The n-Category Café

You no doubt saw my comment here and no doubt could guess what I was thinking :)

I still stand by my old comment here, but extend its (admittedly highly speculative) scope to what you are talking about here as well.

I’m slowly catching up, which means you must be getting slow in your old age ;) Just kidding! :) I am finding that I’m understanding more and more about things discussed here lately, which somehow seems miraculous. Thanks!

I’m glad you’re taking time off now and then to ponder what groupoidification might do for physics. I’m pretty frantic right now — catching a plane in two hours — so it’ll take me a while longer to absorb what you’re doing. But, I’ll think about it.

I can’t tell how far John Baez and Alex Hoffnung have gotten since then…

It turned out Alex’s real talents lie in areas not so directly connected to physics — something more like ‘algebra’. So, my project of understanding path integrals will wait until some other student comes along and helps me out.

Alex hasn’t studied much physics, so it was hard for him to tackle the project of drastically reformulating it. But what really impressed me is how I once said “you know, someone told me that Chen spaces should be an example of a concrete quasitopos, whatever that is” — and then he learned enough about sheaves on sites, topoi and quasitopoi to prove this! He had a lot of help from James Dolan, and a bit from me too — but still, most people find this stuff frighteningly abstract, and he seemed to jump right in and enjoy it.

So, right now our plan is for him to fill in the many holes in HDA7 — some of which involve topos theory. Then, he can use this to groupoidify the usual sort of Hecke algebra we get from any Dynkin diagram. This is part of a bigger project to work out all the details of what was sketched in the Geometric Representation Theory seminar earlier this year. Christopher Walker is working on another side of that project: first giving a nice ‘practical’ introduction to groupoidification — “Look, ma! No topoi!” — and then groupoidifying Hall algebras.

Hall algebras and Hecke algebras are both closely connected to quantum groups, so very roughly you could say this project is about groupoidifying quantum groups. For this reason, Alex plans to spend a lot of time in New York this summer talking to Aaron Lauda.

Meanwhile, as a kind of side project, he wants to work on De Rham cohomology for smooth spaces.

He will be at HOCAT in Barcelona from June 30th to July 5th — but alas, you won’t.

I’n not sure if I’ve asked this before, but how does this related to the groupoid programme of Ian Stewart and Martin Golubitsky, as reviewed in Bulletin (new series) of the American Mathematical Society, vol.42, no.1. Jan 2005, pp.99-103, and the wonderful paper:

Ian Stewart and Martin Golubitsky

NONLINEAR DYNAMICS OF NETWORKS: THE GROUPOID FORMALISM
AMERICAN MATHEMATICAL SOCIETY, 2006
May 3, 2006

Abstract:

A formal theory of symmetries of networks of coupled dynamical
systems, stated in terms of the group of permutations of the nodes that
preserve the network topology, has existed for some time. Global network
symmetries impose strong constraints on the corresponding dynamical systems,
which affect equilibria, periodic states, heteroclinic cycles, and even chaotic
states. In particular, the symmetries of the network can lead to synchrony,
phase relations, resonances, and synchronous or cycling chaos.

Symmetry is a rather restrictive assumption, and a general theory of
networks should be more flexible. A recent generalization of the group-theoretic
notion of symmetry replaces global symmetries by bijections between certain
subsets of the directed edges of the network, the ‘input sets’. Now the
symmetry group becomes a groupoid, which is an algebraic structure
that resembles
a group, except that the product of two elements may not be defined. The
groupoid formalism makes it possible to extend group-theoretic methods to
more general networks, and in particular it leads to a complete classification
of ‘robust’ patterns of synchrony in terms of the combinatorial structure of the
network.

Many phenomena that would be nongeneric in an arbitrary dynamical
system can become generic when constrained by a particular network
topology. A network of dynamical systems is not just a dynamical system with
a high-dimensional phase space. It is also equipped with a canonical set of
observables—the states of the individual nodes of the network. Moreover, the
form of the underlying ODE is constrained by the network topology—which
variables occur in which component equations, and how those equations relate
to each other. The result is a rich and new range of phenomena, only a few of
which are yet properly understood.

Had a busy weekend, celebrating a friend’s wedding. Now I need to run and think about something else.

I should come back here and phrase Freed-Quinn’s discussion of Dijkgraaf-Witten entirely in the above context. But not now.

One thing to notice is: in the above entry I effectively say that what matters as the groupoid of sections of a bundle for the purpose of quantization is the action groupoid of its transport functor.

As it appears, for the fundamental rep., at the very end of the article with David Roberts.

I really want to put some effort into following this idea. I have a lot of catching up to do though. I have made an effort to trace the references back, but after a couple levels became overwhelmed.

I have questions that you will probably find to be quite basic so feel free to ignore them. You have better things to do :) It never hurts to ask though!

The first question is from the first line of the Background/Motivation:

target space is a category $P_1(X)$ generated from a finite graph.

First a warning: I intend to try to either understand or reinterpret everything here in terms of diamonds.

So given a 2-diamond, a.k.a. a binary tree, how exactly do you generate a category? Objects are nodes and morphisms are edges. But then you need to define composition. The obvious thing to do would generate morphisms in $P_1(X)$ that do not correspond to edges in $X$. Is that going to be a problem? Should we try to keep track of how many “time steps” are present in a morphism?

In a binary tree, we can label the nodes with a space index $i$ and time index $j$ via $(i,j)$, where the generating edges are of the form

$(i,j)\to (i\pm 1,j+1)$.

If $P_1(X)$ is generated the way I think it is, do we distinguish between

$(i,j)\to (i+1,j+1)\to (i,j+2)$

and

$(i,j)\to (i-1,j+1)\to (i,j+2)$

? In other words, do we assume there are two morphisms joining $(i,j)$ and $(i,j+2)$?

Baby steps…

I’ll stop there for now although you can imagine with this beginning there are many more questions in store :O

Best wishes

Now that I think I understand $P_1(X)$, i.e. a morphism in $P_1(X)$ corresponds to a “path” in $X$ (which might have something to do with the fact that you use a “$P$” to denote it!), I’ll try to take another baby step.

At one point, I made a sincere effort to learn some basic representation theory, but didn’t get very far. Now you denote a representation as

$\rho :\text{B}G\to \text{Set}$.

This is not like the representations I remember looking at. Having lurked around here long enough, I think you are referring to a more recent understanding of representations? I think I can simply squint and continue, but I think I need to pause and try to get a little bit of a grasp on what $V//G$ is.

How do we think about $V//G$ in this set up?

My next goal is to understand “transgression”. Tracing back references led me to your “Integration wihout Integration” paper, which I remember seeing but couldn’t spend the time to absorb it. Neat idea!

Now that I think about it a litte bit, it reminds me of one of our old conversation from way back when at the String Coffee Table:

Before the Flood

Were we talking about transgression? I was trying to pull back a 1-form on a base manifold to a 0-form on loop space so that integration on the base manifold was evaluation on loop space. Is it at all related?

*glimmer of hope to actually understand this some day*

Thanks

Okay, let’s start going through this in small steps. I propose some steps, and you let me know what you think.

Step 1) A group as a groupoid.

Given any group $G$, there is a groupoid with a single object and one morphism per element of $G$, with composition of morphisms coming from the product in $G$.

This construction is so trivial that often it is taken as being empty. But it is not and we need the distinction between a monoidal set (for instance a group) and the one-object category it gies rise to (for instance a groupoid) later on.

So I write $$BG=\{•\stackrel{g}{\to }•\mid g\in G\}$$

for the one-object groupoid version of the group $G$.

Okay?

Step 2) A representation as a functor.

We eventally want to talk about sections of associated vector bundles. That’ll be easy once we realize that a representation of a group, i.e. a group homomorphism

$$\rho :G\to \mathrm{GL}(V)$$

for some vector space $V$ is nothing but a functor

$$\rho :BG\to \mathrm{Vect}$$

which sends $$\rho :(•\stackrel{g}{\to }•)\mapsto (V\stackrel{\rho (g)}{\to }V).$$

Okay?

Step 3) An action groupoid from a representation.

The construction of the path integral that we are approaching here will be entirely formulated in terms of spans of action groupoids. So we need to understand action groupoids.

Given a representation $\rho :BG\to \mathrm{Vect}$ as above, we define a groupoid to be denoted $$V//G$$ as follows: its objects are the elements of $V$ (i.e. the vectors in $V$) and from each element $v$ there starts precisey one morphism per element in $g$ which goes to $\rho (g)(v)$:

$$V//G:=\{v\stackrel{g}{\to }\rho (g)(v)\mid v\in V,g\in G\}.$$

Composition of morphisms is the obvious one coming from the product in the group.

Okay so far?

As an aside, to be skipped by those who want to stick to the elementary discussion, I mention the more abstract definition of $V//G$ that will eventually provide the big abstract picture to be developed later on:

Over the category of (small) Sets there sits the category of pointed (small) sets $T_{\mathrm{pt}}\mathrm{Set}$ (forming the “universal Set-bundle”)

$$\begin{array}{c}{T}_{\mathrm{pt}}\mathrm{Set}\\ \downarrow \\ \mathrm{Set}\end{array}$$

Using our representation and the forgetful functor $\mathrm{Vect}\to \mathrm{Set}$ we can pull back this universal Set-bundle to $BG$:

$$\begin{array}{ccccc}V//G& \to & T_{\mathrm{pt}}\mathrm{Vect}& \to & T_{\mathrm{pt}}\mathrm{Set}\\ \downarrow & & \downarrow & & \downarrow \\ BG& \stackrel{\rho }{\to }& \mathrm{Vect}& \to & \mathrm{Set}\end{array}.$$

This witnesses $V//G$ as the groupoid incarnation of the $\rho$-associated vector bundle to the universal $G$-bundle. But never mind that.

Step 4) enter $V$-colored sets

Consider the simple situation where there is some finite set $S$ and a map of that set down to the objects of $V//G$:

$$s:S\to V//G.$$

Those objects are just the elements of $V$, so the map $$s:S\to V$$ defines a “$V$-coloring” of $S$. Each element $a$ of $S$ is sent to some vector $s(a)$ in $V$, so we can think of it as being labeled or colored by $s(a)$.

Since $V$ is a vector space we can add elements in $V$. This allows to define a “cardinality of $V$-colored sets”

$$\mid \cdot \mid :\mathrm{SetsOver}(V)\to V$$

by

$$\mid S\mid =\sum _{a\in S}s(a).$$

so our finite set $s:S\to V$ in particular defines a single vector in $V$. (Notice that the empty set maps to the 0-vector.) It is like a single vector equipped with extra information for how that vector was obtained by adding a couple of other vectors.

Okay so far?

Step 5) Finally: vector bundles.

Pick your favorite finite category and allow me to call is $P_1(X)$. That’s because we will think of its objects as points in some space and of its morphisms as paths between these points.

Consider a functor $$\mathrm{tra}:P_1(X)\to BG$$ which labels each path with a group element, such that composition of paths corresponds to multiplication of group elements.

Such a functor we can regard as a $G$-connection on a trivial $G$-bundle over $X$.

Using our representation, we can turn this into a connection on the associated vector bundle, simply by composing functors:

$${\rho }_{*}\mathrm{tra}:P_1(X)\stackrel{\mathrm{tra}}{\to }BG\stackrel{\rho }{\to }\mathrm{Vect}.$$

This functor sends each point of $X$ to the vector space $V$, to be thought of as the fiber of a vector bundle over that point, and sends a path labeled by the group element $g$ to the linear map $V\stackrel{\rho (g)}{\to }V$ between the vector spaces over its endpoints.

Okay?

Step 6) Sections of the vector bundle.

we can combine now the idea which led to the action groupoid with the functor ${\rho }_{*}\mathrm{tra}$ to get something like the action groupoid of $P_1(X)$ acting on $V$ by $\mathrm{tra}$ and $\rho$.

this groupoid I’ll call $${\mathrm{tra}}^{*}V//G.$$

Its objects are pairs $(x,v)$ consisting of a point $x$ in $X$ and a vector $v\in V$. For each path $x\stackrel{\gamma }{\to }y$ in $P_1(X)$ there is one morphism in ${\mathrm{tra}}^{*}V//G$ starting at each $(x,v)$ which goes to $(y,\rho (\mathrm{tra}(\gamma ))(v))$

$$(x,v)\stackrel{(x\stackrel{\gamma }{\to }y)}{\to }(y,\rho (\mathrm{tra}(\gamma ))(v)).$$

You can forget the morphisms for the time being, actually. Doing so, we are left with nothing but the set $X\times V$ of objects: there is the vector space $V$ sitting over each point of $X$.

Then again consider a finite set $$s:S\to {\mathrm{tra}}^{*}V//G$$ sitting over our groupoid. This is hence the same as nothing but a map of sets $$s:S\to X\times V.$$ But this is nothing but a bundle of $V$-colored sets over $X$! It is one $V$-colored set over each point of $X$.

Do you see that?

Taking our cardinality of $V$-colored-sets, we thus obtain from $S\to X\times V$ an assignment of one vector in $V$ over $x$ for each point $x$ of $X$. That’s a section of our vector bundle.

Okay so far?

Here again a more abstract description, to be skipped if you don’t want to see it.

The groupoid ${\mathrm{tra}}^{*}V//G$ is the pullback $$\begin{array}{ccccccc}{\mathrm{tra}}^{*}V//G& \to & V//G& \to & T_{\mathrm{pt}}\mathrm{Vect}& \to & T_{\mathrm{pt}}\mathrm{Set}\\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ P_1(X)& \stackrel{\mathrm{tra}}{\to }& BG& \stackrel{\rho }{\to }& \mathrm{Vect}& \to & \mathrm{Set}\end{array}.$$

Sections of our vector bundle $E\to X$ (at the moment being just the trivial vector bundles $E=X\times V$, but that will change as we proceed) correspond to (finite)set-bundles over this groupoid:

$$\mathrm{FiniteSetsOver}({\mathrm{tra}}^{*}V//G)\to >\Gamma (E\to X).$$

We could also talk about flat sections of $E$, since we have a connection in the game. These would come from full functors down to ${\mathrm{tra}}^{*}V//G$.

Step 7) Transgression

The next step would be transgression of $\mathrm{tra}$ to path space. Then we’d repeat all the above steps there and end up with a span of action groupoids. Pull-pushing sections through that span will realize the path integral.

But before doing that, let me know if you follow all of the steps above. Or else, let me know about what is unclear.