October 11, 2006

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 ${P}_{1}\left(M\right)$ 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 ${P}_{1}\left(M\right)$. 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 ${P}_{1}\left(M\right)$ 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 ${P}_{1}\left(M\right)$? 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 ${P}_{1}\left(M\right)$… 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:

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 ${P}_{1}\left(M\right)$, at least intuitively. Which is why I asked what the classifying space of ${P}_{1}\left(M\right)$ 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

$\mathrm{hol}:{P}_{1}\left(M\right)\to G$

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

$\mathrm{hol}:{P}_{1}\left(M\right)\to G?$

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

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 ${P}_{1}\left(M\right)$, 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:

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 $\varphi$ in the sense of Section 2 there is a category ${C}_{\varphi }$, or more precisely, a groupoid (a category in which all the morphisms are invertible). This is the free groupoid generated by the objects ${V}_{\varphi }$ (the vertices of $\varphi$) and the morphisms ${E}_{\varphi }$ (the edges of $\varphi$). If we fix a trivial $G$-bundle $P$ over ${V}_{\varphi }$, the set ${A}_{\varphi }$ of connections on $\varphi$ consists precisely of functors from ${C}_{\varphi }$ to $G$, where we regard the compact connected Lie group $G$ as a groupoid with one object. Similarly, elements of the set ${G}_{\varphi }$ of gauge transformations on $\varphi$ act as natural transformations between such functors. As we have seen, the set ${A}_{\varphi }/{G}_{\varphi }$ 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 ${L}^{2}\left({A}_{\varphi }/{G}_{\varphi }\right)$ 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 ${P}_{1}\left(M\right)$ to have as objects points of $M$ and as morphisms equivalence classes of piecewise analytic paths in $M$, where two paths $\gamma ,\gamma \prime$ 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 ${P}_{1}\left(M\right)$ 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.

Posted at October 11, 2006 2:04 AM UTC

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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

(1)$\mathrm{tra}:{P}_{1}\left(X\right)\to {\mathrm{Vect}}_{ℂ}\phantom{\rule{thinmathspace}{0ex}}.$

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

(2)

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 $\Sigma \left({M}_{n}\left(ℂ\right)\right)$, and then use the obvious topological and smooth structure on $\Sigma \left({M}_{n}\left(ℂ\right)\right)$ 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

(3)$\left[{P}_{1}\left(X\right),{\mathrm{Vetc}}_{ℂ}\right]$

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 $\left[{P}_{1}\left(X\right),{\mathrm{Vect}}_{ℂ}\right]$.

I think that much is clear.

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

(4)$\left[{P}_{1}\left(X\right),{\mathrm{Vect}}_{ℂ}{\right]}_{\sim }$

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 $\left[{P}_{1}\left(X\right),{\mathrm{Vect}}_{ℂ}\right]$.

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 $\left[{P}_{1}\left(X\right),{\mathrm{Vect}}_{ℂ}\right]$, there is no stopping us - we will want to study $n$-(differential K-cohomology)

(5)$\left[{P}_{n}\left(X\right),n{\mathrm{Vect}}_{ℂ}\right]\phantom{\rule{thinmathspace}{0ex}}.$

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

(6)

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

(7)$\mathrm{tra}:{P}_{0}\left(X\right)\to 0{\mathrm{Vect}}_{ℂ}\phantom{\rule{thinmathspace}{0ex}}.$

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

(8)$0{\mathrm{Vect}}_{ℂ}=ℂ\phantom{\rule{thinmathspace}{0ex}}.$

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 ${\Sigma }^{n}\left(U\left(1\right)\right)$ principal bundles with connection - which we can think of as $n$-line bundles with connection.

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

(9)${P}_{n}\left(X\right)\to T$

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

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

Posted by: urs on October 11, 2006 10:38 AM | Permalink | Reply to this

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

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.

Posted by: urs on October 11, 2006 11:25 AM | Permalink | Reply to this

superpaths

Among the technical points of applying the categorified Gelfand-Naimark theorem to transport functors ${P}_{1}\left(X\right)\to {\mathrm{Vect}}_{ℂ}$ is the presence of gradings.

First of all, we will want to regard hermitean vector bundles, hence really look at ${P}_{1}\left(X\right)\to {\mathrm{Hilb}}_{ℂ}\phantom{\rule{thinmathspace}{0ex}}.$ Next, we can nicely do away with the need to form a group completion of $\left(\left[{P}_{1}\left(X\right),{\mathrm{Hilb}}_{ℂ}\right],\oplus \right)$ by allowing ${ℤ}_{2}$-graded Hilbert spaces ${P}_{1}\left(X\right)\to {\mathrm{SuperHilb}}_{ℂ}\phantom{\rule{thinmathspace}{0ex}}.$ That’s a simple standard trick, of course. We identify the virtual vector space $\left(V,W\right)$ 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 $\left(G,\beta \right)$ is a groupoid $G$ equipped with an involutive automorphism $\beta :{\mathrm{Id}}_{G}\to {\mathrm{Id}}_{G}\phantom{\rule{thinmathspace}{0ex}},$ i.e. $\beta \circ \beta ={\mathrm{Id}}_{{\mathrm{Id}}_{G}}$.

But the ordinary path groupoid ${P}_{1}\left(X\right)$ 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 ${P}_{1}\left(X\right)$, such that we do get a nontrivial involution.

There is a universal such enhancement:

Definition. Let ${P}_{1}\left(X,b\right)$ be the groupoid which is generated from ${P}_{1}\left(X\right)$ together with a collection of morphism $\left\{\left(x\stackrel{{b}_{x}}{\to }x\right)\phantom{\rule{thickmathspace}{0ex}}\mid \phantom{\rule{thickmathspace}{0ex}}\forall \phantom{\rule{thinmathspace}{0ex}}x\in X\right\}$ 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 $\begin{array}{ccc}x& \stackrel{\gamma }{\to }& y\\ {b}_{x}↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}& =& \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓{b}_{y}\\ x& \stackrel{\gamma }{\to }& y\end{array}\phantom{\rule{thinmathspace}{0ex}},$ for all $x\stackrel{\gamma }{\to }y$ in ${P}_{1}\left(X\right)$. The involution $b:{\mathrm{Id}}_{{P}_{1}\left(X,b\right)}\to {\mathrm{Id}}_{{P}_{1}\left(X,b\right)}$ is the obvious one, with $b:x↦\left(x\stackrel{{b}_{x}}{\to }x\right)\phantom{\rule{thinmathspace}{0ex}}.$

Notice that ${P}_{1}\left(X,b\right)$ 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 $\left(E,E\prime \right)$ 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 $\left[{P}_{1}\left(X,b\right),{\mathrm{SuperHilb}}_{ℂ}\right]$ instead of the more naïve $\left[{P}_{1}\left(X\right),{\mathrm{SuperHilb}}_{ℂ}\right]\phantom{\rule{thinmathspace}{0ex}}.$

Be that as it may, using ${P}_{1}\left(X,b\right)$ 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 $\mathrm{Mor}\left({P}_{1}\left(X,b\right)\right)$ distinguish those morphisms that are of the form $x\stackrel{\gamma }{\to }y$ and those that are of the form $\begin{array}{ccc}x& \stackrel{\gamma }{\to }& y\\ {b}_{x}↑\phantom{\rule{thickmathspace}{0ex}}\\ x\end{array}\phantom{\rule{thinmathspace}{0ex}}.$ 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 $\begin{array}{rl}& \begin{array}{c}\\ & & y& \stackrel{\gamma \prime }{\to }& z\\ & & {b}_{y}↑\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\\ x& \stackrel{\gamma }{\to }& y\\ {b}_{x}↑\phantom{\rule{thickmathspace}{0ex}}\\ x\end{array}\\ \\ \\ =& \begin{array}{c}\\ x& \stackrel{\gamma }{\to }& y& \stackrel{\gamma \prime }{\to }& z\\ {b}_{x}↑\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\\ x\\ {b}_{x}↑\phantom{\rule{thickmathspace}{0ex}}\\ x\end{array}\\ \\ \\ =& \begin{array}{ccccc}x& \stackrel{\gamma }{\to }& y& \stackrel{\gamma \prime }{\to }& z\end{array}\phantom{\rule{thinmathspace}{0ex}}.\end{array}$

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 #.

Posted by: urs on October 11, 2006 2:38 PM | Permalink | Reply to this

Re: superpaths

I wrote:

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.

We do want the nontrivial involution of ${P}_{1}\left(X,b\right)$.

By definition 63 our transport functor $\mathrm{tra}$ - in order to qualify as a representation of supergroupoids - must send

(1)$x\stackrel{{b}_{x}}{\to }x$

to the grading involution on the super Hilbert space fiber ${E}_{x}$ over $x$.

That involution is nontrivial, in general, hence $x\stackrel{{b}_{x}}{\to }x$ must be nontrivial.

Even better, functoriality of $\mathrm{tra}$ applied to the relation

(2)$\begin{array}{ccc}x& \stackrel{{b}_{x}}{\to }& x\\ \gamma ↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}& =& \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓\gamma \\ y& \stackrel{{b}_{y}}{\to }& y\end{array}$

says that $\mathrm{tra}$ respects the grading, in the sense that

(3)$\mathrm{tra}\left(x\stackrel{\gamma }{\to }y\right):{E}_{x}\to {E}_{y}$

sends the even part of ${E}_{x}$ to the even part of ${E}_{y}$, and the odd part to the odd part.

This again implies that $\mathrm{tra}$ really comes from two seperate connections on two seperate vector bundles!

That’s exactly what we want. We want $\mathrm{tra}$ to be a virtual vector bundle with connection, i.e. a pair of vector bundles

(4)$\left({E}^{+},{E}^{-}\right)$

equipped with a pair of connections

(5)$\left({\nabla }^{+},{\nabla }^{-}\right)\phantom{\rule{thinmathspace}{0ex}}.$

Conclusion: In order to implement Bruce Bartlett’s observation, we want to study functors

(6)${P}_{1}\left(X,b\right)\to {\mathrm{SuperHilb}}_{ℂ}\phantom{\rule{thinmathspace}{0ex}}.$
Posted by: urs on October 11, 2006 4:05 PM | Permalink | Reply to this

the grading and the universal transition

Here is a comment on the nature of the involutive natural isomorphisms that we are dealing with and the universal nature of the groupoid

(1)${P}_{1}\left(X,b\right)$

that I defined above.

Forget supergroupoids for a moment. Consider just a parallel transport functor

(2)$\mathrm{tra}:{P}_{1}\left(x\right)\to T\phantom{\rule{thinmathspace}{0ex}},$

where ${P}_{1}\left(X\right)$ is the ordinary path groupoid of $X$.

Even without having a nontrivial involution on ${\mathrm{Id}}_{{P}_{1}\left(X\right)}$ we can talk about the condition on $\mathrm{tra}$ that we are interested in:

we demand that there is a nontrivial involutive isomorphism

(3)$\begin{array}{ccc}{P}_{1}\left(X\right)& \stackrel{\mathrm{Id}}{\to }& {P}_{1}\left(X\right)\\ \mathrm{Id}↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}& \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}⇓g& \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓\mathrm{tra}\\ {P}_{1}\left(X\right)& \stackrel{\mathrm{tra}}{\to }& T\end{array}$

from $\mathrm{tra}$ to itself.

By the general logic of transport #, we may regard this as a special degenerate case of a transition of transport.

Accordingly, we can look for the universal transition, such that the above factors through it.

I claim that

a) this universal transition is exactly the groupoid ${P}_{1}\left(X,b\right)$ that I defined before

b) the corresponing unique factorization morphism $\left(\mathrm{tra},g\right)$ is the functor ${P}_{1}\left(X,b\right)\to T$ that I talked about before.

In pictures:

(4)$\begin{array}{ccc}{P}_{1}\left(X\right)& \stackrel{\mathrm{Id}}{\to }& {P}_{1}\left(X\right)\\ \mathrm{Id}↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}& \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}⇓g& \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓\mathrm{tra}\\ {P}_{1}\left(X\right)& \stackrel{\mathrm{tra}}{\to }& T\end{array}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\begin{array}{ccc}{P}_{1}\left(X\right)& \stackrel{\mathrm{Id}}{\to }& {P}_{1}\left(X\right)\\ \mathrm{Id}↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}& \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}⇓\beta & \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓\\ {P}_{1}\left(X\right)& \to & {P}_{1}\left(X,b\right)& \\ & & & {↘}^{\left(\mathrm{tra},g\right)}& \\ & & & & T\end{array}$

The condition in definition 63 is now a consequence of the universal property.

Posted by: urs on October 11, 2006 4:39 PM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

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 ${P}_{1}\left(M\right)$ 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…

Posted by: Bruce Bartlett on October 11, 2006 3:42 PM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

I’ve spoken to Andrew Stacey, who has taught me a bit about the basics of infinite dimensional calculus. He pointed out the notes on his web page , which give a nice introduction to the differential topology of loop spaces. As he puts it, the slogan in infinite dimensional analysis is:

Smooth is as smooth does.

These notes are inspired by the treatment of general infinite dimensional calculus given in the monograph The convenient setting of global analysis by Kriegl and Michor.

John and Urs - you’re probably aware of these references. Interestingly, your definition of smooth spaces via Chen’s setup in the appendix of Higher Gauge Theory II seems to me to be quite similar to the definition of Frolicher spaces, given in Section 23 - page 244 - of “The convenient setting of global analysis”.

I’d be interested to hear your high-level reasoning behind choosing Chen spaces as your model for smooth spaces. It seems to be a general, clean and convenient setup… with the proviso of course that you can’t do stuff like partitions of unity, etc. since smooth spaces don’t specify a local model’ e.g. some kind of Banach space, or something.

But it seems well-suited for the applications you need in this paper.

Posted by: Bruce Bartlett on October 13, 2006 2:29 AM | Permalink | Reply to this

large smooth spaces

I’d be interested to hear your high-level reasoning behind choosing Chen spaces as your model for smooth spaces.

When I started thinking about this stuff I was being mighty naive, thinking like a physicist. From that point of view, loop space is something you access by choosing convenient coordinates, i.e. by looking at different parameterizations of collections of loops.

Luckily, this naive point of view is precisely what Chen/Fröhlicher or diffeology makes precise: you conceive a big fat space in terms of the collection of reasonably-sized maps that map into it.

It’s a bit like with stacks. You probe your space (e.g. loop space) by throwing other things (e.g. ${ℝ}^{n}$s) into it.

[…] you can’t do stuff like partitions of unity

This is alleviated to some degree by that fact that we are not interested in arbitrary smooth maps between smooth space, but just in those which behave functorially.

The space ${P}_{1}\left(X\right)$ of all paths in $X$ is very large. But a smooth functor

(1)$\mathrm{F}:{P}_{1}\left(X\right)\to T$

is determined already by only those paths that are arbitrarily close to identity paths.

The reason is that, by functoriality, the value of $F$ on a long path is already determined by its value on subpaths.

So:

• since $F$ is smooth, it is determined already by its derivative
• since it is functorial, it suffices to know this derivative at all identity paths

That’s how you show that such smooth functors on paths are in bijection with 1-forms on $X$ (taking values in something determined by $T$): such a 1-form encodes the derivative of $F$ at every identity path.

So if you want a partition of unity, one of $X$ might well be sufficient.

Posted by: urs on October 14, 2006 2:05 PM | Permalink | Reply to this

Re: large smooth spaces

I wrote:

It’s a bit like with stacks.

Probably not just a bit. From a smooth space $X$, we get a stack on manifolds by

- assigning to each manifold $M$ the groupoid whose objects are smooth maps (“plots”) $M\to X$ and whose morphisms are given by composition with smooth automorphisms $X\to X$

- assigning to each smooth map $f:M\prime \to M$ of manifolds the pullback of the groupoid over $M$ to that over $M\prime$ along $f$.

The groupoid over $M$ determines the collection of plots, the “restriction” morphisms of the groupoids say that plots composed with ordinary smooth maps must again be plots.

Posted by: urs on October 16, 2006 10:20 AM | Permalink | Reply to this

Re: large smooth spaces

Interesting! Any references for this stacky view of smooth spaces?

Posted by: Bruce Bartlett on October 16, 2006 1:14 PM | Permalink | Reply to this

stacks of plots of smooth spaces

Any references for this stacky view of smooth spaces?

Not that I knew of. What I wrote was my personal observation, triggered by your request # for a “high-level reasoning” behind diffeology.

While I have never seen anyone put it that way, the idea behind “plots” is clearly the same general idea that allows us to conceive other spaces as stacks by “throwing stuff into them”.

I don’t know how to read off from a given stack the composition law which must be hidden if that stack is the stack of plots of a path space. That information would be a necessary prerequisite if we were to replace smooth (path) spaces by their stacks of plots.

Posted by: urs on October 16, 2006 1:49 PM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Bruce wrote:

I’d be interested to hear your high-level reasoning behind choosing Chen spaces as your model for smooth spaces. It seems to be a general, clean and convenient setup…

That’s the main reason. For the stuff Urs and I do in this paper, it’s really crucial that our category of smooth spaces be cartesian closed: we want the space of paths

$\mathrm{hom}\left(\left[0,1\right],X\right)$

to be smooth whenever $X$ is, and to keep things simple we want a smooth map

$f:A\to \mathrm{hom}\left(\left[0,1\right],X\right)$

to be the same as a smooth map

$\stackrel{˜}{f}:\left[0,1\right]×A\to X$

We also want the category of smooth spaces to have colimits, so we mod out the space of paths by thin homotopy and be left with another smooth space.

So, I spent a bunch of time looking for setups with these properties. There are basically three approaches:

• In Chen’s approach we put a smooth structure on $X$ by declaring certain maps from certain standard spaces to $X$ to be smooth.
• In Mostow’s approach we put a smooth structure on $X$ by declaring certain maps from $X$ to certain standard spaces to be smooth.
• In Frolicher’s approach, if I remember correctly, you do both and demand consistency. A map from $X$ into our standard spaces must be smooth iff it’s smooth when composed with all smooth maps to $X$ from our standard spaces, and vice versa.

I think all three approaches give cartesian closed categories with all limits and colimits if you do them correctly. While starting the paper with Urs, I couldn’t decide which approach is “best”. So, I picked Chen’s approach since it seemed easy and good enough for what we were doing.

However, I have a slow-burning project with Dan Christensen to work out this stuff much more carefully and prove some interesting theorems about it. We made a lot of progress when I was visiting the Perimeter Institute this spring. But, it’s top-secret.

One more thing: you definitely don’t want to demand that smooth spaces are locally diffeomorphic to some standard spaces, like ${ℝ}^{n}$ or some infinite-dimensional topological vector space. This kills your chances of getting all limits and colimits! If you glue together things made of standard pieces, they won’t look locally like those standard pieces - not at the point where they’re glued together. Same with taking quotients.

So: manifolds are great, but they should be studied as “specially nice objects” in a bigger category of smooth spaces.

Grothendieck realized the same thing about algebraic varieties: it’s best to study them as specially nice objects in the category of schemes.

What Grothendieck discovered - and he was very explicit about this! - is that it’s better to work in a category with nice properties than a category, all of whose objects have nice properties.

You can always restrict attention to objects with nice properties later, when you’re stating certain theorems. But, it’s crucial to start by setting up a nice environment in which to prove those theorems!

Posted by: John Baez on October 20, 2006 11:24 PM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Thanks for this explanation.

I discussed this stuff a bit with my local guru on infinite dimensional smooth manifolds, Andrew Stacey.

He has the feeling that Frolicher spaces and Chen spaces are the same thing. I’ll explain his argument in a second; let me first for convenience list the relevant references here.

The definition of Chen spaces is given in the appendix of John and Urs’s paper on Higher Gauge Theory II : 2-Connections. It seems that Patrick Iglesias-Zemmour calls these diffeological spaces and is currently writing a big tome on it. To summarize : A Chen space is a set $X$ together with a collection of plots from convex(or open) subsets of ${ℝ}^{n}\to X$, satisfying some properties.

A Frolicher space is defined on Section 23 (page 244) of the downloadable book ‘The convenient setting of Global Analysis’ by Kriegl and Michor. To summarize : it is a set $X$ together with a collection of maps $ℝ\to X$ and $X\to ℝ$ satisfying various conditions.

Here is Andrew’s argument. Firstly, the maps *into* $X$ determine the maps *out of* $X$, and vice-versa (the consistency’ you mentioned above is a misnomer). The definition just includes both in order to have a pleasing symmetrical appearance.

So we can set up a Frolicher space in a similar spirit to a Chen space, i.e. as a space $X$ together with a bunch of maps $ℝ\to X$; these are the smooth curves in $X$.

Then we use Boman’s theorem from page 32 of The convenient setting of Global Analysis’. One of the things it says is that:

For a mapping $f:{ℝ}^{2}\to ℝ$ the following are equivalent:

• $f$ is smooth.
• For all smooth curves $c:ℝ\to {ℝ}^{2}$ the composite $f\circ c$ is smooth.

In other words, you can test’ the smoothness of a function $f:{ℝ}^{2}\to ℝ$ by precomposing with smooth curves. Interestingly, this theorem is false for ${C}^{n}$ functions - they must be ${C}^{\infty }$.

And this seems to be precisely the statement that Frolicher spaces and Chen spaces are equivalent. In other words, one needn’t have all those maps from open subsets of ${ℝ}^{n}\to X$; one need only have the curves $ℝ\to X$.

If this is true, then it seems more elegant to use Frolicher spaces, as you only need to talk about curves into $X$. On the other hand, it *can’t* be true - why would Patrick Iglesias-Zemmour not use it?

Mmmm….

Posted by: Bruce Bartlett on October 25, 2006 1:24 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

I’m glad someone is writing a big tome on diffeological spaces (Chen spaces). I find it mildly irksome to work on this subject, because while it’s very important and foundational, and even sort of fun, it should have been straightened out a long time ago!

It’s a bit like writing papers comparing different definitions of “topological space”. It was very nice that Sierpinski and Hausdorff and all those people spent their time this, and I’m sure it was tremendously fun at the time, proving all those basic facts, but there’s something very early-twentieth-century about it all. You’d think we’d done all that stuff by now. But we haven’t.

As far as I can tell, the study of smooth spaces got severely distracted by the charm of manifolds, and it’s taken a long time to recover - in fact, in some ways noncommutative geometry is ahead of commutative geometry, when it comes to studying general notions of smoothness.

Of course, the theory of topological spaces also got stuck in some ways. Top is not cartesian closed, so algebraic topologists switched to compactly generated weak Hausdorff spaces. Topos theorists switched to locales, which are much simpler in a way. But, kids in grad school still learn about topological spaces as if they were decreed by god to be the correct object of study!

We really need some people with the guts to modernize the creaky worn-out old aspects of point-set topology and especially of differential topology. There’s such enormous inertia when it comes to basic definitions: people learn them in grad school, pass qualifier courses on them, and never want to learn anything new ever again. Instead, they want to inflict the same concepts on their students.

People like to whine about Bourbaki, but at least the Bourbakists had the guts to consider drastic updatings of the basic language of mathematics, the energy to figure out good ways to do this updating, and the muscle to effectively advocate it. I wonder if and when that will ever happen again!

Posted by: John Baez on October 26, 2006 8:21 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Yes, a couple of cheers for Bourbaki. But then the big question: why with Eilenberg around did they forego category theory for so long? For some debate see here and the book by Corry it mentions.

What Grothendieck discovered - and he was very explicit about this! - is that it’s better to work in a category with nice properties than a category, all of whose objects have nice properties.

Presumably this works one level up. Can you give an example of how it is better to work in a 2-category with nice properties than a 2-category, all of whose objects have nice properties?

Posted by: david corfield on October 26, 2006 3:50 PM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

David wrote:

Yes, a couple of cheers for Bourbaki. But then the big question: why with Eilenberg around did they forego category theory for so long?

I don’t know. But you’re right - it’s too bad! I guess you could say there was an “incomplete revolution” in how the discipline of mathematics is structured. Someday we’ll have to finish off that job.

Presumably this works one level up. Can you give an example of how it is better to work in a 2-category with nice properties than a 2-category, all of whose objects have nice properties?

Hmm. Not instantly. The reason it works one level down is that by taking limits, colimits and mapping spaces betwen “nice” objects we get “less nice” ones.

For example, if you take an equation $f\left(x\right)=g\left(x\right)$ where $f,g:X\to Y$ are smooth maps between smooth manifolds, the space of solutions doesn’t need to be a smooth manifold. (That’s an example of a limit: an equalizer.)

Or, if you glue together smooth manifolds using smooth maps you get things more general than smooth manifolds. (That’s an example of a colimit: a pushout.)

Or, the space of maps between smooth manifolds is not a smooth manifold: it’s some sort of infinite-dimensional manifold. (So, the category of manifolds does not have mapping spaces, or an internal hom.)

So, our desire to create a category including smooth manifolds, but with limits, colimits and an internal hom, forces us to throw in spaces that aren’t smooth manifolds.

To get an example one level up, we should think of 2-categories whose objects are “categories with nice extra structure and properties”… but where taking weak limits, weak colimits or mapping spaces gives categories that aren’t so nice.

Suppose you look at $\mathrm{FinProdCat}$, the 2-category of categories with finite products (also known as “algebraic theories”.) Does this 2-category have weak limits, weak colimits and an internal hom? I’m not sure. Somehow I think it does. I think it’s only when you make the extra structure or properties be “very delicate” that you run into trouble.

One level down, for example, the structure of being a smooth manifold is “very delicate” - being locally diffeomorphic to ${ℝ}^{n}$ is like fine china: beautiful, but incredibly easy to break. It’s good for impressing guests, but not for everyday use when you have kids around.

I guess I can somewhat artificially make up an example. In the 2-category of topoi, there’s a sub-2-category of “topoi that are equivalent to a topos of sheaves on a topological space”. I’m no expert on topoi, but I bet all sorts of standard operations on topoi, when applied to these topoi, give topoi that aren’t of this sort.

In fact, I bet even Grothendieck topoi - “topoi that are equivalent to a topos of sheaves on a site” - are a bit less robust than elementary topoi as defined by Lawvere.

But presumably the currently most general topoi (elementary topoi) are sufficiently robust, having been torture-tested by a generation of category theorists.

Maybe someone else can make up better examples.

Posted by: John Baez on October 28, 2006 3:29 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Is that cheating to suggest the extension of the 2-category of groups to the 2-category of groupoids? Groups failing to have internal homs.

Posted by: David Corfield on October 29, 2006 3:31 PM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

[Hmmm. I tried posting this a couple of days ago. Let me try again.]

All this about smooth spaces is really interesting. Does anyone know if there is an approach similar to the way algebraic spaces are constructed in algebraic geometry?

Let me remind/tell you how that goes.

1. Let Aff denote the category of affine schemes, which by definition is just the opposite of the category of (commutative) rings. These are our local models.

2. Put the etale topology on Aff. (I won’t say what this is here.)

3. An algebraic space is a sheaf X on Aff which is locally represented by an object in Aff. (If we want X to be separated– the algebraic version of Hausdorff– this means X has a cover {U_i} by objects in Aff such that the “intersections” U_i x_X U_j are also in Aff. When you want no separatedness assumptions whatsoever, I don’t think anyone has bothered to work out the proper definition, unfortunately.)

So, is there a similar approach to defining some notion of smooth spaces?

Note that the local models above are very far from being non-singular. In fact, they are arbitrary closed subsets (in the algebraic sense) of affine n-dimensional space, for all n.

Posted by: James on October 28, 2006 4:47 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Oops. In the definition of locally representable in the separated case, we also want U_i x_X U_j to be a closed subset of U_i x U_j.

Posted by: James on October 28, 2006 4:52 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

I think the buzzwords you’re looking for are either “analytic stack” or “differentiable stack”.

Posted by: Aaron Bergman on October 28, 2006 4:59 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Mmm… very good point. Trying to relate the concepts of “smooth spaces” and “differentiable stacks” is a real chestnut.

These two approaches are two seemingly different solutions to the same problem.

Namely : the category of differentiable manifolds is awful to work in. So what are we going to do about it?

Answer 1 : Relax the definition of differentiable manifolds, but remain in the paradigm of “sets with structure”. Call the resulting objects “smooth spaces”.

Answer 2 : Be more daring. Conisder a generalized manifold as a stack, i.e. as a special kind of functor from Differentiable manifolds to groupoids.

The interesting thing of course is that both answers have an element of “searching for lost keys under the lamplight”.

The first answer, because it accepts implicitly the notion of “set with structure” - and does not dare to change it. The second answer, because it accepts implicitly the very category of Differentiable manifolds as being fundamental, in that it considers functors from DiffMan -> Groupoids - it does not dare to change the base’ category.

Amusingly, one could adopt a more democratic approach and combine both answers, by declaring a differentiable (pre)stack to be a functor Smooth spaces -> Groupoids.

Posted by: Bruce Bartlett on October 28, 2006 12:02 PM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

In algebraic geometry, one doesn’t really need to work with schemes as a base space; since you can glue sheaves, you might as well work with affine schemes, but that’s just Rings^op. I’m not entirely sure what you’d generalize that to for differentiabe manifolds. Given the plenthora of smooth structures on even R^4, perhaps there’s something nicer that’s not immediately obvious to me. Maybe there’s an answer to this somewhere in the differentiable stack literature.

On the algebraic geometry side, you can generalize things another way, too, by replacing Rings^op with something like simplicial rings. This apparently makes intersection theory much more natural. Of course, if one wants to have real fun, I guess you start talking about derived infinity stacks, but as seems usual I’m talking about things about which I know very little. Reading Jacob Lurie’s book is probably not as high on my list of things to do as I’d like it to be.

Posted by: Aaron Bergman on October 28, 2006 4:47 PM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Probably reading that book is something we all need to do some day. Recently every second interesting question that I have asked somebody was answered by “ah, for that you have to read Jacob Lurie’s book.”

Posted by: urs on October 29, 2006 2:37 PM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Yes, stackiness is completely independent to what I’m asking about. An algebraic space is a plain old space (“schemes done right”), whereas a stack is a categorified thing.

Another way of putting my question:

For any of these generalized notions of smooth space, is it true that smooth spaces are locally some more familiar kind of object? And if so, what are the rules for gluing the local models together to make global spaces?

Another question: Are arbitrary subsets of R^n naturally smooth spaces of some kind?

Posted by: James on October 30, 2006 5:14 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Let me try again:

Are arbitrary *closed* subsets of R^n naturally smooth spaces of some kind?

Posted by: James on October 30, 2006 5:19 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

James writes:

For any of these generalized notions of smooth space, is it true that smooth spaces are locally some more familiar kind of object?

It’s hard to answer your question for any notion of smooth space. But, for Chen or Mostow spaces, the answer to this question is no!

To specify a Chen space, we simply give a set $X$ and specify which maps $f:{ℝ}^{n}\to X$ are taken to be smooth (for all $n\ge 0$). These maps are called plots, and we demand that they satisfy a few obvious properties.

To specify a Mostow space, we simply give a topological space $X$ and specify for each open set $U\subseteq X$ which maps $f:U\to ℝ$ are taken to be smooth. Again, we demand that these collections of smooth maps satisfy a few obvious properties.

Another question: Are arbitrary subsets of ${ℝ}^{n}$ naturally smooth spaces of some kind?

Any subset of a Chen space naturally becomes a Chen space. Any subset of a Mostow space naturally becomes a Mostow space. So, the answer to this question is yes!

Posted by: John Baez on October 30, 2006 5:27 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Even more interesting.

My (hard won) conception of geometry is that there is always a local part and a global part. The local part has a hard nature, while the global part has a soft nature. For example, algebraic geometry is locally commutative algebra, differentiable-manifold geometry is locally the study of smooth maps between n-balls, topological-manifold geometry is locally the study of continuous maps between n-balls, and homotopy theory is locally trivial. The global part is essentially some gluing formalism, such as sheaf theory or homotopy theory.

Perhaps I’m too much of an algebraic-geometry ideologue, but my gut feeling is that *any* notion of geometry should fit into this pattern. Or am I missing some basic idea? My feeling is that spaces that aren’t locally of some standard form would essentially be unknowable—theological spaces, if you will.

On the other hand, it seems like it oughtn’t be too hard to glue together (in some way to be made precise) quite general level sets of smooth maps between manifolds to make a more general notion of smooth space. These spaces would be locally of a known form (by definition). Does anyone know if such an approach has been attempted?

I might offer a opposing point of view to the one of Grothendieck’s mentioned above (which I of course agree with). While we do want a category of spaces which is flexible enough to make lots of categorical constructions, we actually want nothing more than what we need.

Posted by: James on October 30, 2006 8:56 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

My feeling is that spaces that aren’t locally of some standard form would essentially be unknowable

Can we tweak this to “forms”? Otherwise I think you manage to throw out Whitney stratified spaces. Even so, I’m not sure thinking of them directly in terms of local neighborhoods is the right way to go about studying them.

Posted by: John Armstrong on October 30, 2006 11:57 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Yes, it would have been clearer to say “forms”. In algebraic geometry there are tons of things a space can look like locally.

Posted by: James on October 30, 2006 10:06 PM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

James wrote:

On the other hand, it seems like it oughtn’t be too hard to glue together (in some way to be made precise) quite general level sets of smooth maps between manifolds to make a more general notion of smooth space. These spaces would be locally of a known form (by definition). Does anyone know if such an approach has been attempted?

I don’t know. If you just wanted smooth spaces to be things that were obtained by gluing together smooth manifolds via smooth maps - including the possibility of gluing together infinitely many smooth manifolds - then you might work with presheaves on the category of smooth manifolds: that is, functors $F:{C}^{\mathrm{op}}\to \mathrm{Set}$ The reason is that the category of presheaves on a category $C$ is the cocompletion of $C$: loosely, the category whose objects are precisely colimits of objects in $C$. Colimits describe ways of “gluing objects together”.

However, the original colimits in $C$ are no longer colimits in the category of presheaves on $C$. This is rather annoying, since there are certainly cases when gluing together manifolds gives us manifolds.

The usual solution, if I understand this stuff correctly, is to work not with presheaves but sheaves with respect to some Grothendieck topology on $C$. A Grothendieck topology says when we regard an object in $C$ as “covered” by a bunch of other objects - and in the category of sheaves, we then think of it as the result of gluing these other objects together.

This strategy is pretty popular in algebraic geometry, but I find Chen spaces are closer to the kind of thing differential geometers are willing to work with: a set equipped with a bunch of “plots”. So, I decided to use Chen spaces when I was doing differential geometry and started needing smooth spaces more general than manifolds. It was a tactical decision.

Posted by: John Baez on October 31, 2006 3:36 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

I may be wrong, but I have convinced myself that Chen spaces (also known as “diffeologies”) *are* sheaves with respect to a fairly natural Grothendieck topology on the category of open subsets of Euclidean spaces.

Posted by: Eugene Lerman on October 31, 2006 4:01 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Do tell! That would be great. (But for it to be truly wonderful, I would want to understand the issues around their local structure. If Chen spaces really are some natural sheaves, then perhaps there is some subcategory of “locally knowable” Chen spaces which contains everything you really care about when doing geometry. Much like algebraic spaces are the locally knowable sheaves on the etale topology.)

Posted by: James on October 31, 2006 8:07 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Hi, Eugene!

I may be wrong, but I have convinced myself that Chen spaces (also known as “diffeologies”) are sheaves with respect to a fairly natural Grothendieck topology on the category of open subsets of Euclidean spaces.

I think you’re almost right; Urs and I explain this on page 9 of Higher Gauge Theory.

There are two subtleties. First, a Chen space is a special sort of sheaf on the category of convex subsets of Euclidean spaces, equipped with a Grothendieck topology that we describe. Note: convex - and not necessarily open! The importance of this nuance took us a while to discover, but briefly, one needs it to make manifolds with boundary into smooth spaces in a reasonable way.

The second subtlety is hidden in the phrase “special sort”. Only the “concrete” sheaves on the above site are Chen spaces! Here I say a sheaf $S$ on this site is concrete if there exists some set $X$ such that for any convex set $C$, $S\left(C\right)$ is a subset of the maps $\left\{\varphi :C\to X\right\}$ and for any smooth map $f:C\to C\prime$, $S\left(f\right)\varphi =\varphi \circ f.$ There are more fancy ways to say this, but I hope this is clear - we’re demanding that our “plots” $\varphi :C\to X$ really are maps into some set $X$.

Without this condition there would be extra sheaves of a more ethereal sort, for example “Chen spaces with no points” - nontrivial sheaves such that $S\left({ℝ}^{0}\right)=\varnothing .$

Because of this concreteness condition, Chen spaces don’t form a topos. They form a “concrete quasitopos”, which is almost as good in many ways.

But, it might be better, ultimately, to follow Grothendieck’s advice and use all sheaves on this site. The “concreteness” condition mainly serves to reassure lowbrow differential geometers (like me) that a smooth space is just a set with some extra structure.

Posted by: John Baez on October 31, 2006 4:59 PM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

a Chen space is a special sort of sheaf

[…]

for any convex set $C$, $S\left(C\right)$ is a subset of the maps $\left\{\varphi :C\to X\right\}$

[…]

for any smooth map $f:C\to C\prime$ $S\left(f\right)\varphi =\varphi \circ f$

I still think # we might actually want to regard a Chen space as a (special kind of) stack.

Above you use only part of the Chen space axioms. There is a further rule, which says that smooth maps $X\to X\prime$ between smooth spaces are those that become plots of $X\prime$ when pulled back along any plot of $X$.

This turns the set $S\left(C\right)$ of plots from $C$ into $X$ into a groupoid, and the restriction map $S\left(f\right)$ becomes a functor of groupoids.

Posted by: urs on November 1, 2006 9:52 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

An algebraic space is a plain old space (“schemes done right”), whereas a stack is a categorified thing.

It is true that a stack is something like a categorified sheaf, but it is also true that there are stacks which represent ordinary non-categorified spaces. For instance an elegant way to think of ordinary orbifolds is to think of them as stacks.

Posted by: urs on October 30, 2006 8:36 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Has anything precise been written about how to go from orbifolds to stacks? The reason why I ask is that orbifolds form a category, but stacks form a 2-category. It makes sense to me to take the orbifold underlying a certain kind of stack, and likewise for maps of stacks. But the lack of a notion of 2-morphism for orbifolds would seem to make associating a stack to an orbifold an unnatural thing. So I prefer not to think of an orbifold as a kind of stack, but rather as a shadow of a stack.

Posted by: James on October 30, 2006 10:24 PM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

James wrote:

But the lack of a notion of 2-morphism for orbifolds…

I think you want to read this:

• Eugene Lerman, Orbifolds as a localization of the 2-category of groupoids.

Abstract: We build a concrete and natural model for the strict 2-category of orbifolds. In particular we prove that if one localizes the 2-category of proper etale Lie groupoids at a class of 1-arrows that we call “covers,” then the strict 2-category structure drops down to the localization. In our construction the spaces of 1- and 2-arrows admit natural topologies, the space of morphisms (1-arrows) between two orbifolds is naturally a groupoid and the symmetries of an orbifold form a strict 2-group.
Posted by: John Baez on October 30, 2006 11:39 PM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

I think I’m responsible for a semantic mess here. (Is there a philosopher in the house?!) The question was whether orbifolds give rise to stacks which are “plain old spaces”, by which I meant 1-categorical things, not 2-categorical things, which stacks usually are.

I would say you have to decide what you mean by “orbifold”, or rather the “collection of all orbifolds”. (I myself have always had a hard time understanding what people mean because whenever someone tries to explain it to me, all I can hear is the little fish whispering “Stack. Stack. Stack.” in my ear.) So, the way I see it is that if you want your orbifolds to have 2-morphisms, then they’re no longer “plain old spaces” – the orbifold points have automorphisms, if you like. On the other hand, if you want them to be “plain old spaces”, in other words not have 2-morphisms between each other, then there is probably no reasonable way to embed the 1-category of orbifolds in the 2-category of stacks.

Posted by: James on October 31, 2006 7:54 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

As far as I understand it, one can think of orbifolds as special kinds of groupoids, and groupoids naturally form a 2-category Groupoids, thus so do orbifolds.

Alternatively, one might want to think of orbifolds in a stacky way. In that case, the morphisms between the orbifolds will be the morphisms between the stacks represented by their associated groupoids. Working this out explicitly, you arrive at the so-called Hilsum-Skandalis maps. This way you get another 2-category of groupoids, lets call it Bim_Groupoids.

The relation between Groupoids and Bim_Groupoids is precisely the same as the relationship between the ordinary 2-category Alg (of algebras, morphisms and natural transformations - here we are thinking of an algebra as a one object category) and the 2-category Bim of algebras, bimodules and bimodule homomorphisms.

In particular, two groupoids are Morita equivalent if and only if they are equivalent (in the 2-category sense) inside Bim_Groupoids.

This stuff seems to be explained nicely here. Which is not to say that I understand the precise details!

Indeed, thinking about orbifolds as groupoids (or as stacks) seems natural enough, but it apparantly has some drawbacks as outlined in the paper John referenced above, where another 2-category way to think about orbifolds is proposed. Unfortunately I don’t properly understand the issues involved… gulp!

Posted by: Bruce Bartlett on October 31, 2006 1:53 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Answer 1 : Relax the definition of differentiable manifolds, but remain in the paradigm of “sets with structure”. Call the resulting objects “smooth spaces”.

Answer 2 : Be more daring. Conisder a generalized manifold as a stack, i.e. as a special kind of functor from Differentiable manifolds to groupoids.

[…]

The first answer, because it accepts implicitly the notion of “set with structure” - and does not dare to change it. The second answer, because it accepts implicitly the very category of Differentiable manifolds as being fundamental, in that it considers functors from $\mathrm{DiffMan}\to \mathrm{Groupoids}$ - it does not dare to change the ‘base’ category.

I am not sure I would agree with this dichotomy.

To regard a space $M$ as a stack (over manifolds), you assign to each manifold $N$ the groupoid of allowed maps from $N$ into $M$.

That’s precisely what happens for Chen’s smooth spaces: there we assign to each ${ℝ}^{n}$ the groupoid of “plots” into some would-be space.

So Chen even restricts the “base” to manifolds which are diffeomorphic to some ${ℝ}^{n}$ for some $n$.

Hence one could consider generalizing in Chen’s definition the base space from the category of $\left\{{ℝ}^{n}\mid n\in ℕ\right\}$ to that of all manifolds.

Posted by: urs on October 30, 2006 8:46 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

And I think one can go further than thinking of smooth spaces as sheaves on $\mathrm{Diff}$, but think of them as stacks on (a category cooked up from) the category with objects ${R}^{n}$ for each $n$. I saw this once (here, section 2.11), where the author wanted to get around defining manifolds as particularly nice smooth stacks, and smooth stacks as stacks in manifolds. (Oh, I see you’ve said this already, Urs!)

Or perhaps I was mistaken - the above paper makes little effort to pursue the point. Even so, defining smooth stacks without mentioning manifolds is most desirable. Much as in algebraic geometry where varieties can be defined as nice schemes’, without recourse to loci of homogeneous polynomials.

Posted by: David Roberts on October 31, 2006 4:33 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

A question to John and Urs about their draft paper Higher gauge theory II : 2-connections.

I am interested in Proposition 42’, in which, given a Lie group $G$, you show that there is a one-to-one correspondence between smooth connections on a trivial principal $G$-bundle over a manifold $X$, and smooth functors ${𝒫}_{1}\left(X\right)\to G$.

Does it make sense to ask whether this kind of idea can be extended as follows:

The category of vector-bundles-with-connection on a Lie groupoid $M$ is equivalent to the category of smooth functors from $M$ into Vect.

Actually, its clear I’m making the cardinal error of expressing vector bundles classically’ instead of by their transport’. So I suppose my question is really to understand what is the best way of formulating this, i.e. how can you express the category of vector-bundles-with-connection on a Lie groupoid $M$ in terms of smooth functors from such and such to such and such’? Is it just smooth functors from $M$ into the disjoint union of the GL(n)’s?

Posted by: Bruce Bartlett on December 6, 2006 6:04 PM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

how can you express the category of vector-bundles-with-connection on a Lie groupoid $M$ in terms of ‘smooth functors from such and such to such and such’? Is it just smooth functors from M into the disjoint union of the $\mathrm{GL}\left(n\right)$’s?

One would hope this has a simple answer:

A smooth vector bundle with connection is a smooth functor into – $\mathrm{Vect}$.

But what is a smooth functor into $\mathrm{Vect}$, given that there is no chance to put a reasonable smooth structure on $\mathrm{Vect}$??

I propose the following definition, details of whose formulation here are owed to discussion with Konrad Waldorf, with whom I am hoping to write this up one fine day:

Let $C$ be a smooth category, i.e. a category internal to smooth spaces.

For fixed morphism

(1)$i:C\to \mathrm{Vect}$

I say the functor

(2)$\mathrm{tra}:{P}_{1}\to \mathrm{Vect}$

for ${P}_{1}$ some category is $p$-locally $i$-trivial iff there is

(3)$p:{P}_{1}\left(U\right)\to {P}_{1}$

such that

(4)${p}^{*}\mathrm{tra}\simeq {i}_{*}{\mathrm{tra}}_{U}$

for some

(5)${\mathrm{tra}}_{U}:{P}_{2}\left(U\right)\to C\phantom{\rule{thinmathspace}{0ex}}.$

In other words, if

(6)$\begin{array}{ccc}{P}_{1}\left(U\right)& \stackrel{p}{\to }& {P}_{1}\\ {\mathrm{tra}}_{U}↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}& \simeq ⇓t& \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓\mathrm{tra}\\ C& \stackrel{i}{\to }& \mathrm{Vect}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

(details are here)

I say that $\mathrm{tra}$ is $p$-locally $i$-smooth if ${P}_{1}\left(U\right)$ is a smooth category, ${\mathrm{tra}}_{U}$ is a smooth functor of smooth categories and if the transition data

(7)$g:{p}_{1}^{*}{\mathrm{tra}}_{U}\to {p}_{2}^{*}{\mathrm{tra}}_{U}$

obtained from $t$ is smooth.

Notice that $g$ is a $\left(1-1=0\right)$-functor with values in “cylinders” in $\mathrm{Vect}$.

I say such a 0-functor is smooth if it factors smoothly through cylinders in $C$.

You see, this is really the beginning of a recursive definition of what it means for a $n$-functor to be $p$-locally $i$-smooth.

An $n$-functor is $p$-locally $i$-smooth if it has a $p$-local $i$-trivialization whose transition $\left(n-1\right)$-functor is $p$-locally $i$-smooth.

This recursion terminates at 0-functors in an obvious way.

As an example, take ordinary vector bundles with connection on a smooth space $X$.

Take ${P}_{1}\left(X\right)$ to be the groupoid of thin-homotopy classes of paths in $X$. This is a smooth category with the standard Chen-space (“diffeological”) smooth structure on paths in $X$.

Take $C=\Sigma \left(\mathrm{GL}\left(n\right)\right)$ to be the smooth category obtained by suspending the general linear Lie group.

Let

(8)$i:\Sigma \left(\mathrm{GL}\left(n\right)\right)\to \mathrm{Vect}$

the standard representation of that Lie group.

Let $U\to X$ be a surjective submersion. For instance a good covering of $X$ by open contractible sets. Let

(9)$p:{P}_{1}\left(U\right)\to {P}_{1}\left(X\right)$

be the obvious functor on paths induced by this submersion.

Then, I think, $p$-locally $i$-smooth functors

(10)$\mathrm{tra}:{P}_{1}\left(X\right)\to \mathrm{Vect}$

are precisely smooth vector bundles with connection on $X$.

Now, for all this to be helpful for the question you asked, we need to know if the groupoid ${P}_{1}$ that you consider has any natural notion of “covering”

(11)$p:\mathrm{something}\to {P}_{1}$

such that the vector bundles with connection that you are interested in can be trivialized when pulled back to that $\mathrm{something}$.

Posted by: urs on December 6, 2006 7:24 PM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Urs wrote:

One would hope this has a simple answer:

A smooth vector bundle with connection is a smooth functor into $\mathrm{Vect}$.

But what is a smooth functor into Vect, given that there is no chance to put a reasonable smooth structure on Vect??

The problem is not the lack of a smooth structure on Vect. There’s a perfectly nice smooth structure on Vect: the set of objects is discrete (a 0-dimensional manifold), while the set of morphisms is a disjoint union of the vector spaces $\mathrm{hom}\left(V,W\right)$ for all vector spaces, with their usual smooth structures (thus, a union of manifolds of different dimensions).

The problem is that with this smooth structure on Vect, a smooth functor from the category of paths in $M$ to Vect is a connection on a trivial vector bundle.

The solution is to use smooth anafunctors, in the sense of Toby Bartels.

I explained this in the case of principal bundles on page 7 of these notes from a talk I gave. But, you’ll need to read the definition of smooth anafunctor in Toby’s thesis, and ponder it a while, for it to make complete sense.

In my notes, I claim that:

A principal $G$-bundle with connection over a manifold $M$ is the same as a smooth anafunctor from the category of paths in $M$ to $G$.

I believe that similarly:

A vector bundle with connection over a manifold $M$ is the same as a smooth anafunctor from the category of paths in $M$ to $\mathrm{Vect}$.

These should also work whenever $M$ is any smooth space in the sense of Chen.

It’s this sort of thing that convinced me of the importance of smooth anafunctors. They also play a big role in Toby’s work on higher gauge theory.

(By the way, for nitpicky people, I should admit that I’m using Vect to mean any category equivalent to the category of finite-dimensional vector spaces, but which has a mere set of objects instead of a proper class. For example, we can take Vect to be any skeletal subcategory of the category of finite-dimensional vector spaces.)

Posted by: John Baez on December 6, 2006 9:39 PM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

John wrote in part:

The solution is to use smooth anafunctors, in the sense of Toby Bartels.

I didn’t really go through all of Urs’s stuff about p-local i-trivializations, but smooth anafunctors are all about functors’ being locally smooth (or better, locally defined and smooth on each defined patch) when seen as a map between the smooth spaces of morphisms (and between the smooth spaces of objects, for that matter).

By the way, for nitpicky people, I should admit that I’m using Vect to mean any category equivalent to the category of finite-dimensional vector spaces, but which has a mere set of objects instead of a proper class.

(Recall: A category is small if it has a mere set of objects and a set of morphisms.) Given any cardinal number κ, you can always find a small category equivalent to the category of vector spaces of dimension less than κ. If you believe in Grothendieck’s axiom of universes, then you can always reinterpret the category of ‘all’ vector spaces as a small category in a larger universe of sets, by letting κ be the cardinality of your original universe (which is a proper class in the original universe but a set in the larger universe). My point is that we need not limit ourselves to finite-dimensional vector spaces in order to know what a smooth anafunctor to Vect is.

Posted by: Toby Bartels on December 6, 2006 10:44 PM | Permalink | Reply to this

anafunctors

I didn’t really go through all of Urs’s stuff about $p$-local $i$-trivializations,

I guess I am being too verbose, making my point look more involved than it is. Sorry.

All I am saying is this: for a functor $\mathrm{tra}:{P}_{1}\to T$ to be “of the form we want” I require that it fits into a diagram

(1)$\begin{array}{ccc}{P}_{1}\left(U\right)& \stackrel{p}{\to }& {P}_{1}\\ ↓& ⇓t& \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓\mathrm{tra}\\ T\prime & \stackrel{i}{\to }& T\end{array}$

with $t$ “of the form we want”.

From what I have seen of your thesis I seem to conclude that you consider this part of the above diagram:

(2)$\begin{array}{ccc}{P}_{1}\left(U\right)& \stackrel{p}{\to }& {P}_{1}\\ ↓& \\ T& \stackrel{\mathrm{Id}}{\to }& T\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

At least for 1-functors, I have no quarrel with the $\mathrm{Id}$ at the bottom.

But don’t you want to have “$t$” in the game in order to say what it means for the transition data of the anafunctor to be smooth?

Posted by: urs on December 7, 2006 6:03 PM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

A couple of comments above, a couple of weeks ago, I wrote:

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.

#

What I cannot quite discern in Toby’s thesis is where he includes the condition that the transitions have to be smooth.

Anyway, it seems to me that up to some minor differences in the setup, what I am talking about is essentially the same as the concept of an anafunctor.

One of the difference is that in what I described above I don’t initially require to use

$\mathrm{Vect}$ to mean any category equivalent to the category of finite-dimensional vector spaces, but which has a mere set of objects

Instead, something like such a step is, in the setup I described, encoded in the choice of injection

(1)$i:C\to \mathrm{Vect}$

which specifies how I want the typical fiber to look like (namely like an object in $C$).

I find this useful for reasons like the following:

Consider a 2-functor to $\Sigma \left(1d{\mathrm{Vect}}_{ℂ}\right)$, locally trivializable along

(2)$i:\Sigma \left(\Sigma \left(ℂ\right)\right)\to \Sigma \left(1d\mathrm{Vect}\right)\phantom{\rule{thinmathspace}{0ex}}.$

Its transition $g$ will be a 1-functor with values in squares in $\Sigma \left(1d\mathrm{Vect}\right)$ whose top and bottom horizontal morphisms are identities. These squares are hence nothing but morphisms in $\mathrm{Vect}$ and hence we find that the transition is a 1-functor to $\mathrm{Vect}$. But that’s just a vector bundle with connection!

This way we get the transition line bundle of a bundle gerbe from locally trivializing a 2-vector 2-transport.

Now, if I were instead to use the concept of an ana-2-functor as you suggest, wouldn’t I then always find this transition line bundle with connection to be trivial?

Not that this would be a real problem. We can generally assume it to be trivializable anyway.

Posted by: urs on December 7, 2006 9:52 AM | Permalink | Reply to this

anafunctors

I am again looking at Toby’s thesis, trying to better understand how the concept of anafunctor might be related to what I was imagining.

The definition of anafunctor begins at diagram (109).

This diagram, as far as I understand, expresses the concept that I denoted by

(1)$p:{P}_{1}\left(U\right)\to {P}_{1}$

above.

(By the way, it seems to me that in the right column of that diagram the ${X}^{1}$ should read ${X}^{2}$ and the ${X}^{0}$ should read ${X}^{1}$, could that be?)

So, an anafunctor between smooth categories $X$ and $Y$

(2)$\mathrm{R}:X\to Y$

is something that becomes an ordinary smooth functor when pulled back to some cover of the $X$.

And we want this cover to be induced by a cover of $\mathrm{Obj}\left(X\right)$.

So, choose any cover

(3)$U\to \mathrm{Obj}\left(X\right)$

and then use the pullback diagram (109) to canonically construct from this a category $\mid R\mid$ with $\mathrm{Obj}\left(\mid R\mid \right)=U$ and a functor

(4)$\mid R\mid \to X\phantom{\rule{thinmathspace}{0ex}}.$

This is, I think, what I denoted by

(5)$p:{P}_{1}\left(U\right)\to {P}_{1}\phantom{\rule{thinmathspace}{0ex}}.$

Then, diagram (112) expresses the existence of a functor

(6)$\mid R\mid \to Y\phantom{\rule{thinmathspace}{0ex}}.$

Somehow I am left with the feeling that there is more data to an anafunctor than this. (Probably I am not looking at the right part of the text!?) So far, this corresponds in the notation I was talking about to the situation

(7)$\begin{array}{ccc}\mid R\mid & \to & X\\ ↓& & \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓R\\ Y& \stackrel{\mathrm{Id}}{\to }& Y\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

I guess next there is a requirement that this diagram is filled by an isomorphism which itself is smooth in a suitable sense.

If so, then what I was talking about is indeed just that, for the special case where the codomain is already smooth and the morphism I called $i$ is hence the identity.

Posted by: urs on December 7, 2006 12:01 PM | Permalink | Reply to this

Re: anafunctors

Thanks for all this help. I always knew it was essential that I should read Toby Bartels’ thesis… and I guess that day has come!

I was actually being sneaky earlier on. What I’m really interested in is the ‘plain old’ category of vector bundles on a manifold $X$. No connections.

I thought I could use a trick, inspired by reading Lupercio and Uribe. Recall that to any manifold $X$ with its charts $U$, we can associate a Lie groupoid X, whose objects are the (disjoint) elements of the charts and whose morphisms are the gluing identifications.

A vector bundle on $X$ should be the same thing as a vector-bundle-with-connection on the Lie groupoid X.

Does that make sense?

In this way you should be able to express ‘vector bundles on $X$’ using anafunctors.

Posted by: Bruce Bartlett on December 7, 2006 2:17 PM | Permalink | Reply to this

Re: anafunctors

A vector bundle on $X$ should be the same thing as a vector-bundle-with-connection on the Lie groupoid $X$.

Yes. This is included as a special case in what we talked about above. Just take the domain ${P}_{1}\left(X\right)$ to be the category of constant paths in $X$. In other words, set

(1)${P}_{1}\left(X\right):=\mathrm{Disc}\left(X\right)\phantom{\rule{thinmathspace}{0ex}},$

the discrete category on the space $X$.

Then all the functor

(2)$\mathrm{tra}:{P}_{1}\left(X\right)\to \mathrm{Vect}$

does is to assign to each point in $X$ the fiber of a vector bundle over it.

Choosing a cover $U\to X$ of $X$ and a local trivialization of $\mathrm{tra}$ - or the equivalent data in the anafunctor - we get a transition

(3)$g:{p}_{1}^{*}{\mathrm{tra}}_{U}\to {p}_{2}^{*}{\mathrm{tra}}_{U}\phantom{\rule{thinmathspace}{0ex}}.$

This, as always, satisfies a cocycle condition. It makes the obvious triangle commute.

But if you unwrap what this all means, it says precisely that $g$ is a functor on the groupoid you have in mind.

So, just concentrate on constant paths and you get a vector bundle without connection using an anafunctor to $\mathrm{Vect}$.

Posted by: urs on December 7, 2006 4:24 PM | Permalink | Reply to this

Re: anafunctors

I wrote:

But if you unwrap what this all means, it says precisely that g is a functor on the groupoid you have in mind.

I should emphasize more the main point here: more generally, if you also allow non-constant paths, the transition $g$ defines for you a functor on the category of paths in the groupoid ${U}^{•}$. If all paths are constant, this is the same as the groupoid ${U}^{•}$ itself.

I have collected further details on this phenomenon here, explicitly making the connection to what Lupercio and Uribe were talking about.

Posted by: urs on December 7, 2006 4:48 PM | Permalink | Reply to this

Re: anafunctors

Urs wrote:

Somehow I am left with the feeling that there is more data to an anafunctor than this. (Probably I am not looking at the right part of the text!?)

I agree that we need more than what you’ve described. Very roughly, a smooth anafunctor should be like “smooth functors on coordinate patches, related by smooth natural isomorphisms on double overlaps, satisfying a cocycle condition on triple overlaps”. It sounds like you’ve just described the first of these three items.

I hope Toby answers your question. I hope either he convinces you that his definition of anafunctor is okay, or you convince him to add something to the definition!

Posted by: John Baez on December 7, 2006 6:58 PM | Permalink | Reply to this

Re: anafunctors

Urs wrote (in various recent comments):

What I cannot quite discern in Toby’s thesis is where he includes the condition that the transitions have to be smooth.

Every map in my thesis is smooth. (Or rather, every map is a morphism in C, where C miught be a category of smooth spaces, algebraic spaces, or whatever kind of spaces you’re interested in.) So technically, this requirement appears in Section 1.2 (page 10).

(By the way, it seems to me that in the right column of that diagram the ${X}^{1}$ should read ${X}^{2}$ and the ${X}^{0}$ should read ${X}^{1}$, could that be?)

Gosh, you’re right! (I’ve incorporated this correction in the unofficial development version of the paper, which is available at an unadvertised URL that you may know.) Ironically, since I use a slightly idiosyncratic numbering scheme (I have my reasons), most people would say that those superscripts are correct (while all the others are wrong!).

So far, this corresponds in the notation I was talking about to the situation

(1)$\begin{array}{ccc}\mid R\mid & \to & X\\ ↓& & ↓R\\ Y& \stackrel{\mathrm{Id}}{\to }& Y\end{array}.$

I guess next there is a requirement that this diagram is filled by an isomorphism which itself is smooth in a suitable sense.

(How did you make this diagram, Urs? I keep giving itex all kinds of valid LaTeX, using all possible text filters, even encoding & as &amp;, but nothing works.) By the way, that ‘R’ up there is, in my paper, really a Fraktur ‘x’: ‘𝔵’

This isomorphism doesn’t appear because there is no smooth functor from X to Y! Even to say that such a functor (not necessarily smooth) exists requires the axiom of choice (in the category of sets), placing severe restrictions on the category C (like being equipped with a faithful functor to the category of sets, which is false for many categories of generalised smooth spaces). Part of the point of anafunctors is that we never have to actually talk about this functor; in fact, Michael Makkai’s original motivation for inventing anafunctors to avoid using the axiom of choice (or if you prefer, to do category theory internal to a category of sets that doesn’t satisfy the axiom of choice). Part of my motivation is to keep everything internal to C, where C may be the category of smooth manifolds but may be something quite different; since the axiom of choice doesn’t hold in that category, I adapted Makkai’s anafunctors.

So for me, R consists entirely of the cover (defining the smooth category |R| and a smooth functor from |R| to X) and the smooth functor from from |R| to Y; there is no functor from X to Y as far as I’m concerned (since I have no way to talk about non-smooth maps or non-smooth functors). For the categorially inclined, an anafunctor is a span (with a certain structure on the common domain).

I guess I am being too verbose, making my point look more involved than it is.

Well, I haven’t even read Transport, Trivialization, Transition! So I only know my own definition.

John wrote in part:

I agree that we need more than what you’ve described. Very roughly, a smooth anafunctor should be like “smooth functors on coordinate patches, related by smooth natural isomorphisms on double overlaps, satisfying a cocycle condition on triple overlaps”. It sounds like you’ve just described the first of these three items.

This other stuff is all trivial, because of the special way that |R| is constructed. In other words, you start with only a simple cover of X1 (the space of objects of the source of the 2map), not a 2cover of X (the source itself), which in fact you cannot define until you know what a 2map is. What would be interesting is if (already knowing what 2maps are) you consider an arbitrary 2cover of X, build a span like this, and then consider what else you need to make the whole thing collapse to (up to equivalence of spans in the 2category of 2spaces) to a simple 2map. Or you could define 2map badly (as a smooth functor, on the nose), hence define 2cover badly, and then do something like this to get the good notion of 2map.

Posted by: Toby Bartels on December 8, 2006 3:56 AM | Permalink | Reply to this

Re: anafunctors

Toby wrote:

So for me, $R$ consists entirely of the cover (defining the smooth category $\mid R\mid$ and a smooth functor from $\mid R\mid$ to $X$) and the smooth functor from from $\mid R\mid$ to $Y$; there is no functor from $X$ to $Y$ as far as I’m concerned (since I have no way to talk about non-smooth maps or non-smooth functors).

There is one question which arises naturally in my mind (hopefully it hastn’t been answered somewhere in the thread way up there): Is $\mid R\mid$ in some sense equivalent to $X$? Also, I’m sure this doesn’t preclude the notion of have the identity of $X$ as a cover - it’s just not a generic situation.

Posted by: David Roberts on December 8, 2006 5:08 AM | Permalink | Reply to this

Re: anafunctors

Is $\mid R\mid$ in some sense equivalent to $X$?

Very good question!

Not $\mid R\mid$ is, but the transition groupoid $\mid R{\mid }^{•}$ obtained from it is.

Better yet, the category of paths in the transition groupoid of the surjective submersion is equivalent to the category of paths in the original space.

Konrad Waldorf and myself have been busy proving that the same remains true one step higher: the 2-category of 2-paths in the transition 2-groupoid is 2-equivalent to the 2-category of paths in the original space.

The proof is hard if you want to stay in $\mathrm{Gray}$, with all 2-categories strict. However, if you allow yourself non-strictness the proof is rather easy. I have given it here. (Where I don’t say $\mid R\mid$ (which is my misreading # for a fraktur $X$, anyway :-)) but $Y$).

This is the first ingredient you need to show that your $n$-anafunctors form an $n$-stack.

Posted by: urs on December 8, 2006 5:42 AM | Permalink | Reply to this

Re: anafunctors

Is ∣R∣ in some sense equivalent to X?

Yes! Indeed, there is anafunctor from |R| to X, and there is an anafunctor from X to |R|; each is a span of which one leg is an identity functor. And these anafunctors are (up to natural isomorphism) inverses.

Note that when Urs writes below that ∣R∣•, not |R|, is equivalent to X, he is wrong only because is misusing the notation (which is ultimately from my thesis). He is quite right about the content, however; his |R| is my |R|1, and his |R|• is (if I understand his notation correctly) my |R|. (But he did use |R| for my |R| in previous comments!)

I’m sure this doesn’t preclude the notion of have the identity of X as a cover

Not at all. Every functor is an anafunctor.

By the way, people interested in the general idea of anafunctors could hardly do better than to read Makkai’s original paper. My only contribution to the definition of smooth anafunctor was to determine the analogue of a surjective function (which I took to be a surjective local diffeomorphism in my thesis, although a sujrective submersion is actually good enough).

Posted by: Toby Bartels on December 8, 2006 7:44 PM | Permalink | Reply to this

Re: anafunctors

he is wrong only because is misusing the notation

Yes, right, sorry. I missed the point of Toby’s diagram (109) for a while.

It became clear to me only after I saw Makkai’s definition, that the anafunctor is a span whose left leg is a surjection such that every morphism has at most one lift with specified source and target.

That’s precisely the property of the map from what I was calling the “category of paths in the transition groupoid” down to that of paths in base space.

I hope I am correctly describing the situation here.

Posted by: urs on December 9, 2006 12:41 PM | Permalink | Reply to this

Re: anafunctors

What I cannot quite discern in Toby’s thesis is where he includes the condition that the transitions have to be smooth.

Every map in my thesis is smooth. […] So technically, this requirement appears in Section 1.2 (page 10).

Okay, but you don’t put in any transitions at all it seems, smooth or not!

From what I understand from what John says, an anafunctor is exactly what other people would call descent data for a functor, or what I sometimes call the transition data for a locally trivialized functor.

I am sure that’s what you mean, too. Just that in your thesis I don’t see that you mention the presence of the transitions. (Maybe you do, and I don’t see it.)

With that functor on the cover given, we want to demand that on double intersections there is an isomorphism which makes a triangle commute on triple intersections.

You either have to require this by hand, or you require a trivialization $t$ as I did, then this data follows (though I do understand that this last choice is not what you want to do).

I think I will write an entry on this, putting the entire discussion here in a clean form, pointing out what is going on.

Posted by: urs on December 8, 2006 5:36 AM | Permalink | Reply to this

Re: anafunctors

I wrote:

you don’t put in any transitions at all it seems

I am now (finally) looking at Makkai’s definition. The way he puts it, the transitions exist due to 1.(ii) and behave as transitions due to 1.(v).

I guess what I was not seeing in your writeup is the analog of 1.(ii). But now I understand that this is a consequence of how you build the cover using the pullback in diagram (109)!

(Sorry for the confusion I am causing in trying to understand this.)

Posted by: urs on December 8, 2006 1:27 PM | Permalink | Reply to this

Re: anafunctors

Here is the diagram that should have appeard in my previous comment (copied from Urs’s comment before it):

(1)$\begin{array}{ccc}\mid R\mid & \to & X\\ ↓\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}& & \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}↓R\\ Y& \stackrel{Id}{\to }& Y\end{array}.$
Posted by: Toby Bartels on December 8, 2006 7:49 PM | Permalink | Reply to this
Read the post What is the categorified Gelfand-Naimark theorem?
Weblog: The n-Category Café
Excerpt: A lightning review of the categorified Gelfand-Naimark theorem.
Tracked: October 11, 2006 6:15 PM
Read the post Algebras as 2-Categories and its Effect on Algebraic Geometry
Weblog: The n-Category Café
Excerpt: A question by Bruce Bartlett about categories of algebras, algebras as categories and the possible implications for non-commutative algebraic geometry.
Tracked: October 19, 2006 12:39 PM
Read the post Local Transition of Transport, Anafunctors and Descent of n-Functors
Weblog: The n-Category Café
Excerpt: Conceps and examples of what would be called transition data or descent data for n-functors.
Tracked: December 8, 2006 7:28 AM
Read the post History of Understanding Bundles with Connection using Parallel Transport around Loops
Weblog: The n-Category Café
Excerpt: A list of some papers involved in the historical development of the idea of expressing bundles with connection in terms of their parallel transport around loops.
Tracked: March 23, 2007 6:42 PM
Weblog: The n-Category Café
Excerpt: Paolo Bertozzini, Roberto Conti and Wicharn Lewkeeratiyutkul on categorified spaces and the many-object version of C-star algebras.
Tracked: January 17, 2008 8:00 PM

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Just learned that Israel Gelfand is no longer with us.

Posted by: Successful Researcher on October 6, 2009 4:04 AM | Permalink | Reply to this

Re: Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection

Posted by: Toby Bartels on October 6, 2009 9:13 AM | Permalink | Reply to this

RIP

RIP, Professor Gelfand.

Seeing how he was one of the great giants of twentieth-century mathematics, a living legend really, it may seem a little out of place for me to share a few memories I have of him. I can’t say I knew him well when I was a graduate student at Rutgers, but he left a lasting imprint.

As a teacher he was kind as well as tough and scary. His principle: teach through the simplest examples that would reveal the importance of a concept. The things he tried to teach me back in those days (Coxeter groups, quivers, quasideterminants, among other things) have had a strange way of coming back to me much later, especially as I rediscover them in talking with people like James Dolan, and his manner of sometimes saying to me, “You won’t understand anything of this talk, but it will be very important to you” was often strangely prescient.

His seminar, over which he was in complete control, was a high point of department life. The way he would call on people was frequently unnerving, but his personality and amazing mathematical intuition exerted a fascination which was pretty hard for people to resist.

I believe life in the United States suited him admirably. I’ll always remember and treasure my final memory of him: almost ninety years old, his vigor seemed only to have grown from the time I first knew him; he stood taller and straighter, his bearing more animated than ever.

My condolences to his family, even though they probably won’t be reading these lines.

Posted by: Todd Trimble on October 6, 2009 7:17 PM | Permalink | Reply to this

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