The n-Category Café

Very interesting question. I have thought about aspects of this before, but the way Bruce puts it rightly indicates that this deserves to be understood more systematically.

In essence, Bruce points out that a rather elegant arrow-theoretic formulation of differential K cohomology # - and, in fact, of higher versions of differential K-theory (which is for instance expected to include “differential elliptic cohomology” of sorts #) - is almost forced upon us by the transport # point of view on $n$-vector bundles with connection.

As recalled above already, a topological or smooth vector bundle with connection on $X$ is a continuous or smooth functor

Here I define such a functor to be continuous or smooth precisely if it has a continuous or smooth local $i$-trivialization, for $i$ the embedding

This is a recursive definition. I can provide the details if desired, but the idea is to iteratively locally identify diagrams in ${\mathrm{Vect}}_{ℂ}$ with those in $\Sigma (M_n(ℂ))$, and then use the obvious topological and smooth structure on $\Sigma (M_n(ℂ))$ to define continuity and smoothness of the respective maps taking values in it.

I am guessing that this way of defining continuous/smooth $n$-functors by the property of having continuous/smooth local trivializations is more or less equivalent to the anafunctor way to describe this, invented by Toby Bartels (slide 7 here gives the rough idea, more details in Toby’s thesis). But I haven’t tried to check.

Anyway. One can rather easily see that of course even more is true: morphisms of vector bundles with connection are precisely morphisms in the category

of continuous or smooth functors from paths to vector spaces.

Hence classes in differential K-cohomology (like K-theory #, but for vector bundles with connection) are precisely isomorphism classes in $[P_1(X),{\mathrm{Vect}}_{ℂ}]$.

I think that much is clear.

Bruce’s point is that the way of looking at differential K-theory as

should allow us to apply a rather beautiful theorem - the categorified Gelfand-Naimark # theorem.

As he indicates, this theorem, applied to the present setup, should reduce to a statement about which properties of $X$ can be re-obtained from just knowing $[P_1(X),{\mathrm{Vect}}_{ℂ}]$.

Indeed, somebody should look at that.

Here I shall be content with adding two related observations:

Once we start looking at differential K-cohomology as $[P_1(X),{\mathrm{Vect}}_{ℂ}]$, there is no stopping us - we will want to study $n$-(differential K-cohomology)

For instance for $n=2$ we might # take $2{\mathrm{Vect}}_{ℂ}$ to be

But before getting into that, it pays to quickly follow John’s advice on how to get definitions right, and check that what we are doing is natural for $n=0$.

For $n=0$ we are looking at 0-transport

Given that usually $n{\mathrm{Vect}}_{ℂ}\simeq {}_{(n-1){\mathrm{Vect}}_{ℂ}}\mathrm{Mod}$ we find that

Hence a 0-vector bundle with connection is a (continuous or smooth) function from $X$ to $ℂ$.

That’s precisely what the ordinary Gelfand-Naimark theorem # applies to.

So our definition is indeed good, even in the degenerate case.

The other remark I would like to add, in closing, is that we can of course also handle ordinary differential cohomology # (differential integral cohomology, if you like) along the above lines. Bruce has already mentioned that, too.

In order to do so, we simply need to change the target $T$ for our transport functor. Instead of general $n$-vector bundles, we use ${\Sigma }^n(U(1))$ principal bundles with connection - which we can think of as $n$-line bundles with connection.

These are $n$-functors (continuous or smooth)

such that they have a full local ${\Sigma }^n(U(1))$-trivialization.

Equivalence classes in the $n$-category of these functors are # nothing but Deligne ($n+1$) cohomology - or equivalently, Cheeger-Simons differential character # cohomology .

Jones, Cohen and Segal figured out how to associate a topological category $M_f$ to a manifold $M$ equipped with a Morse function $f$:

• Ralph L. Cohen, John D. S. Jones, Graeme B. Segal, Morse theory and classifying spaces.

The objects of $M_f$ are just the critical points of $f$, while the morphisms between two critical points are the flow lines which connect them.

Interesting. This is the same procedure by which conformal surfaces are decomposed in the context of Hilbert uniformization of moduli space of Riemann surfaces.

In that context people only ever seem to realize the graph formed of critical points and flow lines for some reason. (A certain hostility towards categories might play a role here and there.)

I have proposed # to take the (2-) category of critical points, flow lines (and faces between these) as the domain 2-category for the 2-transport describing 2D conformal field theory. But I haven’t gotten around making this precise. I should have a look at the Cohen-Jones-Segal paper.

Maybe this suggests a way to solve the question that came up recently:

What is a holomorphic $n$-bundle with connection?

Maybe the answer is: An $n$-functor with domain the category $M_f$ - where $f$ is required to be a holomorphic Morse function.

Among the technical points of applying the categorified Gelfand-Naimark theorem to transport functors $$P_1(X)\to {\mathrm{Vect}}_{ℂ}$$ is the presence of gradings.

First of all, we will want to regard hermitean vector bundles, hence really look at $$P_1(X)\to {\mathrm{Hilb}}_{ℂ}.$$ Next, we can nicely do away with the need to form a group completion of $([P_1(X),{\mathrm{Hilb}}_{ℂ}],\oplus )$ by allowing ${\mathbb{Z}}_2$-graded Hilbert spaces $$P_1(X)\to {\mathrm{SuperHilb}}_{ℂ}.$$ That’s a simple standard trick, of course. We identify the virtual vector space $$(V,W)$$ with the graded vector space $$V\oplus W$$ with $V$ in degree 0 and $W$ in degree 1.

So that’s how K-theory makes us want to superize the codomain of our transport.

But then, in order to have a chance of applying Baez’ theorem, we also need the domain to be super. More precisely, to be a supergroupoid in the sense of def. 62:

Definition. A (“$N=1$”) supergroupoid $(G,\beta )$ is a groupoid $G$ equipped with an involutive automorphism $$\beta :{\mathrm{Id}}_G\to {\mathrm{Id}}_G,$$ i.e. $\beta \circ \beta ={\mathrm{Id}}_{{\mathrm{Id}}_G}$.

But the ordinary path groupoid $P_1(X)$ is a supergroupoid only for the trivial choice of $\beta$.

I’d need to have a closer look at the details to have a chance of knowing if this is an issue or not. Maybe we do want to use that trivial involution. But I feel a little worried about that.

In any case, it might be interesting to try to enhance $P_1(X)$, such that we do get a nontrivial involution.

There is a universal such enhancement:

Definition. Let $$P_1(X,b)$$ be the groupoid which is generated from $P_1(X)$ together with a collection of morphism $$\{\left(x\stackrel{b_x}{\to }x\right)\mid \forall x\in X\}$$ modulo the relations $$x\stackrel{b_x}{\to }x\stackrel{b_x}{\to }x=x\stackrel{Id}{\to }x$$ for all $x\in X$, as well as $$\begin{array}{ccc}x& \stackrel{\gamma }{\to }& y\\ b_x\downarrow & =& \downarrow b_y\\ x& \stackrel{\gamma }{\to }& y\end{array},$$ for all $x\stackrel{\gamma }{\to }y$ in $P_1(X)$. The involution $$b:{\mathrm{Id}}_{P_1(X,b)}\to {\mathrm{Id}}_{P_1(X,b)}$$ is the obvious one, with $$b:x\mapsto (x\stackrel{b_x}{\to }x).$$

Notice that $P_1(X,b)$ is essentially the path groupoid associated to a “stack” (in the sense of physicists) consisting of a brane $X$ and its antibrane $\overline{X}$ in the functorial formulation of bundles with connections on stacks of branes #.

And that should make good sense, since also virtual vector bundles $(E,E\prime )$ are well known # to be interpretable as an ordinary vector bundle $E$ on a stack of branes and an ordinary bundle $E\prime$ on a stack of anti-branes.

So I am suggesting that, following Bruce Bartlett’s observation, we might maybe want to look at the category $$[P_1(X,b),{\mathrm{SuperHilb}}_{ℂ}]$$ instead of the more naïve $$[P_1(X),{\mathrm{SuperHilb}}_{ℂ}].$$

Be that as it may, using $P_1(X,b)$ one finds a fun simple way to express the basic idea of spacetime supersymmetry, namely that

a supertranslation is a “square root” of a translation

in an arrow-theoretic way:

In $\mathrm{Mor}(P_1(X,b))$ distinguish those morphisms that are of the form $$x\stackrel{\gamma }{\to }y$$ and those that are of the form $$\begin{array}{ccc}x& \stackrel{\gamma }{\to }& y\\ b_x\uparrow \\ x\end{array}.$$ Let the former have grade 0 and the latter have grade 1. Then composition of morphisms is an operation of grade 0, which sends in particular two grade 1 paths to a grade 0 path $$\begin{array}{rl}& \begin{array}{c}\\ & & y& \stackrel{\gamma \prime }{\to }& z\\ & & b_y\uparrow \\ x& \stackrel{\gamma }{\to }& y\\ b_x\uparrow \\ x\end{array}\\ \\ \\ =& \begin{array}{c}\\ x& \stackrel{\gamma }{\to }& y& \stackrel{\gamma \prime }{\to }& z\\ b_x\uparrow \\ x\\ b_x\uparrow \\ x\end{array}\\ \\ \\ =& \begin{array}{ccccc}x& \stackrel{\gamma }{\to }& y& \stackrel{\gamma \prime }{\to }& z\end{array}.\end{array}$$

It’s not particularly deep - but sort of nice. Maybe one can get an arrow-theoretic handle on supersymmetric physics in such a manner.

With regard to that, I just remark that when describing K-theory in terms of quantum mechanics, one also needs supersymmetric quantum mechanics #.

Thanks for these informative comments. I’ve had some TeXnical problems with this post, so please excuse its untidy nature.

John wrote:

A smooth functor of this type is precisely a smooth connection on the trivial principal G-bundle over M - this result is proved here in “Higher gauge theory II: 2-connections”, draft version in the section on “Connections on smooth spaces”.

The definition of “smooth spaces” given in this draft is fascinating - I have never encountered it before. I will bounce it off some of the folks here at my department. I must say, the whole paper is looking great…

Urs wrote:

Bruce’s point is that the way of looking at differential K-theory as [P 1(X), Vect] should allow us to apply a rather beautiful theorem - the categorified Gelfand-Naimark theorem.

Yeps! Except it should be emphasized that one needs the full information of the category of vector-bundles-with-connection in order to recover P_1(M). One can’t just use the isomorphism classes, in the same way that one can’t recover a group from its isomorphism classes of representations - but you *can* recover it abstractly from its representation category, by applying Spec.

Ultimately, the slogan should be:

Question : What information about X can you recover from the vector-bundles-with-connection which live on it? Answer : The path groupoid P_1(X).

One problem I’ve subsequently noticed though, is that for the categorified Gelfand Naimark theorem from HDA II to work, one needs to start with groupoids whose hom-sets are compact topological spaces… and I doubt $P_1(M)$ has this property.

However, in the ordinary Gelfand-Naimark theorem, one can remove the need for compactness by passing to a slightly different construction which works for locally compact spaces. Perhaps this can be done in the categorified setting too.

Urs wrote:

Interesting. This is the same procedure by which conformal surfaces are decomposed in the context of Hilbert uniformization of moduli space of Riemann surfaces.

Indeed! That is certainly something worth thinking about…