The n-Category Café

A good approach would be to try to first handle all known lattice models of AQFT, then understanding the continuum limit. But work in this direction is scarce. One relevant article I am aware of is

Florian Nill, Kornél Szalachányi
Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry
arXiv:hep-th/9509100

They are considering a 1-dimensional lattice where to every second site is assigned a finite dimensional $C^*$-Hopf algebra $H$ and to every other site its dual $C^*$-Hopf algebra $\hat H$.

A possible 2-functorial interpretation would be to read this as the Heisenberg picture version of what in the Schrödinger picture is a 2-functor $$Z : P_2(X) \to D$$

on a causal lattice 2-path 2-category $P_2(X)$ of the form

$$\array{ && \nearrow \searrow && \nearrow \searrow && \nearrow \searrow \\ \cdots & x_i &\Downarrow& x_{i+1} &\Downarrow& x_{i+2} &\Downarrow& x_{i+3} & \cdots \\ && \searrow \nearrow && \searrow \nearrow && \searrow \nearrow & }$$

such that the 1-morphisms assigned to the lower paths

$$\array{ \cdots & Z(x_i) && Z(x_{i+1}) && Z(x_{i+2}) && Z(x_{i+3}) & \cdots \\ && {}_{Z(i,d)}\searrow \nearrow_{Z({i,u})} && {}_{Z({i+1,d})}\searrow \nearrow_{Z({i+1,u})} && {}_{Z({i+2,d})}\searrow \nearrow_{Z({i+2,u})} & }$$

are such that $$H = End_D(Z({i,u})\circ Z(i,d))$$ and $$\hat H = End_D(Z({i+1,d})\circ Z(i,u)) \,.$$

Given that, if we furthermore assume the time evolution propagator

$$Z\left( \array{ & \nearrow \searrow \\ x_i &\Downarrow& x_{i+1} \\ & \searrow \nearrow } \right)$$

on every elementary 2-path be in $End(Z(i,d)) \times End(Z(i,u))$, the local net obtained from this 2-functor restricted to a horizontal zig-zag path would be a net of the kind discussed by the above authors in section 2.2.

I wasn’t entirely sure how I had to choose the codomain 2-category $D$ and the action of the 2-functor $Z$ on 1-morphisms to achieve that. But luckily, Pasquale Zito kindly provided very helpful comments. He has written

Pasquale A. Zito,
2-$C*$-categories with non-simple units
arXiv:math/0509266

First of all, quite generally for many common classes of applications the right choice of codomain 2-category $D$ should be such that the endomorphism monoid of any 1-morphism is a $C^*$-algebra. This strongly suggests that the codomain be a 2-$C^*$-category: a category enriched over $C^*$-categories. You can find the definition of $C^*$-category for instance in the entry Spaceoids. A $C^*$-category is a $C^*$-algebroid. The definition of a 2-$C^*$-category is for instance on p. 7 of Pasquale Zito’s article.

So that much is clear. Less immediate is the following nice fact, which Pasquale Zito discusses in section 5, from p. 39 on:

Let $D$ be a 2-$C^*$-category and let $$a \stackrel{\rho}{\to} b$$ be a 1-morphism in there which has a conjugate $$b \stackrel{\bar \rho}{\to} a$$ i.e. such that we have an ambidextrous adjunction.

Then $$H := End_D( \bar \rho\circ\rho)$$ is a Hopf algebra and $$\hat H := End_D( \rho\circ \bar\rho)$$ is its dual. Together they satisfy a bunch of nice properties, for which the source seems to be

Michael Mueger
From Subfactors to Categories and Topology I. Frobenius algebras in and Morita equivalence of tensor categories
arXiv:math/0111204

around p. 45.

So it seems that the “Hopf spin chain” nets discussed by Nill and Szalachányi can be regarded as the nets obtained from 2-functor $Z$ as above which come from a choice of ambidextrous adjunction $\rho, \bar \rho$ in a $2-C^*$-category under the assignment $$Z \left( \array{ x_i \\ & \searrow } \right) = a \stackrel{\rho}{\to} b$$ and $$Z \left( \array{ & x_{i+ 1} \\ \nearrow } \right) = b \stackrel{\bar \rho}{\to} a \,.$$