The n-Category Café

Yes, your construction of $T_{n+1}$ is precisely analogous to the Koszul-Tate construction.

Oh, that’s interesting! I’ll have to think about this. Do you think this can be made precise?

Does the definition of covariant derivative come before parallel sections? Even though parallel transport comes first?

Right, so from the point of view of parallel transport functors, the logical order of some of these constructions is different from what one is ordinarily used to.

So given just the parallel transport $$\mathrm{tra} : \mathcal{P}_n(X) \to T$$ I can already say what flat sections are, namely morphisms into this $$e : \mathrm{tra}_o \to \mathrm{tra} \,.$$ I haven’t even introduced the covariant derivative at this point yet. But when one looks at what these constructions mean in detail, one sees that the existence of the morphism $e$ above is nothing but the “integrated” version of the usual differential condition $\nabla e = 0$.

This relation is then made explicit and precise by the realization that arbitrary sections come from parallel sections of the curvature transport $$(e,\nabla e) : \mathrm{curv}_o \to \mathrm{curv}_{\mathrm{tra}} \,.$$ Indeed, as a morphism of $(n+1)$-functors, this transformation is an $n$-functor on $\mathcal{P}_n(X)$ itself. As such, if smooth, it comes from $p$-form data (which encodes the differential description of the “integrated” $n$-functor). This $p$-form data is the covariant derivative of the section (by my definition for $n\gt 1$, and by inspection for $n=1$).

p.2 - The diagram with the diagonal double arrow says the diagram commutes up to the double arrow (I’ll think of it as a homotopy) but is something stronger implied by labeling it with ∼ $\sim$??

If you think homotopy anyway, then the $\sim$ doesn’t add anything. That symbol is supposed to indicate that the 2-arrow here is an equivalence, hence that it may be inverted up to higher stuff.

p.3 - $e : \mathrm{tra}_o \to \mathrm{tra}$ is defined for every $o$, so we really have $\mathrm{Obj}(T) \times \mathcal{P}_n(X) \to T$?

Hm, interesting point. I’ll think about this.

Right now, I’d fix one $o \in \mathrm{Obj}(T)$ once and for all and then consider flat sections with respect to this given $o$. For principal 1-bundles and choosing $o = G$ one obtains the ordinary notion. Same for vector bundles and $o = \mathbb{C}$.

If the vector bundle is rank $n$, one might be tempted to set $o = \mathbb{C}^n$. The a morphism $e : \mathrm{tra}_o \to \mathrm{tra}$ is not just one flat section, but $n$ of them. On the other hand, it then makes sense to ask if $e$ is an isomorphism, in which case it would trivialize the vector bundle.

So, different purposes are served by looking at flat sections for different $o$. I am not sure yet that I fully understand what this is trying to tell me. Somehow having to choose such an $o$ is a nuisance.

So maybe you have good point here, and I should be looking at all $o$ at once. I’ll think about this.

- what’s $T'$

Oh, a typo. Usually this denotes my “catgeory of typical fibers”, but here it’s a typo. I’ll correct that.

- $\nabla = d + A$

Well, that’s at least the idea. $\nabla$ denotes the covariant derivative associated with a connection. Locally and with $A$ acting suitably, it does look like $d + A$, yes. And $d$ is the just the ordinary exterior derivative.

Defn 1 - if there is an $n$-functor and a transformation (the diagonal one in the diagram) ??

Yes, that’s what I mean.

- except for one sentence in line 6 I see no need for weak cats

Right, you could do all this just for strict $n$-categories and everything else strict.

But for many applications that won’t be interesting enough. So we would want to have a notion of $n$-curvature that applies also to the weak case.

Right now I don’t address this in detail, really. I spell out the cases $n= 0,1,2$ where I consider all categories to be strict, all $n$-functors to be strict and all 2-transformations to be pseudo.

But from my first entry on curvature here at the $n$-café I know that the kind of 2-connection that Breen-Messing considered is really a flat 3-transport with values in $\mathrm{INN}(G_2)$ – and the latter is not strict but has a nontrivial isomorphism in the exchange law. I also know that for $G_2 = \mathrm{String}_k(G)$ this is intimately related to Chern-Simons transport.

And this means that I am interested in non-strict parallel transport. :-)

- If I’m right, a 0-transport is just a function $X \to T$,

Yes!

p.6 - I’m tempted to draw these diagrams to look more cubical either in the usual 3D perspective or the square within a square

Yes, that’s probably a good idea. There are a couple of ways one could defomr these diagrams without changing them, but possibly making them look more suggestive. I wouldn’t be surprised if it turned out that these diagrams are a special case of some general construction. My hope is that somebody will see them here and recognize them as what they really are.

But first I must master your pullback. It consists of pairs $(x_1,x_2)$ with both in the same connected component so this is a subthing of the usual $(x_1,x_2)(x_2,x_3) \to (x_1,x_3)$ ??

Yes, exactly. The groupoid here that appears for curvature of 0-transport is not the full pair groupoid of $X$, but the disjoint union of the pair groupoids of the connected components of $X$.

(And, by the way, is different from the fundamental groupoid of $X$, except when $X$ is simply connected.)

p. 7 - why $e^{f(x)}$ just to get $F_1 = df$?

Yes. First I didn’t have that. Then I realized that if I don’t assume the 0-transport to be non-vanishing in this example, I’d need to be more careful with the corresponding 1-curvature. So for simplicity I then just assumed that it is non-vanishing. Just to get the main point across.

homotopy classes of surfaces - so you really want cobordisms i.e. allow the surfaces to have handles as opposed to just disks?

Well, currently I am still thinking of $\Pi_n(X)$ as having as objects separate points (not disjoint unions of them), and having as $p$-morphisms $p$-surfaces cobounding their source and target $(p-1)$-morphisms. That impliees that everything here is $p$-disk-shaped.

But I believe this is not all that essential. A similar discussion of $n$-curvature should go through if I allow my domain $n$-category to be more like extended $n$-cobordisms in $X$.

But in that case the codomain must be able to mimick the required tensor product structures. So then we can no longer talk about principal transport anymore, but are restricted to vector transport.

You seem to be assuming that these pullbacks must exist

That’s true. One would need to think about under which conditions they actually do exist.

My point in these notes was that in the cases we care about, where these functors go from paths to group elements, for instance, the pullback happens to exist.

On the other hand, notice that the curvature of a functor the way I define it in Arrow-theoretic differential theory always exists, for every functor whatsoever.

And in those “cases of interest”, the result happens to coincide (up to some canonical identifications) with the result I get here using a pullback construction.

So, now I tend to think of the “arrow-theoretic exterior differential” of a functor as the general concept, and as the description in terms of pullbacks as a useful alternative point of view for certain applications.