The n-Category Café

I think I want to work inside the $(n+1)$-category whose

• - objects are $n$-categories equipped with a sub-$n$-graph of their underlying $n$-graph of $n$-morphisms
• - 1-morphisms are $n$-functors that respect these sub-$n$-graphs

Probably I am thinking of strict $n$-categories here and take an $n$-graph to be the same as an $n$-category but without any rules for composition.

Of course, the choice of sub-$n$-graph is supposed to describe a sub-collection of those $n$-morphisms that have “a small extension”.

Being the physicist that I am, I have a couple of immensely naive examples for this in mind for illustrating (and motivating) all of my constructions below. I am hoping, though, that what I do here is formulated in a manner general enough to accomodate sophisticated topos-theoretic technology as used in synthetic differential geometry - even though this tends to scare me, admittedly.

Mostly, I allow myself to be immensely naive and usually think of categories that are freely generated from their graph of small morphisms. That’s like what you’d be content with for applications in lattice field theory.

But I am hoping that there are good implementations in between this very crude model and the very high-brow technology. For instance: let $X$ be a $d$-dimensional Riemannian manifold. Let $\epsilon \in \mathbb{R}$ be a small real number such that all points $x \in X$ have an $\epsilon$-ball around them in which geodesics through $x$ provide good coordinates. Choose an orthonormal system of $d$ vector fields in $TX$. Now identify in the category of thin-homotopy classes of paths in $X$ the subgraph consisting of all those geodesic paths that have length $\epsilon$ and are tangent in their source point to one of the chosen orthonormal vectors.

Presumably, I should now make a definition like this:

Definition. Denote by $d^r$ for $r \in \mathbb{N}$ the $r$-category obtained from the smallest $r$-fold category with a single nontrivial $r$-morphism.

The first three of these look like

Let $P X$ be my default symbol for some $n$-category with specified sub-$n$-graph.

I should then say that a tangent $p$-vector to some point in $P X$ is an $n$-functor

It should be noteworthy that a morphism

in $[d^p,P X]$ looks like a $p$-dimensional cobordism in $PX$ which starts tangential to $v$ and ends tangential to $v'$. For QFT applications it seems # we want to turn $p$-transport functors on globular $p$-paths in $P X$ into “generalized phases” on $p$-dimensional cobordisms, and it looks to me that it’s the structure of morphisms in $[d^p,P X]$ which makes this happen naturally.

And in fact, there is now nothing more immediate than

Definition. For $T$ a $p$-category, a $T$-valued $p$-form $\omega$ on $PX$ is a $p$-functor

In particular, every $n$-transport $n$-functor

gives rise to a $T$-valued $p$-form on $P X$

We will want to have an idea of wedge products and exterior differentials of $T$-valued $p$-forms. I am not yet completely sure about the general definition. So I start by giving a crucial motivating example:

Let $\tilde \mathrm{Vect}$ be the 2-category which has finite-dimensional hermitian vector spaces as objects, has linear maps between these as morphisms, and has a unique 2-morphisms for every ordered pair of 1-morphisms (for reasons described here). We should think of these 2-morphisms as labeled by the source morphisms and the difference of the target and the source morphism

Let the 1-functor

be a hermitian vector bundle $E$ with connection. We might address $d\mathrm{tra}$ as the corresponding $\tilde \mathrm{Vect}$-valued connection 1-form on $X$.

Now let

be an endomorphism-valued 0-form on $X$. We may “wedge” $d\mathrm{tra}$ with $e$ in order to get the covariant derivative 1-form of $e$ with respect to $d\mathrm{tra}$:

Here the right hand side is supposed to denote the functor which sends $d^1 = (a \to b)$ to the morphism $E_x \to E_y$ given by

Notice that this does indeed yield the ordinary covariant exterior derivative to linear order, when $x\to y$ is a short path in an ordinary manifold.

In a similar fashion, we can “wedge” $d\mathrm{tra}$ with itself, to obtain a $\tilde \mathrm{Vect}$-valued 2-form, the curvature 2-form of $\mathrm{tra}$:

This, too, does indeed yield the familiar curvature 2-form in cases where it makes sense.

What is important here is that the (covariant) exterior derivation involves something like a graded commutator of a $1$-form with a $p$-form. This is just the first example of a general pattern #, which needs to be systematized.

Also, there is a nice relation to Quillen’s notion of superconnections here #.

For nice $T$, our $T$-valued differential forms will naturally be elements of a Hilbert space.

With target being hermitian vector spaces as in the above example, we should equip our space of $T$-valued 0-forms with a scalar product of the form

Similarly for our $\tilde \mathrm{Vect}$-valued 1-forms

The integral sign here denotes some operation of “summing” over all points of $X$, which I haven’t defined yet in general.

But, recall, in case of trouble I’ll always resort to the most naive situation: that of lattice gauge theory. Then $P X$ is just generated from a (finite) square lattice and the sum is literally a sum. And in this case for instance the scalar product

for the above example is the Lagrangian for lattice Yang-Mills theory coupled to a boson in the adjoint rep.

So much for today.

The basic concept I am describing here goes back to work I did a while ago with Eric Forgy. But at that time I was a category illiterate. Part of what I am doing here is trying to find the truly natural setting for these ideas.