The n-Category Café

Thanks a lot for your comments!

Right now it’s rather late here, and I am somewhat tired. Therefore I won’t try to reply to all of your comments at once, but instead reply to them piece by piece.

Here, I’ll reply to this question:

I’m not up on ‘adjoint equivalence’ - why not just equivalence

For other readers, let me recall what this question is about: in my notes I made the obvious statement that we say a 2-transport is trivializable if there is a certain morphism from it to a trivial 2-transport (with a certain notion of trivial).

One would expect that “a certain morphism” means “an equivalence”.

This is a very interesting issue. I, too, did expect in the beginning that I want an equivalence. But after working out various examples, it turned out that in fact what is really needed in all of this business are special ambidextrous adjunctions.

This is a concept slightly weaker than that of equivalence, in fact. But every equivalence can be turned into an adjoint equivalence, which is a special case of a special ambidextrous adjunction. Hence nothing is lost by passing from equivalences to special ambidextrous adjunctions.

Moreover, in the crucial motivating examples, like that of line bundle gerbes, the trivialization morphism $t$ that we naturally want to choose, happens to be an adjoint equivalence, anyway. Hence everything works as expected.

The necessity of working with special ambidextrous adjunctions in this business arises when we want to construct the ($n$-)categories of local trivialization data ${\mathrm{Triv}}_{i,p}$ and that of local transition data ${\mathrm{Trans}}_{i,p}$ of transport $n$-functors, as well as the forgetful functors

(For instance, a bundle gerbe with “connection and curving” is nothing but an object in ${\mathrm{Trans}}_{i,p}$, for suitable choice of $i$ and $p$.)

If you look at the later pages of my notes, you’ll see that in order for this functor to exist we do need to make use of the zig-zag identity of the special ambidextrous adjunction various times.

This is why it is necessary to demand the trivialization morphism to fit into a special ambidextrous adjunction.

And it turns out to also be sufficient. In all general considerations and all examples that I have looked at so far, I have never encountered the need to demand that the trivialization morphism is more than a special ambidextrous adjunction. Though in some cases it of course happens to be.

That’s the technical necessity. There is also an interesting consequence of allowing local trivializations which are not necessarily adjoint equivalences, but may be just special ambidextrous adjunctions.

Namely, the monad induced by a special ambidextrous adjunction is precisely a special Frobenius algebra (or “Frobenius monad”, if we are not $\mathrm{Vect}$-enriched).

That’s in fact where the “special” in “special ambidextrous adjunction” comes from. For Frobenius algebras, the corresponding terminology “special” for the respective property is well established. (Even though it is, I think, not a very good choice of terminology, because it is not suggestive enough. But since it is established…)

What is interesting about this, is that it implies that locally trivializing a 2-transport on a surface amounts to (dual-)triangulating that surface and labeling the edges by a special Frobenius algebra, and the vertices by products and coproducts of that special Frobenius algebra.

More precisely, it’s something like a Frobenius algebroid, because the product may change from point to point.

For ordinary abelian bundle gerbes, for example, this product operation at the point $x$ is what is usually called $f_{\mathrm{ijk}}(x)$ (or $g_{\mathrm{ijk}}(x)$) . One does not really notice the Frobenius structure here because it is so trivial: the product $f_{\mathrm{ijk}}$ is in fact an isomorphism in this case.

But there are examples of 2-transport of kinds that are very different from what has been considered in the context of gerbes. For some of them, the Frobenius algebra structure of the transitions on triple overlaps plays a crucial role.

One such application is the description of certain constructions in 2-dimensional conformal field theory in terms of transport 2-functors. There, the special Frobenius algebra coming from the special ambidextrous adjunction of the trivialization morphism is nothing but the operator product algebra of open string states. If you like, you can find more information on that in my recent comment on the FRS formalism.

A quick way to see the basic mechanism here is to look at the special case where the 2-dimensional CFT degenerates to a topological field theory. For this case, one can see that the formalism discovered by Fukuma-Hososo-Kawai (described here) is just a special case of a local trivialization of a certain vector 2-transport functor. I have some notes with more details on that, too.

Fukuma-Hosono-Kawai postulate that we should (dual-)triangulate surfaces and label the edges by a semisimple algebra and the vertices by product and coproduct. We may regard this procedure as writing down a certain flat vector 2-transport in terms of local transition data. The fact that the local transition comes from a special ambidextrous adjunction then implies the Fukuma-Hosono-Kawai prescription.

how about commenting on $i$-trivial for $T\prime$ the one object, one morph cat? as in ordinary bundle speak?

I recall for other readers what this question is about.

We are talking about $(n-)$ functors that we write as

Given a morphism of codomains

I want to call the functor $\mathrm{tra}$ $i$-trivial if it factors through $i$, i.e. if there is

such that

(And I want to say $\mathrm{tra}$ is $i$-trivializable if it not equal to $i_{*}{\mathrm{tra}}_{T\prime }$, but related by a certain morphism).

A standard example is this:

let $\Sigma (G)$ be some group $G$, regarded as a category with a single object. Let

be the functor which sends the single object of $\Sigma (G)$ to the group $G$, regarded as a torsor over itself, and which sends every group element to itself, but now regarded as a torsor morphism of $G$.

Let $X$ be some space and let $P\to X$ be a principal $G$-bundle with connection over $X$. This connection gives rise to a parallel transport functor

where here $P(X)$ denotes the groupoid of thin-homotopy classes of paths in $X$.

Now, what does it mean for $\mathrm{tra}$ to be $i$-trivial? Evidently, it means that for every $x\in X=\mathrm{Obj}(P(X))$, we have $P_x=\mathrm{tra}(x)=G$.

But this simply means that $P$ is a trivial $G$-bundle in the ordinary sense!

That’s the idea.

So what happens if we let $T\prime$ be the category with a single object and the only morphism being the identity on that?

In the above example, this would correspond to setting $G$ equal to the trivial, 1-element group. Accordingly, in this case an $i$-trivial transport would be “completely trivial”, in that it not only associates the typical fiber (the trivial group) to every point, but also the identity morphism from the trivial group to itself to each path in $X$.

A similar conclusion holds no matter which morphism $i$ we choose with domain the one-object, one-morphism category. In such a case an $i$-trivial transport must, by definition, send every path in $X$ to the identity morphism on a fixed object in $T$.

from your point of view in TTT or elsewhere how does the local to global bit go?

We are still working on writing this out in detail.

The idea is this:

As I discuss in TraTriTra, we have functors

which send globally defined transport functors first to their local trivialization data and then to the transition data obtained from that.

The question you are asking is whether this is an equivalence - whether we can construct a functor

which sends a bunch of local trivial transport functors related by transition data on $n$-fold overlaps to a single globally defined transport functor.

This should be possible precisely if the global functor admits a proper local trivialization.

I spell out some (but not all) details on that idea in the examples of this text.

The crucial ingredient to make this work is the 2-category of 2-path in the Čech 2-groupoid corresponding to a good covering $U\to X$.

This Čech 2-groupoid has as objects points $(x,i)$ in the good covering, has 1-morphisms generated from unique morphisms

between points whenever these project to the same point in $X$, and has unique 2-morphisms filling any triangle of these elementary 1-morphisms.

The 2-category of 2-paths in this 2-groupoid is that generated from 2-paths in $U$ combined with the Čech 2-morphisms.

Now, a globally defined transport functor gives rise to a transport on this 2-category of paths in the Čech groupoid precisely if it admits a proper local trivialization factoring through the latter.

On the other hand, from an object in ${\mathrm{Trans}}_{i,p}$ we also obtain a 2-functor on 2-paths in the Čech 2-groupoid by assigning local transport and transition data in the obvious way to the 2-morphisms.

The claim is that this way one gets an equivalence

whenever $p$ admits a proper local trivialization.

But this hasn’t been written down in detail yet.

For 2-paths, you consider only ‘surfaces’ between paths with same source and same target. Why not ordered or oriented 2-simplices

Nothing stops me from considering such a case. It all depends on which application one has in mind.

In some situations, for instance, like when one is thinking of categorifications of splittings of the Atiyah sequence, it is desireable to have

- not strict 2-functors from thin-homotopy classes of 2-paths $I^2\to X$

- but pseudofunctors from the pair groupoid of $X$ (or the fundamental groupoid, more precisely).

A pseudofunctor on the fundamental groupoid of $X$ with values in some $n$ category $T$ is pretty much what you describe above:

first of all, it associates 1-morphisms to pairs of points in $X$.

This will respect composition only up to a “compositor”. But this compositor is nothing but an assignment of a 2-morphism in $T$ to triangles

in $X$.

For $n=2$, the compositor will satisfy a coherence law on tetrahedra. For $n>2$ this law is replaced by a sort of associator, which is an assignment of a 3-morphism to tetrahedra.

And so on.

I think for $X$ a smooth manifold one can show that a strict 2-functor from thin-homotopy classes of 2-paths in $X$ to a strict 2-group is the same as a pseudofunctor on the fundamental 1-groupoid with values in that 2-group.

So here it is a matter of which of two equivalent descriptions one happens to find more convenient.

But I think the point of view of pseudofunctors with domain the fundamental groupoid of $X$, which leads to the simplicial assignments which you were asking about, is indeed the more general and more generalizable point of view.

So I completely agree that it is important to look at the case you were asking about. All I can offer in this direction at the moment, though, are these very brief notes.

in re: the motivating toy example

(1)$$i:\Sigma (\mathrm{End}({ℂ}^n))\to {\mathrm{Vect}}_{ℂ}$$

is required to have image in the sub cat in which objects are n-dim vector spaces?

Yes. The idea here is simply to emded the endomorphisms of ${ℂ}^n$ into the category of $ℂ$-vector spaces.

You could in principle consider other morphisms $i$ than this obvious embedding. But if we want to understand ordinary vector bundles from the point of view of locally trivialized $n$-transport, then you would want to choose the canonical embedding.

why in the vector bundle case will any connection do while for the gerbes it needs to be fake-flat? (why do we need to retain “fake”?)

That’s a very interesting question.

What is fake curvature, really?

The general answer can very nicely be understood in terms of $n$-algebroid morphisms.

Given a morphism of Lie $n$-groupoids, like our transport $n$-functor is an example of

we can differentiate and obtain a morphism of the underlying Lie $n$-algebroids

Morphisms of $n$-algebroids can conveniently and equivalently be thought of as morphisms of the corresponding Koszul dual free differential algebras.

John explains the general theory behind this fact here. Worked examples relevant for the discussion of $n$-transport can be found elsewhere #.

Anyway, thinking of algebroid morphisms like

in terms of morphisms of free differential graded algebras makes the nature of “fake curvatures” very manifest.

Namely, such a morphism of FDAs is, first of all, a chain map (between the complexes of vector spaces corresponding to the graded differential algebras corresponding to ${\mathrm{dP}}_n(X)$ and $\mathrm{dT}$).

By definition of chain maps, these have to be assignments that make lots of little squares commute

For ${\wedge }^{•}A$ the deRham complex, one finds that the failure of the $p$-th square to commute is measured by a $p$-form with values in $p$-morphisms of the target $n$-algebroid.

This $p$-form is the $p$-form curvature of the connection represented by the algebroid morphism.

More precisely, only the top form is what is ordinarily known as the curvature form.

For $n=1$ (ordinary bundles) there is no intermediate curvature form, just the top level curvature 2-form $F_A=\mathrm{dA}+A\wedge A$.

For $n=2$, however, there is the top level curvature 3-form

but there is now in addition the intermediate curvature 2-form

This “intermediate curvature form” has been called (in the paper by Breen and Messing, where it arises by a very different line of reasoning), the “fake curvature” - to distinguish it from the “true” (= top level) curvature $H$.

Since, as I said, this $\beta$ measures the failure of the first of the above squares to commute, and since these squares are required to commute in order to describe a morphism of free differential graded-commutative algebras, it follows that $\beta$ has to vanish!

This is what I call “fake flatness”. It is to be distinguished from true flatness, which would be $H=0$.

In short: the constraint of “fake flatness” is precisely the constraint that ensures that a morphism

respects the $n$-algebroid structure - or, equivalently in the integrated picture, that the morphism

does respect the $n$-groupoid structure, i.e. that it really is an $n$-functor.

The description of fake curvatures in terms of algebroid morphisms has in particular been emphasized by Thomas Strobl, for instance in his hep-th/0406215.

If you ever encounter a fake flatness constraint which is too strict for your needs, you should consider passing from the target $n$-category $T$ to the $n+1$-category $\mathrm{AUT}(T)$. I describe this here and here.

Deligne cocycles occur for all $n$ or only $n+2$? Same question for nonab gerbes with connection

At the level of finite $n$-transport, i.e. at the level of $n$-functors from $n$-path groupoids, I have checked this up to Deligne 3-cocycles. The explicit computation is here:

Deligne 3-Cocycles and 2-Transport Transition

This document contains a discussion how the category of transitions of 2-transport functors on some $X$ with values in the 2-group $\Sigma \Sigma (U(1))$ is, when equivalence classes are taken, nothing but 3rd Deligne cohomology of $X$.

This certainly goes through for all $n$, but drawing the diagrams for morphisms and morphisms-of-morphisms of $n$-functors with $n>2$ becomes very cumbersome.

But there is a shortcut for how to do it. Instead of talking about smooth morphisms of Lie $n$-groupoids, we may talk about morphisms of Lie $n$-algebroids.

In this differential picture it is rather easy to check that transitions of $n$-functors with values in ${\Sigma }^{n+1}(U(1))$ yield $(n+1)$st Deligne cohomology. This is discusses elsewhere #.

In principle, the same argument holds also for the nonabelian case. The only problem here is that the nonabelian version of Deligne cohomology which one gets by differentiating transitions of $n$-transport with values in nonabelian $n$-groups is (as is to be expected from a differential description) restricted to linear order in the transition data.

However, nonabelian cocycles tend to be quadratic and even higher order in the transition data. For instance, where the cocycle condition for abelian gerbes reads

that for nonabelian gerbes reveals a certain twist

The term $g_{\mathrm{ij}}(f_{\mathrm{jkl}})$ is second order on the transition data $g_{\cdot \cdot }$ and $f_{\cdot \cdot \cdot }$. Hence a linearized treatment (like with morphisms of algebroids) won’t see the contribution by $g_{\mathrm{ij}}$ here.

what’s (a reference for) orientifolding? does it mean and oriented orbifold?

String theorists have invented the word orientifolding for an operation where you consider a 2-dimensional $\sigma$-model of maps

with target a space $X$ on which some finite group $G$ acts; and then divide out by a combined operation where you act with $G$ on $X$ and by orientation reversal on $\Sigma$.

So the word “orientifolding” is a combination of “orbifolding” (namely globaly orbifolding on a target space $X$) and “orientation reversal” (namely on a 2-dimensional parameter space $\Sigma$).

As usually defined, it is an operation acting on the space of 2-dimensional quantum field theories with certain target spaces.

What I was trying to say is that, when you restrict attention to that part of the $\sigma$-model which just involves the surface holonomy of the gerbe of the Kalb-Ramond field over the worldsheet of the string, then there is a nice definition of “orientifolding” simply in terms of a $G$-equivariant structure on that gerbe.

One way to understand this is to realize that an abelian gerbe (2-bundle) with structure 2-group that coming from the crossed module

should really be thought of as having the structure 2-group

which is

From this point of view it becomes clear that a $G$-equivariant structure on an abelian gerbe may involve, in addition to the usual data considered in the study of discrete torsion, certain data that takes values in the nontrivial automorphism of $U(1)$. It turns out to be this additional freedom which implements the “worldsheet orientation involution” for unoriented strings coupled to the Kalb-Ramond field.

It might be noteworthy that the 2-group $U(1)\to {\mathbb{Z}}_2$ is not braided. It is indeed non-abelian. (Both $U(1)$ and ${\mathbb{Z}}_2$ are of course abelian groups, but the semidirect product $U(1)⋉{\mathbb{Z}}_2$ is not, which makes the 2-group $U(1)\to {\mathbb{Z}}_2$ non-abelian).

This has the maybe remarkable consequence that we have hence identified a gerbe appearing in (string-)physics which is indeed (even if only “slightly”) non-abelian.

Unoriented (type I) strings do couple to a nonabelian 2-bundle with 2-connection.

Since the non-abelianness here is only very mild, one can manage to ignore this fact by adding additonal signs to the theory of bundle gerbes “by hand”. But a systematic understanding of these additional signs is available in terms of equivariant structures on nonabelian $(U(1)\to {\mathbb{Z}}_2)$-2-bundles.

the product operation for abelian gerbes you denote $f_{\mathrm{ijk}}$ - do you mean the structure function so really $f_{\mathrm{ij}}^k$ in terms of a basis or do you mean the fudge factor I am more familiar with as $c_{\mathrm{ijk}}$ which is a 2-cocycle?

The transition functions of an abelian gerbe, which are $U(1)$-valued functions $f_{\mathrm{ijk}}$ on triple overlaps $U_i\cap U_j\cap U_k$ (satisfying some condition) can indeed be understood as defining an associative product operation

(something like an algebroid product, in fact).

This doesn’t mean that $f_{\mathrm{ijk}}$ denote the structure constants of an algebra.

The best I can do to explain in more detail what I really mean is to refer to the explanation of a general transition of a 2-transport, as given in TraTriTra, which clearly does involve a product operation (in fact a monad coming from an ambidextrous adjunction), and to the fact that Deligne 3-cocycles can indeed by understood as a special case of that #.

The construction of the global from the local trivializations and transitions is indeed what Wirth makes an axiom for his ‘fibration theory’.

The point you make here, or rather the problem you are addressing, is one which I only have partial answers to.

It is a problem that has concerned me for quite a while now. Personally I feel I have made a little progress here and there, but certainly not of the general sort that one would hope for.

In fact, after you pointed me to Wirth’s work, I immediately wondered what - or if - this might tell us about the reconstruction of any 2-bundle, 2-fibration, 2-transport, or gerbe, etc. from its local trivializations.

How can we categorify ! a mapping cylinder?

Would we need to do that? My understanding was that Wirth’s result already applied to the $n=2$ case, only that it isn’t formulated in the world of categories, but in the world of topological spaces.

It seemed to me - but please correct me if that’s wrong - that in order to understand what Wirth’s construction has to say about 2-bundles, one would need to see if his result could be understood as applying to the topological space obtained by taking something like the realization of the nerve of a 2-bundle.

This is a problem that I don’t know the answer to. There is however something closely related which I think I do understand. I’ll talk about that in the following.

Namely, looking at parallel transport changes the perspective - by 180 degrees in a sense, and this makes the problem appear in a somewhat different light.

What I mean is this, illustrated for the simple case of principal (1-)bundles with connection:

A principal 1-bundle is, of course, a map

with certain properties. Remarkably, when we put a connection $\nabla$ on that bundle and pass to its parallel transport, we instead obtain a morphism which goes the other way:

Now, the morphism $\mathrm{tra}$ going from right to left contains the same information as the morphism $p$ going from left to right (together with the connection $\nabla$). Still, the total space $B$ is only indirectly visible from the point of view of ${\mathrm{tra}}_{\nabla }$.

In fact, we can forget the smooth structure on the transport groupoid $B{\times }_{g}B$ and consider ${\mathrm{tra}}_{\nabla }$ as a morphism

We can do so if we agree that we call a transport functor with codomain a category not carrying a smooth structure smooth iff it has local transition data that is smooth. (I guess you can analogously replace all occurences of “smooth” here by “continuous”, using the relevant formalism.)

But if we make that agreement, then there is a way to “reconstruct” ${\mathrm{tra}}_{\nabla }$ from its local transition data which nicely categorifies.

I’ll now explain what I have in mind here (this is related to what I called “proper” local trivializations.):

Say we have chosen a good covering

of $X$ by open contractible sets. Say we have trivialized ${\mathrm{tra}}_{\nabla }$ in the obvious sense over each $U_i$, obtaining on each $U_i$ a functor

From composing on double overlaps the de-trivialization on $U_i$ with the re-trivialization on $U_j$ we obtain the transition functions $g_{\mathrm{ij}}$.

Now consider the groupoid of paths within the groupoid defined by $U$

This is, by definition, the groupoid whose objects are points $(x,i)$ of $U$, and whose morphisms are generated from

- smooth paths in $U$

- and the unique morphisms of the form $(x,i)\to (x,j)$.

Two things are interesting about this groupoid $P_1(U)$:

1) the collection of local trivial functors ${\mathrm{tra}}_i:P_1(U_i)\to \Sigma (G)$ together with the transition data $g_{\mathrm{ij}}$ naturally yields a functor

which acts on the generators of the morphisms of $P_1(U)$ in the obvious way.

2) we can always find a lift

by taking any path in $X$, decomposing it into many small paths if necessary, lifting each of the small paths to one of the $U_i$ and inserting the morphisms $(x,i)\to (x,j)$ whenver two neighbouring pieces of path get sent to different patches.

The point is that by composing these two morphisms, we do get again a globally defined functor

from $P_1(X)$ to $G\mathrm{Tor}$.

This functor is indeed isomorphic to the functor

that we started with.

Hence - from the point of view of the “right to left”-picture of parallel transport, the above procedure provides a perfectly satisfactory notion of reconstructing a bundle with connection from its local transition data.

And everything I said so far has a more or less obvious categorification to 2-transport.

So in this sense (which I used to refer to as proper local trivialization) one can in fact reconstruct every parallel 2-transport from its local transition data.

On the other hand, this might not be quite the sense one is looking for. Nowhere in this reconstruction of the transport functor $\mathrm{tra}$ from its local trivializations do I explicitly consider the total space $B$.

That works, because from the point of view of transport the total space (total $n$-bundle) is not a primary object.

So we can say that the difficult problem is not to reconstruct a globally defined functor with values in $G\mathrm{Tor}$ from the local trivializations of a given $\mathrm{tra}$, but to reconstruct a globally defined functor which is in addition required to factor through the transport ($n$-)groupoid of some ($n$-)bundle.

I have no good idea about the general answer to this more difficult question. All I know is one special example # where I think I know the answer.