The n-Category Café

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

and a codomain

and we want to talk about $n$-functors

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

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

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

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

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

on triple intersections an equivalence

on quadruple intersections an equivalence

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

from $A$ to $B$ is a span

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

we can form the category

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

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

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

given by the span

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

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

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

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

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

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

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

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

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

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

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:

Morphisms of Anafunctors

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

On Anafunctors and Transitions

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.