### Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

#### Posted by John Baez

Here’s a question from Bruce Bartlett which really deserves to be a post of its own… it’s about path groupoids, the categorified Gelfand-Naimark theorem, and vector bundles!

Hi John,

Lately I’ve been thinking of loads of things, but I can’t get the following out of my head. If it would be more convenient for you to answer this question through the medium of the n-Category Café, then that would suit me fine.

Consider the path groupoid $P_1(M)$ of a manifold M. By forgetting the smooth structure on the space of objects, we can think of this as a topological groupoid, in the sense of HDA II: 2-Hilbert spaces. In other words, it’s a groupoid whose hom-sets are topological spaces.

But HDA II told us how to perform the categorified Gelfand-Naimark transform… take the category of representations of $P_1(M)$. What is this category? It’s `basically’ the category of vector-bundles-with-connection on M!

I say `basically’, because I understand that there are some technical differences between a vector-bundle-with-connection and a functor from paths into Vect… the latter should satisfy some smoothness conditions, as expressed in the Connections as functors homework of the Fall 2004 QG Seminar. On the other hand, HDA II explained that we could recover $P_1(M)$ as a topological groupoid by taking the Spec of the category of vector-bundles-with-connection on M!

All in all, it’s an intriguing picture: that the information in the topological groupoid of the space of paths on M is the same information as in the vector bundles with connection on M… I suppose this is obvious, but at least the categorified Gelfand-Naimark theorem makes it explicit.

It’s kind of a generalization of K-theory and the Cheeger-Simons group of differential characters at the same time. K-theory deals with `bare’ vector bundles (not equipped with a connection), while the Cheeger-Simons group deals with line-bundles-with connection.

A related question is: what is the classifying space of $P_1(M)$? Be warned - here I mean the classifying space construction which takes into account the topology on the hom-sets, not just the `bare-bones’ classifying space.

That gives an indication as to how much topological information is contained inside $P_1(M)$… but by now you’re tired of reading this, so I will explain below what I am getting to here.

Best,

Bruce Bartlett

P.S. 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. They proved that the classifying space of $M_f$ (the construction which sees the topology) is in fact homeomorphic to M, for a nice enough function f.

The point is that $M_f$ is very similar to $P_1(M)$, at least intuitively. Which is why I asked what the classifying space of $P_1(M)$ is.

I’ll have to ponder this question before writing a real answer. In the meantime, let me just say some random stuff. For example, what’s the difference between functors

$\mathrm{hol}: P_1(M) \to G$

for a manifold $M$ and Lie group $G$, and *smooth* functors

$\mathrm{hol}: P_1(M) \to G?$

A smooth functor of this type is precisely a smooth connection on the trivial principal $G$-bundle over $M$ - this result is proved here:

- John Baez and Urs Schreiber, Higher gauge theory II: 2-connections, draft version.

in the section “Connections on smooth spaces”. (Section numbers may change, so I won’t give those.)

An arbitrary functor of this type is almost the same as what people in loop quantum gravity call a “generalized connection”. The only difference is that in loop quantum gravity, people use a somewhat different version of $P_1(M)$, where the morphisms are piecewise-analytic or piecewise-smoothly-embedded paths (modulo reparametrization).

I think the quickest explanation of these generalized connections is in the Conclusions here:

- John Baez, Spin network states in gauge theory, Adv.Math. 117 (1996) 253-272.

It just takes two paragraphs; let me paraphrase it to eliminate some notation that was defined earlier in the paper:

Associated to any abstract graph $\phi$ in the sense of Section 2 there is a category $C_\phi$, or more precisely, a groupoid (a category in which all the morphisms are invertible). This is the free groupoid generated by the objects $V_\phi$ (the vertices of $\phi$) and the morphisms $E_\phi$ (the edges of $\phi$). If we fix a trivial $G$-bundle $P$ over $V_\phi$, the set $A_\phi$ of connections on $\phi$ consists precisely of functors from $C_\phi$ to $G$, where we regard the compact connected Lie group $G$ as a groupoid with one object. Similarly, elements of the set $G_\phi$ of gauge transformations on $\phi$ act as natural transformations between such functors. As we have seen, the set $A_\phi/G_\phi$ of connections modulo gauge transformations - or `functors modulo natural transformations’ - inherits the structure of a measure space from $G$, and Lemma 3 gives an explicit description of $L^2(A_\phi/G_\phi)$ in terms of the category of finite-dimensional unitary representations of $G$.

Similarly, given a real-analytic manifold $M$ and a smooth principal $G$-bundle $P$ over $M$, we may define the

holonomy groupoid$P_1(M)$ to have as objects points of $M$ and as morphisms equivalence classes of piecewise analytic paths in $M$, where two paths $\gamma,\gamma'$ are regarded as equivalent if they give the same holonomy for all connections $A$ on $P$. This has as a subgroupoid the `holonomy loop group’ of Ashtekar and Lewandowski [4]. If we fix a trivialization of $P_x$ for all $x \in M$, any connection on $P$ determines a functor from $P_1(M)$ to $G$, while conversely any such functor can be thought of as a `generalized connection’ [8,9]. Similarly, any gauge transformation determines a natural transformation between such functors, and any natural transformation between such functors can be thought of as coming from a `generalized gauge transformation’.

I wrote this before I thought about the categorified Gelfand-Naimark theorem… I like your idea of trying to clarify the relationships! Alas, I’m too tired to figure out them out tonight.

## n-(differential K theory)

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 $\mathrm{Vect}_\mathbb{C}$ with those in $\Sigma(M_n(\mathbb{C}))$, and then use the obvious topological and smooth structure on $\Sigma(M_n(\mathbb{C}))$ 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

anafunctorway 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 $[P_1(X),\mathrm{Vect}_\mathbb{C}]$.

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 $[P_1(X),\mathrm{Vect}_\mathbb{C}]$.

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 $[P_1(X),\mathrm{Vect}_\mathbb{C}]$, there is no stopping us - we will want to study

$n$-(differential K-cohomology)For instance for $n=2$ we might # take $2\mathrm{Vect}_\mathbb{C}$ 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 $n\mathrm{Vect}_\mathbb{C} \simeq {}_{(n-1)\mathrm{Vect}_\mathbb{C}}\mathrm{Mod}$ we find that

Hence a

0-vector bundle with connectionis a (continuous or smooth)functionfrom $X$ to $\mathbb{C}$.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 .