## July 27, 2007

### Arrow-Theoretic Differential Theory

#### Posted by Urs Schreiber

Using the concept of tangent categories (derived from that of supercategories) I had indicated how to refine my previous discussion of $n$-curvature. Here are more details.

Abstract: We propose and study a notion of a tangent $\left(n+1\right)$-bundle to an arbitrary $n$-category. Despite its simplicity, this notion turns out to be useful, as we shall indicate.

1 Introduction … 1
2 Main results … 2
2.1 Tangent $\left(n+1\right)$-bundle … 3
2.2 Vector fields and Lie derivatives … 4
2.3 Inner automorphism n-groups … 4
2.4 Curvature and Bianchi Identity for functors … 5
2.4.1 General functors … 5
2.4.2 Parallel transport functors and differential forms … 6
2.5 Sections and covariant derivatives … 6
3 Differential arrow theory … 8
3.1 Tangent categories … 8
3.2 Differentials of functors … 11
4 Parallel transport functors and their curvature … 12
4.1 Principal parallel transport … 12
4.1.1 Trivial G-bundles with connection … 12

Introduction

Various applications of ($n$-)categories in quantum field theory indicate that ($n$-)categories play an important role over and above their more traditional role as mere organizing principles of the mathematical structures used to describe the world: they appear instead themselves as the very models of this world.

For instance there are various indications that thinking of configuration spaces and of physical processes taking place in these as categories, with the configurations forming the objects and the processes the morphisms, is a step of considerably deeper relevance than the tautological construction it arises from seems to indicate.

While evidence for this is visible for the attentive eye in various modern mathematical approaches to aspects of quantum field theory – for instance [FreedQuinn], [Freed] but also [Willerton] – the development of this observation is clearly impeded by the lack of understanding of its formal underpinnings.

If we ought to think of configuration spaces as categories, what does that imply for our formulation of physics involving these configuration spaces? In particular: how do the morphisms, which we introduce when refining traditional spaces from 0-categories to 1-categories, relate to existing concepts that must surely secretly encode the information contained in these morphisms. Like tangent spaces for instance.

Possibly one of the first places where this question was at all realized as such is [Isham]. That this is a piece of work which certainly most physicists currently won’t recognize as physics, while mathematicians might not recognize it as interesting mathematics, we take as further indication for the need of a refined formal analysis of the problem at hand.

Several of the things we shall have to say here may be regarded as an attempt to strictly think the approach indicated in Isham’s work to its end. Our particular goal here is to indicate how we may indeed naturally, generally and usefully relate morphisms in a category to the wider concept of tangency.

For instance his “arrow fields” on categories we identify as categorical tangents to identity functors on categories and find their relation to ordinary vector fields as well as to Lie derivatives, thereby, by the nature of arrow-theory, generalizing the latter concepts to essentially arbitrary categorical contexts.

While there is, for reasons mentioned, no real body of literature yet, which we could point the reader to, on the concrete question we are aiming at, the reader can find information on the way of thinking involved here most notably in the work of John Baez, the spiritus rector of the idea of extracting the appearance of $n$-categories as the right model for the notion of state and process in physics. In particular the text [BaezLauda] as well as the lecture notes [Baez] should serve as good background reading.

The work that our particular developments here have grown out is described in [S1, S2]. Our discussion of the Bianchi identity for $n$-functors should be compared with the similar but different constructions in the world of $n$-fold categories given in [Kock].

Main results

Our working model for all concrete computations in the following is $2\mathrm{Cat}$, the Gray category whose objects are strict 2-categories, whose morphisms are strict 2-functors, whose 2-morphisms are pseudonatural transformations and whose 3-morphisms are modifications of these. It is clear that all our statements ought to have analogs for weaker, more general and higher $n$ versions of $n$-categories. But with a good general theory of higher $n$-categories still being somewhat elusive, we won’t bother to try to go beyond our model $2\mathrm{Cat}$.

So we shall now set $n=2$ once and for all and take the liberty of using $n$ instead of 2 in our statements, to make them look more suggestive of the general picture which ought to exist.

Tangent $\left(n+1\right)$-bundle

We define for any $n$-category $C$ an $n$-category $\mathrm{TC}$ which is an $\left(n+1\right)$-bundle $p:\mathrm{TC}\to \mathrm{Obj}\left(C\right)$ over the space of objects of $C$. This we address as the tangent bundle of $C$.

The definition of this tangent bundle is morally similar to but in detail somewhat different form the way tangent bundles are defined in synthetic differential geometry and in supergeometry:

we consider the category $\mathrm{pt}:=\left\{•\stackrel{\sim }{\to }\circ \right\}$ as an arrow-theoretic model for the “infinitesimal interval” or the “superpoint” in that it is a puffed-up version of the mere point $\mathrm{pt}:=\left\{•\right\}$ to which it is equivalent, by way of the injection $\mathrm{pt}↪\mathrm{pt}\phantom{\rule{thinmathspace}{0ex}},$ but not isomorphic. This suble difference, rooted deeply in the very notion of category theory, we claim usefully models the notion of tangency as “extension which hardly differs from no extension”. Concretely, we consider $\mathrm{TC}\subset {\mathrm{Hom}}_{n\mathrm{Cat}}\left(\mathrm{pt},C\right)$ to be that subcategory of morphisms from the fat point into $C$ which collapses to a 0-category after pulled back to the point $\mathrm{pt}$.

The characteristic property of the tangent $\left(n+1\right)$-bundle is that it sits inside the short exact sequence $\mathrm{Mor}\left(C\right)\to TC\to C\phantom{\rule{thinmathspace}{0ex}}.$

Finally, for later use notice that dual to its realization as a projection $TC\to \mathrm{Obj}\left(C\right)$ the tangent bundle may be thought of as an $n$-functor $TC:{C}^{\mathrm{op}}\to n\mathrm{Cat}$ which sends objects $a$ to the tangent categories ${T}_{a}C$ over them and sends morphisms the the pullback of these along them $TC\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\left(a\stackrel{f}{←}b\right)\phantom{\rule{thickmathspace}{0ex}}↦\phantom{\rule{thickmathspace}{0ex}}\left({T}_{a}C\stackrel{{T}_{f}C}{\to }{T}_{b}C\right)\phantom{\rule{thinmathspace}{0ex}}.$

Vector fields and Lie derivatives

Let $X$ be a smooth manifold and let ${P}_{1}\left(X\right)$ be the groupoid of thin homotopy classes of paths in $X$.

Then ordinary vector fields $v\in \Gamma \left(\mathrm{TX}\right)$ on $X$ are in canonical bijection with smooth 1-parameter families of categorical tangent vectors to the identity map on ${P}_{1}\left(X\right)$: $\Gamma \left(\mathrm{TX}\right)\phantom{\rule{thickmathspace}{0ex}}\stackrel{\sim }{\to }\phantom{\rule{thickmathspace}{0ex}}\left\{\Sigma ℝ\to \Sigma {T}_{{\mathrm{Id}}_{{𝒫}_{1}\left(X\right)}}\left(\mathrm{Cat}\right)\right\}\phantom{\rule{thinmathspace}{0ex}}.$

On a general category $C$, it may be useful to consider generalizations of this where $ℝ$ is replaced by some other group $G$. We speak of $G$-flow on a category, in this general case.

The “arrow fields” on a category $C$, considered by Isham in [Isham], are $ℤ$-flows $\left\{\Sigma ℤ\to \Sigma {T}_{{\mathrm{Id}}_{C}}\left(\mathrm{Cat}\right)\right\}$ on $C$.

On the other hand, the identitfication of $TC$ itself (as opposed to ${T}_{{\mathrm{Id}}_{C}}\mathrm{Cat}$) with ordinary vectors for suitable choice of $C$ is both more subtle and more interesting than the above. This will be discussed elsewhere, once fully worked out.

Inner automorphism $n$-groups

Of particular importance are the tangent bundles, in our sense, to $n$-categories which are 1-object $\left(n-1\right)$-groupoids $\Sigma {G}_{\left(n\right)}$, hence $n$-groups ${G}_{\left(n\right)}$. In our context these $\left(n-1\right)$-groupoids must be thought of as 1-point orbifolds. Accordingly, they have just a single “tangent space” (tangent $n$-category) ${T}_{•}\Sigma {G}_{\left(n\right)}\phantom{\rule{thinmathspace}{0ex}}.$

This turns out to have interesting properties [Roberts-S]:

$•$

For $G$ an ordinary group, one finds that ${T}_{•}\Sigma G\simeq {T}_{{\mathrm{Id}}_{\Sigma G}}\left(\mathrm{Cat}\right)$ is a 2-group, which we call $\mathrm{INN}\left(G\right)$. It sits inside the exact sequence $Z\left(G\right)\to \mathrm{INN}\left(G\right)\to \mathrm{AUT}\right)\left(G\right)\to \mathrm{OUT}\left(G\right)$ of 1-groupoids. Here $Z\left(G\right)$ is the categorical center of $\Sigma G$ (which coincides with the ordinary center of $G$), regarded as a 1-object groupoid. This identifies $\mathrm{INN}\left(G\right)$ as the 2-group of inner automorphisms of $G$. But $\mathrm{INN}\left(G\right)$ also sits inside the exact sequence $\mathrm{Disc}\left(G\right)\to \mathrm{INN}\left(G\right)\to \Sigma G\phantom{\rule{thinmathspace}{0ex}}.$ Moreover, it is equivalent to the trivial 2-group, hence “contractible”. This identifies $\mathrm{INN}\left(G\right)$ as the categorical version of the universal $G$-bundle.

$•$

For ${G}_{\left(2\right)}$ a strict 2-group, one finds that ${T}_{•}\Sigma {G}_{\left(2\right)}\subset {T}_{{\mathrm{Id}}_{\Sigma {G}_{\left(2\right)}}}\left(2\mathrm{Cat}\right)$ is a 3-group, which we call $\mathrm{INN}\left({G}_{\left(2\right)}\right)$. It sits inside the exact sequence $Z\left({G}_{\left(2\right)}\right)\to \mathrm{INN}\left({G}_{\left(2\right)}\right)\to \mathrm{AUT}\right)\left({G}_{\left(2\right)}\right)\to \mathrm{OUT}\left({G}_{\left(2\right)}\right)$ of 2-groupoids. Here $Z\left({G}_{\left(2\right)}\right)$ is the 2-categorical center of $\Sigma G$, regarded as a 1-object 2-groupoid. This identifies $\mathrm{INN}\left({G}_{\left(2\right)}\right)$ as the 3-group of inner automorphisms of $G$. But $\mathrm{INN}\left({G}_{\left(2\right)}\right)$ also sits inside the exact sequence $\mathrm{Disc}\left({G}_{\left(2\right)}\right)\to \mathrm{INN}\left({G}_{\left(2\right)}\right)\to \Sigma {G}_{\left(2\right)}\phantom{\rule{thinmathspace}{0ex}}.$ Moreover, it is equivalent to the trivial 3-group, hence “contractible”. This identifies $\mathrm{INN}\left({G}_{\left(2\right)}\right)$ as the categorical version of the universal ${G}_{\left(2\right)}$-2-bundle.

Curvature and Bianchi Identity for functors

General functors

Using the functorial incarnation $\mathrm{TC}:{C}^{\mathrm{op}}\to n\mathrm{Cat}$ of the tangent bundle, we may push forward any $n$-functor $F:C\to D$ to a connection on the tangent bundle of $C$, simply by postcomposing $dF:C\stackrel{F}{\to }D\stackrel{TD}{\to }n\mathrm{Cat}\phantom{\rule{thinmathspace}{0ex}}.$ The crucial point of this construction is that it extends uniquely (up to equivalence) to an $\left(n+1\right)$-functor $dF:{C}_{\left(n+1\right)}\to n\mathrm{Cat}$ on the $\left(n+1\right)$-category ${C}_{\left(n+1\right)}:=\mathrm{Codisc}\left(C\right)$ which is obtained from $C$ by replacing all Hom-($n-1$)-categories by the corresponding codiscrete $n$-groupoids over them.

By introducing the terminology

$•$ $dF$ is the curvature of $F$
$•$ $F$ is flat if $dF$ is degenerate (sends all $\left(n+1\right)$-morphisms to identities)

we obtain the technically easy but conceptually important generalization of the Bianchi identity: for any functor $F$

$•$ $dF$ is flat

or equivalently

$•$ $ddF$ is degenerate.

Parallel transport functors and differential forms

When $F:C\to D$ is the smooth parallel transport functor \cite{transport} in an $n$-bundle with connection [Bartels,Baez-S,S3], the arrow-theoretic notion of curvature described above does reproduce the theory of curvature forms of connection forms. The general Bianchi identity we have discussed then reduces to the ordinary Bianchi identity familiar from differential geometry.

More precisely, let $C:={𝒫}_{3}\left(X\right)$ be the strict 3-groupoid of thin homotopy classes of $k$-paths in a smooth manifold $X$. And let ${G}_{\left(2\right)}$ be a strict Lie 2-group coming from the Lie crossed module $H\stackrel{t}{\to }G\stackrel{\alpha }{\to }\mathrm{Aut}\left(H\right)$.

Then, according to [S-Waldorf,Baez-S,S3,Roberts-S] we have the following bijections of smooth $n$-functors with differential forms

$•$ $\left\{\mathrm{smooth}\phantom{\rule{thickmathspace}{0ex}}1-\mathrm{functors}{𝒫}_{1}\left(X\right)\to \Sigma G\right\}\stackrel{\sim }{\to }\left\{A\in {\Omega }^{1}\left(X,\mathrm{Lie}\left(G\right)\right)\right\}$

$•$

$\left\{\mathrm{smooth}\phantom{\rule{thickmathspace}{0ex}}2-\mathrm{functors}{𝒫}_{2}\left(X\right)\to \Sigma {G}_{\left(2\right)}\right\}\stackrel{\sim }{\to }\left\{\left(A,B\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{F}_{A}+{t}_{*}\circ B=0\right\}$

$•$

$\left\{\mathrm{smooth}\phantom{\rule{thickmathspace}{0ex}}3-\mathrm{functors}{𝒫}_{3}\left(X\right)\to \Sigma \mathrm{INN}\left({G}_{\left(2\right)\right)}\right\}\stackrel{\sim }{\to }\left\{\left(A,B,C\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}C={d}_{A}B\right\}$

Now let $\mathrm{tra}:{P}_{1}\left(X\right)\to \Sigma G$ be a smooth 1-functor with values in the Lie group $G$. Then, under these bijections, we find that its curvatures correspond to the following differential forms at top level $\begin{array}{rl}\mathrm{tra}& ↦A\\ \mathrm{curv}=d\mathrm{tra}& ↦{F}_{A}:=dA+A\wedge A\\ d\mathrm{curv}=dd\mathrm{tra}& ↦{d}_{A}{F}_{A}=0\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

This way the ordinary Bianchi identity for the curvature 2-form ${F}_{A}$ of $A$ is reproduced. Notice that for this result come out the way it does, just by turning our abstract crank for differential arrow-theory, the result of \ref{Inner automorphism $n$-groups} is crucial, which says that the curvature $\left(n+1\right)$-functor of a ${G}_{\left(n\right)}$-transport is itself an $\mathrm{INN}\left({G}_{\left(n\right)}\right)$-transport.

Sections and covariant derivatives

The curvature $\mathrm{curv}=d\mathrm{tra}$ of a parallel transport $n$-functor is typically trivializable, in that it admits morphisms $e:I\stackrel{\sim }{\to }\mathrm{curv}$ for $I$ some “trivial” $\left(n+1\right)$-transport. As with the inner automorphism $\left(n+1\right)$-groups $\mathrm{INN}\left({G}_{\left(n\right)}\right)$, this trivializability, far from making these objects uninteresting, turns out to control the entire theory.

(Compare this to the contractibility of the universal $G$-bundle: while equivalent to a point, it is far from being an uninteresting object, due to the morphisms which go into and out of it. According to the above this comparison is far more than an mere analogy.)

A basic fact of $n$-category theory has major implications here:

recall that for $F$ and $G$ $n$-functors, a transformation $G\to F$ is given in components itself by an $\left(n-1\right)$-functor. Now if $G$ is trivial in some sense to be made precise, and if the transformation is an equivalence $I\stackrel{{f}_{\sim }}{\to }F\phantom{\rule{thinmathspace}{0ex}},$ then this implies that the $n$-functor $F$ is entirely encoded in the $\left(n-1\right)$-functor $f$.

We show that for $F=\mathrm{curv}=d\mathrm{tra}$ the curvature $\left(n+1\right)$-functor of a transport $n$-functor $\mathrm{tra}$, the latter essentially encodes the component map of the transformation $I\stackrel{{\mathrm{tra}}_{\sim }}{\to }\mathrm{curv}\phantom{\rule{thinmathspace}{0ex}}.$ In components this is nothing but a generalization of Stokes’ law ${\int }_{X}d\omega ={\int }_{\partial X}\omega \phantom{\rule{thinmathspace}{0ex}}.$

Moreover, it turns out that there may be other trivializations of $\mathrm{curv}$, not by isomorphisms but by mere equivalences. On objects, the component functions of these correspond to sections of the original bundle. On morphisms it corresponds, under the identification of smooth functors and differential forms mentioned above to the covariant derivative of these sections.

In [S1] it is indicated how all these statements have a quantum analogue as we push our $n$-functors forward. There it is indicated how the fact that transport $n-\mathrm{functors}$ have sectins which are themselves transport $\left(n-1\right)$-functors translates in the context of extended functorial quantum field theory to essentially what is known in physics as the holographic principle. This needs to be discussed elsewhere, clearly.

Posted at July 27, 2007 4:10 PM UTC

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### Re: Arrow-Theoretic Differential Theory

Hi,
So just as a background I have a beginning graduate level knowledge of diff geo, am half-way through Saunders Mac Lane, and read Baez and Dolan’s Categorification only once. So I’m not exactly the target audience of this…yet.

That being said, there’s a couple of things early on that made my brain grind to a halt.

First, in what way are pt and pt equivalent? It looks like you are just including pt in as the first point of pt and then the equivalence is just given by lifting up or projecting down the first point. The context makes it sound more subtle and complicated than that though, so what am I missing?

Also, I feel like I’ve read the first section a few times and that I’m still not seeing the connection between tangent vectors/bundles from diff geo with what you’re doing here. I’m used to a vector field being a map from the smooth functions on a manifold to itself. I don’t really see how what we’re defining is equivalent to that for 0-categories (is it supposed to?). Is the answer related to the requirement of Mor(C) -> TC -> C being a short exact sequence? What exactly is Mor(C)?

Sorry for the questions, but I appreciate any help.

Posted by: Creighton Hogg on July 27, 2007 6:51 PM | Permalink | Reply to this

### Re: Arrow-Theoretic Differential Theory

Oops. I think some of my questions may be answered by looking through older posts. My apologies.

Posted by: Creighton Hogg on July 27, 2007 7:31 PM | Permalink | Reply to this

### Re: Arrow-Theoretic Differential Theory

Hi,

Why are $\mathrm{pt}$ and $\mathrm{pt}$ equivalent? It looks like you are just including $\mathrm{pt}$ in as the first point of $\mathrm{pt}$ and then the equivalence is just given by lifting up or projecting down the first point. The context makes it sound more subtle and complicated than that though, so what am I missing?

It’s not supposed to be subtle or complicated. In fact, it’s supposed to be a very elementary fact. I am just stressing it because my goal is to use that elementary fact for something which seems to be kind of interesting.

Just for those reading this and wondering:

the two categories which I called $\mathrm{pt}$ and $\mathrm{pt}$ are equivalent as categories in that there is the injection functor $\mathrm{pt}↪\mathrm{pt}$ and the unique projection functor $\mathrm{pt}\to \mathrm{pt}$ such that $\mathrm{pt}↪\mathrm{pt}\to \mathrm{pt}$ is strictly the identity functor on $\mathrm{pt}$ (the only endomorphism of $\mathrm{pt}$ there is, of course). Moreover, the composition $\mathrm{pt}\to \mathrm{pt}↪\mathrm{pt}$ is, while not equal to the identity on $\mathrm{pt}$, isomorphic to the identity functor on $\mathrm{pt}$, as one easily checks.

So that’s all I mean: these two categories are equivalent. Which is to say: the inclusion $\mathrm{pt}↪\mathrm{pt}$ is essentially surjective, full and faithful. Trivially.

I’m still not seeing the connection between tangent vectors/bundles from diff geo with what you’re doing here. I’m used to a vector field being a map from the smooth functions on a manifold to itself. I don’t really see how what we’re defining is equivalent to that for 0-categories (is it supposed to?).

I give a characterization of ordinary vector fields in terms of tangents to the identity functor on the path groupoid of a manifold.

But here is another way to say this, which is maybe more easily compared to the usual definition:

Take $X$ to be a smooth manifold and ${P}_{1}\left(X\right)$ the groupoid whose objects are the points of $X$ and whose morphisms are thin-homotopy classes of paths in $X$.

Then form the tangent 2-bundle $T{P}_{1}\left(X\right)$ and look at its sections $\mathrm{Obj}\left(X\right)=X\to T{P}_{1}\left(X\right)$. The crucuial point is that the space of these sections of the categorical tangent bundle, $\Gamma \left(T{P}_{1}\left(X\right)\right)$ has a monoidal structure (since it is rather a “path bundle”. The monoidal structure comes from composing these paths.)

The claim is that ordinary vector fields are smooth 1-parameter families of such categorical sections, respecting this monoidal structure. In formulas: $\Gamma \left(TX\right)\simeq \mathrm{Hom}\left(ℝ,\Gamma \left(T{P}_{1}\left(X\right)\right)\right)\phantom{\rule{thinmathspace}{0ex}},$ where the $\mathrm{Hom}\left(\cdots \right)$ here denotes group homomorphisms under the monoidal structure on both sides. So I’d also write this ${\mathrm{Hom}}_{\mathrm{Cat}}\left(\Sigma \left(ℝ\right),\Sigma \left(\Gamma \left(T{P}_{1}\left(X\right)\right)\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$ This works by sending each vector field $v$ to the flow lines $t↦\mathrm{exp}\left(v\right)\left(t\right)$ starting at each point of $X$.

Somewhere in my notes I remark that I would like to give a more detailed discussion of this particular issue later on. So I am hoping to get back to this point. The thing is, I would like to see not just the tangent bundle showing up, but also the fact that its “dual” in the present context is the algebra of differential forms on $X$. I am beginning to see how it works, but I still need to think more about it.

Posted by: Urs Schreiber on July 30, 2007 11:54 AM | Permalink | Reply to this

### Re: Arrow-Theoretic Differential Theory

The claim is that ordinary vector fields are smooth 1-parameter families of such categorical sections, respecting this monoidal structure. In formulas:

(1)$\Gamma \left(\mathrm{TX}\right)\cong \mathrm{Hom}\left(ℝ,{\mathrm{TP}}_{1}\left(X\right)\right).$

I am growing to understand and appreciate this tangent category concept. Let’s see… a section $\gamma \in \Gamma \left({\mathrm{TP}}_{1}\left(X\right)\right)$ of the categorical tangent bundle is precisely a smooth choice of a path emanating from each point $x\in X$, for each $x$. By the way, isn’t that the same thing as saying that the space of sections of the categorical tangent bundle is the space of Isham’s arrow fields,

(2)$\Gamma \left(T{P}_{1}\left(X\right)\right)=\mathrm{Arr}\left({P}_{1}\left(X\right)\right)?$

Anyhow, what you are telling us is that a vector field on a manifold $X$ can be thought of as a homomorphism from $ℝ$ into $\Gamma \left(T{P}_{1}\left(X\right)\right)$.

It seems you left out the “space of sections” symbol $\Gamma \text{Unknown character}\mathrm{above}...\mathrm{as}I\mathrm{understand}\mathrm{it},\mathrm{you}\mathrm{meant}\mathrm{to}\mathrm{write}\text{Unknown character}/p\text{Unknown character}\text{Unknown character}\mathrm{div}\mathrm{class}=\text{Unknown character}\mathrm{numberedEq}\text{Unknown character}\text{Unknown character}\text{Unknown character}\mathrm{span}\text{Unknown character}\left(3\right)\text{Unknown character}/\mathrm{span}\text{Unknown character}$$\Gamma \left(\mathrm{TX}\right)\cong \mathrm{Hom}\left(ℝ,\Gamma \left({\mathrm{TP}}_{1}\left(X\right)\right)\right).$$\text{Unknown character}/\mathrm{div}\text{Unknown character}\text{Unknown character}p\text{Unknown character}\mathrm{In}a\mathrm{slogan},\text{Unknown character}/p\text{Unknown character}\text{Unknown character}p\text{Unknown character}\text{Unknown character}i\text{Unknown character}A\mathrm{vector}\mathrm{field}\mathrm{on}a\mathrm{manifold}$X$\mathrm{is}\mathrm{the}\mathrm{same}\mathrm{thing}\mathrm{as}a\mathrm{field}\mathrm{of}\mathrm{germinating}\mathrm{paths}\prime \mathrm{on}$X$.\text{Unknown character}/i\text{Unknown character}\text{Unknown character}/p\text{Unknown character}\text{Unknown character}p\text{Unknown character}\mathrm{Lol}!I\mathrm{don}\prime t\mathrm{know}\mathrm{how}\mathrm{to}\mathrm{write}\mathrm{it}\mathrm{properly}\mathrm{in}a\mathrm{slogan}.I\mathrm{am}\mathrm{tring}\mathrm{to}\mathrm{deliver}\mathrm{the}\mathrm{point}\mathrm{that}\mathrm{it}\mathrm{should}\mathrm{be}a\text{Unknown character}i\text{Unknown character}\mathrm{homomorphism}\text{Unknown character}/i\text{Unknown character}\mathrm{from}$\mathbb{R}$\mathrm{into}$\Gamma (T P_1 (X)))$.A\mathrm{germinating}\mathrm{path}\prime \mathrm{is}\mathrm{like}a\mathrm{little}\mathrm{flower}\mathrm{that}\mathrm{grows}\mathrm{at}a\mathrm{point}$x \in X$.\mathrm{At}\mathrm{time}$t=0$, it’s just a point, but as time increases, the length of the path grows. Posted by: Bruce Bartlett on July 30, 2007 3:48 PM | Permalink | Reply to this ### Re: Arrow-Theoretic Differential Theory isn’t that the same thing as saying that the space of sections of the categorical tangent bundle is the space of Isham’s arrow fields, Indeed! I really tried to emphasize precisely this in my notes. Maybe I should have emphasized it even more! :-) It seems you left out the “space of sections” Oh, right. Thanks for catching that! I have corrected it now. Posted by: Urs Schreiber on July 30, 2007 3:56 PM | Permalink | Reply to this ### Re: Arrow-Theoretic Differential Theory By the way: the exchange with David Ben-Zvi over in the thread on Tangent Categories made me realize that there is a maybe much better way to say what I have been saying about how sections of the tangent category of the path groupoid reproduce ordinary vector fields: as we said, the point is that sections of the tangent category have (if everything is in the world of groupoids) a group structure on them, which can be seen to be that inherited from the embedding $\Gamma \left(TC\right)\subset {T}_{{\mathrm{Id}}_{C}}\left(\mathrm{End}\left(C\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$ So instead of saying that tangent fields are 1-parameter subgroups of this group, we can just say that sections of the ordinary tangent bundles are the Lie algebra elements of the sections of the tangent category $\Gamma \left(TX\right)=\mathrm{Lie}\left(\Gamma \left(T{P}_{1}\left(X\right)\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$ Posted by: Urs Schreiber on August 1, 2007 9:35 AM | Permalink | Reply to this ### Re: Arrow-Theoretic Differential Theory $\Gamma \left(TX\right)=\mathrm{Lie}\left(\Gamma \left(T{P}_{1}\left(X\right)\right)\right)$ Formulating it this way, looks like it gets us one more step closer to see the supergeometry in the categorical formulation: for ${ℝ}^{0\mid 1}$ the superpoint, we have that $\Pi TX:=\mathrm{Hom}\left({ℝ}^{0\mid 1},X\right)$ is the odd tangent bundle, in that the “algebra of functions” on $\Pi TX$ is ${C}^{\infty }\left(\Pi TX\right)={\Omega }^{•}\left(X\right)$ the graded commutative algebra of differential forms on $X$. Now, of course, as we consider the $\left(n+1\right)$-group structure on sections $\Gamma \left(TC\right)$ of the tangent category, and then pass to its Lie $\left(n+1\right)$-algebra $\mathrm{Lie}\left(\Gamma \left(TC\right)\right)$, we run into the familiar situation that these Lie $\left(n+1\right)$-algebras are themselves canonically isomorphic to codifferential garded co-commutative coalgebras, which have a dual description in terms of graded commutative differential algebras. It’s precisely this step to the dual of the Lie $\left(n+1\right)$-algebra which indeed produces the algebra of differential forms here, if the underlying Lie $\left(n+1\right)$-algebra involves the Lie algebra of vector fields. So it seems that in order to find in which sense $TC\subset n\mathrm{Func}\left(\mathrm{pt},C\right)$ produces the categorical analogue of the odd tangent bundle, one needs to find the categorical analogue of the passage from codifferential coalgebra encoding Lie $n$-algebras to their dual differetial algebras. Hm… Posted by: Urs Schreiber on August 1, 2007 11:05 AM | Permalink | Reply to this ### Re: Arrow-Theoretic Differential Theory So instead of saying that tangent fields are 1-parameter subgroups of this group, we can just say that sections of the ordinary tangent bundles are the Lie algebra elements of the sections of the tangent category $\Gamma \left(\mathrm{TX}\right)=\mathrm{Lie}\left(\Gamma \left({\mathrm{TP}}_{1}\left(X\right)\right)\right)$. Nice. Can you explain the relationship between the group $\mathrm{Diff}\left(X\right)$ and $\Gamma \left({\mathrm{TP}}_{1}\left(X\right)\right)$? The lie algebras of both groups are the vector fields on $X$. When I try to understand geometrically the difference between them by drawing pictures in my head, I get a little confused. Posted by: Bruce Bartlett on August 1, 2007 1:39 PM | Permalink | Reply to this ### Re: Arrow-Theoretic Differential Theory Can you explain the relationship between the group $\mathrm{Diff}\left(X\right)$ and $\Gamma \left(T{P}_{1}\left(X\right)\right)$? Yes, I can try. I shoud admit, though, that I can see the details of the claim about the Lie algebra of the space of sections of the tangent category $TC$ in detail for $C=\Sigma {G}_{n}$ a suspended Lie $n$-group. For $C={P}_{1}\left(X\right)$ there might be subtleties lurking here which I am glossing over. I am not quite sure yet. But here is the main idea: the crucial thing to realize is the embedding $\Gamma \left(TC\right)\subset {T}_{{\mathrm{Id}}_{C}}\left(\mathrm{End}\left(C\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$ For 1-categories $C$ this is fairly easy. For 2-categories $C$ this is proposition 3 in the paper by David Roberts in myself. Do you see how it works for 1-categories? It’s really kind of trivial, but important: each “arrow field” on $C$ you regard as the component map of a natural transformation ${\mathrm{Id}}_{C}\to F\phantom{\rule{thinmathspace}{0ex}},$ where $F:C\to C$ is some automorphism of $C$. So: this way each section of the tangent category of the path groupoid ${P}_{1}\left(X\right)$ is two things: 1) an automorphism of ${P}_{1}\left(X\right)$ which is required to be connected to the identity. On objects this is a diffeomorphism of $X$, if we require everything to be smooth. 2) A specified choice of isomorphism connecting $F$ to the identity – that’s the precise choice of path field in which each path connects a point with its image under the given diffeomorphism of $X$. So $\Gamma \left(T{P}_{1}\left(X\right)\right)$ is actually a little larger than ${\mathrm{Diff}}_{0}\left(X\right)$ (diffeomorphisms of $X$ connectable to the identity) in a subtle way. This point I am not quite sure about yet, as I mentioned before. Posted by: Urs Schreiber on August 1, 2007 1:55 PM | Permalink | Reply to this ### Re: Arrow-Theoretic Differential Theory The following comes from an email to Urs he asked me to make public, in the hope of triggering some associations in other people’s minds Some comments: Section 2.1 - the bundle $\mathrm{TC}\to \mathrm{Obj}$ is a fibred category, and if we are saying it can be written as a presheaf ${C}^{\mathrm{op}}\to \mathrm{nCat}$. This connects nicely with stacks. 2.2 I presume the notion “$\Sigma {T}_{{\mathrm{Id}}_{P}}\left(\mathrm{Cat}\right)$” will be explained. Is it really meant to be $\mathrm{Cat}$ in brackets there? If we are looking at the tangent category at ${\mathrm{Id}}_{P}$, then I suppose $\mathrm{Funct}$ (=all functors) would be more appropriate. Also, writing $2\mathrm{Funct}$ for the (2-)category of all 2-functors is a little less bracketing than $\mathrm{Mor}\left(2\mathrm{Cat}\right)$. 2.4.1 The notion of extending an n-category to an (n+1)-category by adding an extra, codiscrete, layer is precisely analogous to taking the thin n-path groupoid (dividing only by thin homotopy at top dimension) and throwing in homotopy classes of (n+1)-paths between parallel n-paths. This is implicit in the interpretation as the Bianchi identity, so I think you know this. Having an example of this operation $C↦{C}_{n+1}$ (as above, say) gives a nice intuition why one wants to, or indeed, can legitimately do this at all. I suppose by saying not all flat functors F are of the form $\mathrm{dA}$ we can invent a cohomology ;) More realistically, we would expect a flat functor to be of the form $\mathrm{dA}$ *locally* - for some notion of locally’. Perhaps one could use the covering’ $\mathrm{TC}\to C$, and some coherence conditions on $\mathrm{TTC}⇒\mathrm{TC}$ to get local transport functors whose differentials paste together to a flat functor. 2.5 $F,G:D\to C$ n-functors, a natural transformation $a:F⇒G$ is an (n-1)-functor $a:D\to \mathrm{Mor}\left(C\right)$. I don’t know if this has any relation to the use of $\mathrm{Mor}\left(C\right)$ before, but given such an $a:D\to \mathrm{Mor}\left(C\right)$, we can compose with $\mathrm{Mor}\left(C\right)\to \mathrm{TC}$, to hopefully get something interesting. Posted by: David Roberts on August 1, 2007 6:22 AM | Permalink | Reply to this ### Re: Arrow-Theoretic Differential Theory Hi David, thanks a lot for these very good comments! Section 2.1 - the bundle $TC\to \mathrm{Obj}\left(C\right)$ is a fibred category, and if we are saying it can be written as a presheaf ${C}^{\mathrm{op}}\to n\mathrm{Cat}$. This connects nicely with stacks. Okay, I was wondering about that. I have only a vague recollection of the details of the Grothendieck construction and the like. So while I was aware that I was switching here from a fibered category to a presheaf, I wasn’t sure if this presheaf is indeed precisely the one belonging to that fibered category under the Grothendieck construction. You see what I mean? Maybe it’s some other presheaf, which happens to be somehow related? Probably not, but I am not sure how to see this. Do you? 2.2 I presume the notion “$\Sigma {T}_{{\mathrm{Id}}_{P}}\left(\mathrm{Cat}\right)$” will be explained. Is it really meant to be $\mathrm{Cat}$ in brackets there? Oh, no, a typo. It should be $\mathrm{Mor}\left(\mathrm{Cat}\right)$, or rather $\mathrm{Funct}$, as you rightly point out. Actually, it only hits $\mathrm{End}\left({P}_{1}\left(X\right)\right)\subset \mathrm{Funct}$ of course, so maybe I should write that, to better direct the reader’s attention. If we are looking at the tangent category at ${\mathrm{Id}}_{P}$, then I suppose $\mathrm{Funct}$ (=all functors) would be more appropriate. Also, writing $2\mathrm{Funct}$ for the (2-)category of all 2-functors is a little less bracketing than $\mathrm{Mor}\left(2\mathrm{Cat}\right)$. Yes, true. Thanks. And, by the way, there is a more general statement lurking here, which generalizes the corresponding statement from our paper: the category of sections of the tangent $\left(n+1\right)$-bundle sits inside the tangent space to the identity $n$-functor, and this way the $n$-category of sections inherits an $n$-group structure. 1-parameter smooth sub-1-groups of this $n$-group are ordinary vector fields. But we can have more exotic sub-$n$-groups. I am expecting that something like 2-parameter sub 2-groups actually correspond to exterior products of two vectors, hence to dual 2-forms. But I am not quite sure yet about some details. 2.4.1 The notion of extending an $n$-category to an $\left(n+1\right)$-category by adding an extra, codiscrete, layer is precisely analogous to taking the thin $n$-path groupoid (dividing only by thin homotopy at top dimension) and throwing in homotopy classes of $\left(n+1\right)$-paths between parallel $n$-paths. Yes. This is implicit in the interpretation as the Bianchi identity, so I think you know this. Having an example of this operation $C↦{C}_{\left(n+1\right)}$ (as above, say) gives a nice intuition why one wants to, or indeed, can legitimately do this at all. Yes, that’s certainly the example I am having in mind. But I was pleased to find that it in fact makes good sense to speak of curvature and Bianchi identity of a completely arbitrary functor, too. I suppose by saying not all flat functors $F$ are of the form $dA$ we can invent a cohomology ;) Indeed, this is a question very much on my mind. In a way, this is precisely what brought me to consider $\mathrm{INN}\left({G}_{\left(2\right)}\right)$ back when in the first place: you will recall that we used to be slightly puzzled that transport 2-functors with values in a strict 2-group exactly reproduce Breen-Messing’s differential cocycle data for gerbes with connection – except that the 2-functors impose that funny “fake flatness” constraint. So there are more Breen-Messing cocycles than come from such 2-functors. Then, in that original document on $\mathrm{INN}\left({G}_{\left(2\right)}\right)$, I found, as you know, that the Breen-Messing data actually describes flat 3-functors with values in the 3-group $\mathrm{INN}\left({G}_{\left(2\right)}\right)$. So, apparently, there is an interesting cohomology here: there are flat 3-functors with values in $\mathrm{INN}\left({G}_{\left(2\right)}\right)$ which do not arise as the curvature of some 2-functor. Put this way, it sounds like a very plausible statement. But that took a couple of years to materialize! In that work on ${\mathrm{String}}_{k}\left(G\right)$-connections which I mentioned recently I argue that this cohomology problem here for ${G}_{\left(2\right)}$ the ${\mathrm{String}}_{k}\left(G\right)$-2-group corresponds to Stolz-Teichner’s statement that “Chern-Simons theory is trivialized by String bundles”. Because $\mathrm{Lie}\left(\mathrm{INN}\left({\mathrm{String}}_{k}\left(G\right)\right)\right)\simeq \mathrm{cs}\left(g{\right)}_{k}$ is isomorphic to the Chern-Simons Lie 3-algebra. More realistically, we would expect a flat functor to be of the form $dA\text{Unknown character}\text{Unknown character}\mathrm{em}\text{Unknown character}\mathrm{locally}\text{Unknown character}/\mathrm{em}\text{Unknown character}-\mathrm{for}\mathrm{some}\mathrm{notion}\mathrm{of}\prime \mathrm{locally}\prime .\text{Unknown character}/p\text{Unknown character}\text{Unknown character}/\mathrm{blockquote}\text{Unknown character}\text{Unknown character}p\text{Unknown character}\mathrm{That}\prime s\mathrm{true}.\mathrm{The}\mathrm{construction}$C_n \mapsto C_{n+1}$\mathrm{which}I\mathrm{mention}\left(\mathrm{simply}\mathrm{passing}\mathrm{to}\mathrm{the}\mathrm{codiscrete}\mathrm{Hom}-\mathrm{categories}\right)\mathrm{is}\mathrm{only}\mathrm{the}\text{Unknown character}\mathrm{universal}\text{Unknown character}\mathrm{one},I\mathrm{think}.\mathrm{It}\mathrm{factors}\mathrm{through}\mathrm{all}\mathrm{others},\mathrm{where}\mathrm{we}\mathrm{don}\prime t\mathrm{include}\mathrm{some}\mathrm{of}\mathrm{the}\mathrm{higher}\mathrm{morphisms}\left(i.e.\mathrm{where}\mathrm{there}\mathrm{are}\text{Unknown character}\mathrm{holes}\text{Unknown character}\mathrm{around}\mathrm{which}\mathrm{we}\mathrm{may}\mathrm{have}\mathrm{nontrivial}\mathrm{holonomy}.\right)\text{Unknown character}/p\text{Unknown character}\text{Unknown character}\mathrm{blockquote}\text{Unknown character}\text{Unknown character}p\text{Unknown character}\mathrm{Perhaps}\mathrm{one}\mathrm{could}\mathrm{use}\mathrm{the}`\mathrm{covering}\prime$T C \to C$,\mathrm{and}\mathrm{some}\mathrm{coherence}\mathrm{conditions}\mathrm{on}$ T T C \Rightarrow T C$\mathrm{to}\mathrm{get}\mathrm{local}\mathrm{transport}\mathrm{functors}\mathrm{whose}\mathrm{differentials}\mathrm{paste}\mathrm{together}\mathrm{to}a\mathrm{flat}\mathrm{functor}.\text{Unknown character}/p\text{Unknown character}\text{Unknown character}/\mathrm{blockquote}\text{Unknown character}\text{Unknown character}p\text{Unknown character}\mathrm{Hm},\mathrm{that}\prime s\mathrm{an}\mathrm{idea}.I\prime \mathrm{ll}\mathrm{think}\mathrm{about}\mathrm{it}.\text{Unknown character}/p\text{Unknown character}\text{Unknown character}\mathrm{blockquote}\text{Unknown character}\text{Unknown character}p\text{Unknown character}2.5$F,G:D \to C$\text{}$n$-\mathrm{functors},a\mathrm{natural}\mathrm{transformation}$a : F \Rightarrow G$\mathrm{is}\mathrm{an}$(n-1)$-\mathrm{functor}$a : D \to \mathrm{Mor}(C)$.I\mathrm{don}\prime t\mathrm{know}\mathrm{if}\mathrm{this}\mathrm{has}\mathrm{any}\mathrm{relation}\mathrm{to}\mathrm{the}\mathrm{use}\mathrm{of}$\mathrm{Mor}(C)$\mathrm{before},\mathrm{but}\mathrm{given}\mathrm{such}\mathrm{an}$ a : D \to \mathrm{Mor}(C)$,\mathrm{we}\mathrm{can}\mathrm{compose}\mathrm{with}$\mathrm{Mor}(C) \to T C$, to hopefully get something interesting.

That, too, is an interesting observation. I’ll think about it.

Posted by: Urs Schreiber on August 1, 2007 9:06 AM | Permalink | Reply to this
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### Re: Arrow-Theoretic Differential Theory

I am about to write an entry with more details on how sections of an $n$-bundle with connection given by a transport $n$-functor $\mathrm{tra}$ are really certain morphisms into $\delta \mathrm{tra}\phantom{\rule{thinmathspace}{0ex}},$ where $\delta$ is the differential on functors from section 3.2. When thinking about this with an eye on quantization of higher dimensional objects, one runs into the issue that one needs the sections of a transport after that is pulled back to some configuration space. Then the question might seem to arise whether we compute the sections (hence the space of quantum states) by first pulling back and then forming the differential $\delta \left({p}^{*}F\right)$ or the other way round ${p}^{*}\left(\delta F\right)\phantom{\rule{thinmathspace}{0ex}}.$ As befits the operator $\delta$ which (as discussed in section 2.4.2) plays the role of the exterior differential, both these expressions should be equal! And in fact they are. This is a triviality in the present context: $\delta$ acts on a functor by postcomposition with another functor, while pullback acts by precomposition with another functor. So that’s good.
Posted by: Urs Schreiber on August 14, 2007 4:47 PM | Permalink | Reply to this
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Excerpt: More on the canonical quantization of the charged n-particle for the case of a 1-particle propagating on a lattice.
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### Re: Arrow-Theoretic Differential Theory

I have just upoloaded a new version of Arrow-theoretic differential theory.

The concept of a $G$-flow on a category and hence that of generalized vector fields is stressed a little more now and it is discussed explicitly how the relevant aspects of the work of Roberts&Ruzzi (concerning inner automorphism $n+1$-groups) and of Lazaroiu (concerning supercategories) drops out as a special case.

Posted by: Urs Schreiber on August 17, 2007 1:57 PM | Permalink | Reply to this

### Re: Arrow-Theoretic Differential Theory

I know that this concept of thinking of vector fields and their generalizations as suitable families of inner automorphisms of a category takes a little getting used to. But then it proves to be a good idea.

It might help to emphasize this:

Posted by: Urs Schreiber on August 17, 2007 3:19 PM | Permalink | Reply to this
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Excerpt: How to get the bundle governing Generalized Complex Geometry from abstract nonsense and arrow-theoretic differential theory.
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