### 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.

Arrow-theoretic differential theory

Abstract: We propose and study a notion of a tangent $(n+1)$-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 $(n+1)$-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 $(n+1)$-bundle*

We define for any $n$-category $C$ an $n$-category $TC$ which is an $(n+1)$-bundle $p : TC \to \mathrm{Obj}(C)$ 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 $\mathbf{pt} := \{ \bullet \stackrel{\sim}{\to} \circ \}$ as an arrow-theoretic model for the “infinitesimal interval” or the “superpoint” in that it is a puffed-up version of the mere point $\mathrm{pt} := \{ \bullet \}$ to which it is equivalent, by way of the injection $\mathrm{pt} \hookrightarrow \mathbf{pt} \,,$ 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 $TC \subset \mathrm{Hom}_{n\mathrm{Cat}}(\mathbf{pt}, C)$ 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 $(n+1)$-bundle is that it sits inside the short exact sequence $\mathrm{Mor}(C) \to T C \to C \,.$

Finally, for later use notice that dual to its realization as a projection $T C \to \mathrm{Obj}(C)$ the tangent bundle may be thought of as an $n$-functor $T C : C^{\mathrm{op}} \to n\mathrm{Cat}$ which sends objects $a$ to the tangent categories $T_a C$ over them and sends morphisms the the pullback of these along them $T C \; : \; ( a \stackrel{f}{\leftarrow} b ) \; \mapsto \; ( T_a C \stackrel{T_f C}{\to} T_b C ) \,.$

*Vector fields and Lie derivatives*

Let $X$ be a smooth manifold and let $P_1(X)$ be the groupoid of thin homotopy classes of paths in $X$.

Then ordinary vector fields $v \in \Gamma(TX)$ on $X$ are in canonical bijection with smooth 1-parameter families of categorical tangent vectors to the identity map on $P_1(X)$: $\Gamma(TX) \; \stackrel{\sim}{\to} \; \{ \Sigma \mathbb{R} \to \Sigma T_{\mathrm{Id}_{\mathcal{P}_1(X)}}(\mathrm{Cat}) \} \,.$

On a general category $C$, it may be useful to consider generalizations of this where $\mathbb{R}$ 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 $\mathbb{Z}$-flows $\{ \Sigma \mathbb{Z} \to \Sigma T_{\mathrm{Id}_{C}}(\mathrm{Cat}) \}$ on $C$.

On the other hand, the identitfication of $T C$ 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 $(n-1)$-groupoids $\Sigma G_{(n)}$, hence $n$-groups $G_{(n)}$. In our context these $(n-1)$-groupoids must be thought of as 1-point orbifolds. Accordingly, they have just a single “tangent space” (tangent $n$-category) $T_\bullet \Sigma G_{(n)} \,.$

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

$\bullet$

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

$\bullet$

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

*Curvature and Bianchi Identity for functors*

*General functors*

Using the functorial incarnation $TC : C^{\mathrm{op}} \to n\mathrm{Cat}$
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
$d F :
C \stackrel{F}{\to} D \stackrel{T D}{\to} n\mathrm{Cat}
\,.$
The crucial point of this construction is that it extends *uniquely*
(up to equivalence) to an $(n+1)$-functor
$d F : C_{(n+1)} \to n\mathrm{Cat}$
on the $(n+1)$-category
$C_{(n+1)} := \mathrm{Codisc}(C)$
which is obtained from $C$ by replacing all Hom-($n-1$)-categories by the
corresponding codiscrete $n$-groupoids over them.

By introducing the terminology

$\bullet$ $d F$ is the curvature of $F$

$\bullet$ $F$ is flat if $d F$ is degenerate (sends all $(n+1)$-morphisms to identities)

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

$\bullet$ $d F$ is flat

or equivalently

$\bullet$ $d d F$ 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 $C := \mathcal{P}_3(X)$ be the strict 3-groupoid of thin homotopy classes of $k$-paths in a smooth manifold $X$. And let $G_{(2)}$ be a strict Lie 2-group coming from the Lie crossed module $H \stackrel{t}{\to} G \stackrel{\alpha}{\to} \mathrm{Aut}(H)$.

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

$\bullet$ $\left\lbrace \mathrm{smooth}\;1-\mathrm{functors} \mathcal{P}_1(X) \to \Sigma G \right\rbrace \stackrel{\sim}{\to} \left\lbrace A \in \Omega^1(X,\mathrm{Lie}(G)) \right\rbrace$

$\bullet$

$\left\lbrace \mathrm{smooth}\;2-\mathrm{functors} \mathcal{P}_2(X) \to \Sigma G_{(2)} \right\rbrace \stackrel{\sim}{\to} \left\lbrace (A,B) \;\; F_A + t_*\circ B = 0 \right\rbrace$

$\bullet$

$\left\lbrace \mathrm{smooth}\;3-\mathrm{functors} \mathcal{P}_3(X) \to \Sigma \mathrm{INN}(G_{(2))} \right\rbrace \stackrel{\sim}{\to} \left\lbrace (A,B,C) \;\; C = d_A B \right\rbrace$

Now let $\mathrm{tra} : P_1(X) \to \Sigma G$ 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 $\begin{aligned} \mathrm{tra} &\mapsto A \\ \mathrm{curv} = d \mathrm{tra} &\mapsto F_A := d A + A \wedge A \\ d \mathrm{curv} = d d \mathrm{tra} &\mapsto d_A F_A = 0 \end{aligned} \,.$

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 $(n+1)$-functor of a $G_{(n)}$-transport is itself an $\mathrm{INN}(G_{(n)})$-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” $(n+1)$-transport. As with the inner automorphism $(n+1)$-groups $\mathrm{INN}(G_{(n)})$, 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 $(n-1)$-functor. Now if $G$ is trivial in some sense to be made precise, and if the transformation is an equivalence $I \stackrel{f_\sim}{\to} F \,,$ then this implies that the $n$-functor $F$ is entirely encoded in the $(n-1)$-functor $f$.

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

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 $n-functors$ have sectins which are themselves transport $(n-1)$-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.

## 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

ptequivalent? It looks like you are just including pt in as the first point ofptand 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.