullet) &\stackrel{ \mathrm{Id}\times(\mathrm{tra}_U,g) }{\to}& P_2(U^\bullet) \times \Sigma(G_2) \\ \downarrow \\ P_2(X) } \,.$

I guess that this means that if we are after reconstructing the corresponding “total space structure”, we need to contemplate suitably weakened 2-pushouts of this diagram.

When do these exist? How do we construct them?

Whatever the answer is, I know for sure that we won’t find it this year.

So I conclude by wishing everybody

a happy new year!

Posted by: urs on December 31, 2006 6:35 PM | Permalink | Reply to this

Re: 2-bundles

[…] briefly sketch how […]

What I wrote is not yet in its final shape. It looks slightly awkward. The reason is that I did not follow my own advice, which says that we should always be looking at the magic square of $p$-local $i$-trivialization

(1)$\array{ P_1(U) &\to& P_1(X) \\ \downarrow &\Downarrow& \downarrow \\ T' &\to& T } \,.$

So, what I should have said is this:

Given an anafunctor

(2)$\array{ P_1(U^\bullet) &\stackrel{(\mathrm{tra}_u,g)}{\to}& \Sigma(G) \\ \downarrow \\ P_1(X) }$

we form the weak pushout

(3)$\array{ P_1(U^\bullet) &\stackrel{(\mathrm{tra}_u,g)}{\to}& \Sigma(G) \\ \downarrow &\Downarrow& \downarrow \\ P_1(X) &\stackrel{\mathrm{tra}_\nabla}{\to}& P \times_G P } \,,$

where

(4)$\Sigma(G) \to P \times_G P$

identitfies $G$ with the automorphisms of any one fiber of $P$.

The natural isomorphism filling this square is then given by the morphisms $\tilde j$ of Toby’s thesis.

Posted by: urs on January 1, 2007 11:59 AM | Permalink | Reply to this

Re: Michael Makkai’s text on anafunctors

Toby has been chillin’ in Nebraska.

But I missed that storm, being in California. Instead, I was snowed in for two days in Denver earlier.

He plans to add extra material to his thesis to cover the case when B is a full-fledged 2-space instead of just a space… in order to please you, he said!

I already cover that. What needs more detail is the properties of 2-covers (which are special 2-maps, which in turn are simply smooth anafunctors), including the case where B is a 2-space. It is these properties that are used to construct the total 2-space of a 2-bundle out of the 2-transition data. This is to please Urs (not only myself), as you can see by the complaints in his comment above.

So, if anyone out there […] has attempted to read Toby’s thesis and found that certain sections would be easier to understand if he added more explanations, please let Toby or me know what sort of explanations you would have liked.

Yes, please do! Make public comments here, or send private ones to me by email.

Posted by: Toby Bartels on December 30, 2006 9:48 PM | Permalink | Reply to this

Re: Michael Makkai’s text on anafunctors

John wrote:

Ultimately we expect to find:

For any smooth 2-group G, principal G-2-bundles with 2-connection over a smooth space B are classified by smooth 2-anafunctors from the path 2-groupoid of B to G.

Please alert me directly either to the solution when it occurs or to the problem/difficulty.

Posted by: jim stasheff on December 29, 2006 11:22 PM | Permalink | Reply to this

Re: Michael Makkai’s text on anafunctors

Jim Stasheff wrote:

John wrote:

Ultimately we expect to find:

For any smooth 2-group $G$, principal $G$-2-bundles with 2-connection over a smooth space $B$ are classified by smooth 2-anafunctors from the path 2-groupoid of $B$ to $G$

Please alert me directly either to the solution when it occurs or to the problem/difficulty.

The precise answer to this depends a little on what exactly one is asking for.

But one sensible solution would be this:

First: what is a $G$-2-bundle with connection?

Possible answer: something that assigns to each point of base space a category with a $G$-action on it and which assigns to each path in path space a morphism of these $G$-categories.

When is such a 2-bundle with connection principal, smooth and locally trivializable?

Possible answer: precisely if the above assignment is locally equivalent to a smooth 2-functor that assigns $G$ itself to every point, and if the transitions between these local equivalences are smooth themselves.

If this is the case, then these local 2-functors with the transitions between them form an instance of smooth descent data of a 2-functor.

Above I proposed # that we define such 2-functor descent data to be a 2-anafunctor.

But I expect # that, if one works out the details, such a 2-functor descent also provides an example for what M. Makkai calls an “anafunctor of anabicategories”.

In summary, it follows, almost by definition, that every $G$-2-bundle with connection is characterized by a $G$ 2-anafunctor.

What is more subtle is the converse statement.

Posted by: urs on December 30, 2006 2:08 PM | Permalink | Reply to this

Re: Michael Makkai’s text on anafunctors

Jim wrote:

Please alert me directly either to the solution when it occurs or to the problem/difficulty.

Okay. Most likely Urs will solve this problem.

Posted by: John Baez on December 30, 2006 6:32 PM | Permalink | Reply to this
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