I started thinking about formalizing more the structure that is used here to turn a transport 2-functor, i.e. a differential nonabelian cocycle, into a local net. So about formalizing the role played by the Minkowski structure on base space in this construction.

Maybe, just as one can consider differential cocycles equivariant with respect to a groupoid, here we may have to talk about “poset equivariance”, encoding the lightcone structure of the underlying Minkowski manifold in a poset structure.

I am not sure yet what exactly to do, but started playing around with something like this:

our Minkowski (or globally hyperbolic Lorentzian) space naturally inherits the structure of a poset, as we know, by taking $x \leq y$ precisely if the point $y$ is in the future of the point $x$, equivalently, if and only if $x$ is in the past of $y$ (meaning that there exists smooth curve connecting $x$ with $y$ the Minkowski norm of whose tangent is everywhere non-negative).

So let me write $\mathbf{X}$ for the smooth poset we have, which can be thought of as a category enriched in (-1)-categories, i.e. enriched in the monoid $(\{true, false\}, \otimes = and)$.

Then
$IsFuture(x) := \mathbf{X}(x,-)
: \mathbf{X}_0 \to \{true, false\}$
is the subobject classifyer for the future of $x$, in that the future $Future(x)$ is the subset arising as the pullback
$\array{
Future(x)
&\to&
\{true\}
\\
\downarrow && \downarrow
\\
\mathbf{X}_0
&\stackrel{\mathbf{X}(x,-)}{\to}&
\{true, false\}
}$
and
$IsPast(y) := \mathbf{X}(-,y)
: \mathbf{X}_0 \to \{true, false\}$
similarly is the subobject classifier for the past of $y$.

The crucial structures in the business of local nets are those *causal subsets* $O_{x,y}$: these are precisely the intersections of the future of one point $x$ with the past of another point $y$. So their subobject classifier is
$(a \mapsto
\mathbf{X}(a,y)\otimes\mathbf{X}(x,a)
)
:
\mathbf{X}_0
\to
\{true, false\}$
so that the causal subset $O_{x,y}$ is the pullback
$\array{
O_{x,y}
&\to&
\{true\}
\\
\downarrow && \downarrow
\\
\mathbf{X}_0
&\stackrel{\mathbf{X}(-,y)\otimes \mathbf{X}(x,-)}{\to}&
\{true, false\}
}
\,.$

So suppose we start with something like a poset-covariant 2-functor on 2-paths $P_2(X)$, maybe a 2-functor from
$Codesc(N(\mathbf{X}), P_2)
=
\int^{[n]\in \Delta}
P_2(N(X)_n)\otimes O(\Delta^n)
\,,$
where $N(\mathbf{X})$ denotes the nerve of the poset $\mathbf{X}$ and otherwise I am following the notation used here.

Then somehow the task is to naturally obtain from that
a co-presheaf on the poset of $O_{x,y}$s using just abstract nonsense as above.

Okay, let’s see. What’s the natural expression for the poset structure on the causal subsets $O_{x,y}$ themselves. We simply have
$(O_{x',y'}
\subset
O_{x,y})
\Leftrightarrow
(
(y' \leq y) and (x \leq x')
)
\Leftrightarrow
(\mathbf{X}(-_2,y)\otimes \mathbf{X}(x,-_1))(x',y')$
assuming that both $O_{x,y}$ and $O_{x',y'}$ are non-empty, in that $x \leq y$ and $x'\leq y'$.

Hm, so what now? Can anyone see how to proceed from here along the abstract-nonsense route?

## Re: Talk: Local Nets from Parallel Transport 2-Functors

I received some very useful feedback. One open question for me is/was this:

I describe how 2-functors on 2-paths in Minkowski give rise to their “endomorphism co-presheaves” on causal subsets, which are necessarily local nets satisfying the time slice axiom.

The open question is: how does this work the other way around? given a local net, can we construct the 2-functor that it is the endomorphism co-presheaf of?

One helpful suggestion from the audience was this:

assume the local net $A$ has the following two properties:

- the algebra $A(O)$ assigned to any double cone $O$ is

maximal: with $W^O_L$ the wedge region left of $O$ and $W^O_R$ the wedge region right of $O$, the inclusion $A(O) \hookrightarrow A(W_L^O)' \cap A(W_R^O')$ (prime denotes commutant, as usual), expressing the locality of the net is actually an equality $A(O) = A(W_L^O)' \cap A(W_R^O)' \,.$ Similarly then for $W^x_L$ and $W^x_R$ the left and right wedge at any point $x$, $A(W^x_L) = A(W^x_R)' \,.$- the net satisfies a

split propertywhich says that for $y$ spacelike right of $x$ the inclusion of wedge algebras $W^y_R \subset W^x_R$ always factors through a type I factor $N$ as $W^y_R \subset N \subset W^x_R \,.$Under some extra assumption which is apparently discussed in

S. J. Summers,

On the independence of local algebras in quantum field theory, Rev. Math. Phys. 2 (1990), 201-247, this implies that then $W^x_R \vee (W^y_R)' \simeq W^x_R \otimes (W^y_R)'$ (where $A \vee B$ is the vN algebra generated by $A$ and $B$, as usual).This can be found disucssed at the beginning of section 2.2 of

G. Lechner, On the construction of quantum field theories with Factorizing S-matrices

p. 20-21.

So, assuming the net $A$ satisfies all this we have for $O = O_{x,y}$ the causal subset with left corner at $x$ and right corner at $y$ the identity $\begin{aligned} A(O) &= A(W^x_L)' \cap A(X^y_R)' \\ &= (A(W^x_L) \vee A(X^y_R))' \\ &= (A(W^x_R)' \vee A(X^y_R))' \\ &= (A(W^x_R)' \otimes A(X^y_R))' \,. \end{aligned}$ Now take $H_{x,y}$ to be the total Hilbert space but regarded as a module just for $A(W^x_R)' \otimes A(X^y_R)$. Then the module endomorphisms should be $End(H_{x,y}) = (A(W^x_R)' \otimes A(X^y_R))' = A(O) \,.$

So the idea is that the 2-functor $Z$ of which $A$ might be the endomorphism co-presheaf assigns to paths $x \to y$ this $H_{x,y}$. To a point $x$ it might assign the vN algebra $A(W^x_R)$.

I am not sure yet exactly if I can see the full 2-categorical structure this may be hinting at, but it does look suggestive.