### Isham on Arrow Fields

#### Posted by Urs Schreiber

In

Chris J . Isham
*A New Approach to Quantising Space-Time: I. Quantising on a General Category*

gr-qc/0303060

the author considers the concept of an **arrow field on a category**.

I recall his definition, reformulate it in arrow-theoretic terms, discuss the connection of “arrow fields” to ordinary vector fields and describe how to generalize it to a notion of covariant transport of sections of bundles with connection.

Chris Isham defines (beginning of section 3.1) an **arrow field** on a category $Q$ as (my paraphrasing) a *section of the source map*. So it is an assigmnet
$v : \mathrm{Obj}(Q) \to \mathrm{Mor}(Q)$
which sends each object $o$ in $Q$ to a morphism
$o \stackrel{v(o)}{\to} t(v(o))$ starting at $o$.

Here is comment on that:

we know that something which assigns morphisms to objects is likely to be a *natural transformation*.

Since we want this assignment to be a section of the source map, it is likely a natural transformation starting at the identity functor $\mathrm{Id}_Q$ on $Q$ $\array{ & \nearrow \searrow^{\mathrm{Id}\;\;} \\ Q & \;\,\Downarrow^v & Q \\ & \searrow \nearrow_{\mathrm{exp}(v)} }$ and ending at some endofunctor of $Q$, which I have given the name $\mathrm{exp}(v)$.

Chris Isham goes on to note that there is a natural **monoidal structure** on such “arrow fields”: given two arrow fields $v_1$ and $v_2$, we obtain a new one by assigning to each object $o$ of $Q$ the composite arrow
$o \stackrel{v_1(o)}{\to} t(v_1(o)) \stackrel{v_2(t(v_1(o)))}{\to} t(v_2(t(v_1(o))))
\,.$

Here is a comment on that:

this monoidal structure is the obvious monoidal structure on natural transformations of the above form $\array{ & \nearrow \searrow^{\mathrm{Id}\;\;} & & \nearrow \searrow^{\mathrm{Id}\;\;} \\ Q & \;\,\Downarrow^{v_1} & Q & \;\,\Downarrow^{v_2} & Q \\ & \searrow \nearrow_{\mathrm{exp}(v_1)} & & \searrow \nearrow_{\mathrm{exp}(v_2)} }$

Finally, in section 3.2, the author defines an **action of arrow fields on objects** in the obvious way: each arrow field maps an object $o$ to the target of its value at that object:
$v : o \mapsto t(v(o))
\,.$

Here is a comment on that:

notice that this is the action of the restriction of the functor $\mathrm{exp}(v) : Q \to Q$ that is the target of the natural transformation $v$.

In the last paragraph of section 3.2, Chris Isham remarks that the action of arrow fields on objects of $Q$ is like the **action of the diffeomorphism group** $\mathrm{Diff}(\mathrm{Obj}(Q))$.

Using the above re-formulation in terms of natural transformations, we can make this precise as follows:

Let’s consider the identity component of $\mathrm{DIff}(\mathrm{Obj}(Q))$ and look at “smooth families” of arrow fields that contain the trivial arrow field.

So let $Q = P_1(X)$ be the path groupoid of a smooth space $X$, and let $\Sigma(\mathbb{R})$ be the the additive group of real numbers, regarded as a category with a single object.

Write $\mathrm{Flow}(X) \subset \Sigma(\mathrm{End}_\mathrm{Cat}(P_1(X)))$ for the sub-category of endomorphisms of $P_1(X)$ of the form $\array{ & \nearrow \searrow^{\mathrm{Id}\;\;} \\ P_1(X) & \;\,\Downarrow^v & P_1(X) \\ & \searrow \nearrow_{\mathrm{exp}(v)} } \,,$ where everything is smooth.

As we have just seen, each of these can be regarded as defining an “arrow field” on $P_1(X)$.

Notice how these arrow fields $v$ look like in this case: they assign to each point $x$ in $X$ a path
$x \stackrel{v(x)}{\to} t(v(x))$
in $X$. Moreover, $\mathrm{exp}(v) : P_1(X) \to P_1(X)$ is the functor which sends any path $x \stackrel{\gamma}{\to} y$ to
$\mathrm{exp}(v) :
\left(
\array{
x
\\
\;\;\downarrow\gamma
\\
y
}
\right)
\;\;\;\;\;
\mapsto
\;\;\;\;\;
\left(
\array{
x &\stackrel{v(x)^{-1}}{\leftarrow}& t(v(x))
\\
{}^\gamma \downarrow\;\;
\\
y &\stackrel{v(y)}{\to}& t(v(y))
}
\right)
\,.$
We might want to think of this as the *adjoint action* of arrow fields on morphisms.

Now, in order to get a smooth family of such, consider smooth functors $v : \Sigma(\mathbb{R}) \to \mathrm{Flow}(X) \,.$ These send $v : t \mapsto \array{ & \nearrow \searrow^{\mathrm{Id}\;\;} \\ P_1(X) & \;\,\Downarrow^{v(t)} & P_1(X) \\ & \searrow \nearrow_{\mathrm{exp}(v(t))} }$ in a way that respects translation on the real line: $v(t_1 + t_2) : x \mapsto x \stackrel{v(t_1)(x)}{\to} t(v(t_1)) \stackrel{v(t_2)(t(v(t_1)(x)))}{\to} t(v(t_2)(t(v(t_1)(x)))) \,.$ And smoothly so.

I think such smooth functors $v : \Sigma(\mathbb{R}) \to \mathrm{Flow}(X)$ are precisely in bijection with ordinary vector fields on $X$.

To me, this is an arrow-theoretic notion of vector field alternative to that used in synthetic differential geometry.

If “arrow fields” on $Q$ are to be thought of as vector fields, they should act on functions on $\mathrm{Obj}(Q)$ by some kind of translation.

At the beginning of section 4.2, Chris Isham considers the obvious **action of an arrow field on a function**:
$v : C(\mathrm{Obj}(X)) \ni \psi \mapsto (o \mapsto \psi(t(v(o))))
\,.$

My final comment here shall be that this action of arrow fields on functions also has a nice arrow-theoretic formulation in terms of the above natural transformations. This formulation allows in particular to readily see how **arrow fields act by covariant transport on sections of fiber bundles** over $\mathrm{Obj}(Q)$, in the case where a bundle with connection over $Q$ is present. Such a covariant translation along arrow fields is conidered around equation (67) in Chris Isham’s text.

The relevant diagram, however, is a little hard to draw here. I am discussing it in section 1.2,

of the document which accompanies the discussion here.

Here is a snapshot of the end of that section:

## Re: Isham on Arrow Fields

I’ll incorporate the following comments tomorrow, when I am less tired:

The entire discussion in section 4.3 of Chris Isham’s paper amounts to saying that with the morphisms of $Q$ regarded as paths between its objects (which is the relevant interpretation for Isham’s purposes), a functor $\mathrm{tra} : Q \to \mathrm{Hilb}$ is a hermitean vector bundle with connection on $\mathrm{Obj}(Q)$.

However, this is not the way Chris Isham puts it. The connection part he calls a “multiplier” and instead of thinking of a parallel transport functor he thinks of this functor as a

presheaf.I don’t really follow the reasoning behind this. Thinking of the above functor as a presheaf would imply thinking of $Q$ as a site. But this is

notthe role played by $Q$ in the rest of the discussion:$\mathrm{Obj}(Q)$ rather plays the role of the

configuration spaceof some physical system (and not of a site of open sets on that configuration space).The discussion around equation (83) then amounts to noticing that a transformation $1 \to \mathrm{tra}$ is not just a section of this bundle, but a

flatsection.This is a general issue, which I also do discuss for instance in that Rosetta stone section 1.2:

when we want to form the space of states in our arrow-theoretic formulation of quantum mechanics, we need to pass from the category of “configurations and paths between configurations”, which in my terminology is $\mathrm{cob} = [\mathrm{par},\mathrm{tar}] \,,$ the category of functors from parameter space into target space (for the present case the parameter space is simply the point $\mathrm{par} = \bullet$ so that $[\mathrm{par},\mathrm{tar}] \simeq \mathrm{tar} := Q$)

to the subcategory $\mathrm{conf} \subset [\mathrm{par},\mathrm{tar}] \,,$ the true

configuration space, which contains only morphism between configurations that we want to regard as physically equivalent.That’s the difference between all “cobordisms” (processes) between configurations (morphisms in $\mathrm{cob}$) and the morphisms in $\mathrm{conf}$, which are the

processes that are “pure gauge”.So, this is why I keep going on about why specifying a physical system arrow-theoretically involves not just choosing a target space $\mathrm{tar}$ and a bundle with connection $\mathrm{tra} : \mathrm{tar} \to \mathrm{phas}$ on that, but also a sub-category $\mathrm{conf}$ of the category of all processes in target space. That’s why I define # a physical system as a situation of the form $\left( \mathrm{par} \stackrel{\gamma \in \mathrm{conf}}{\to} \mathrm{tar} \stackrel{\mathrm{tra}}{\to} \mathrm{phas} \right)$ and then give a prescription for computing the space of states which involves

firstpulling back $\mathrm{tra}$ (Isham’s “multiplier”!) back to configuration space andthentaking its sections.All this is, secretly, the issue discussed around equations (83)-(84) in Chris Isham’s paper.

It’s a not a particularly complicated issue, but I think it pays to try to extract a clear picture here.

I am making this comparison with Isham’s ideas here also in the hope that it will help our mutual understanding, in the light of John’s remarks at the end of this comment.