Combinatorial quantization of 4d 2-Chern-Simons theory I: the Hopf category of higher-graph states

Abstract: 2-Chern-Simons theory, or more commonly known as 4d BF-BB theory with gauged shift symmetry, is a natural generalization of Chern-Simons theory to 4-dimensional manifolds. It is part of the bestiary of higher-homotopy Maurer-Cartan theories. In this article, we present a framework towards the combinatorial quantization of 2-Chern-Simons theory on the lattice, taking inspiration from the work of Aleskeev-Grosse-Schomerus three decades ago. The central geometric input is the 2-truncation $\Gamma^2$ of the $\infty$-groupoid of simplices formed by the underlying lattice $\Gamma$. On such a "2-graph", we model states of 2-Chern-Simons holonomies as Crane-Yetter's \textit{measureable fields}. We show that the 2-Chern-Simons action endows the 2-graph states -- as well as their quantum 2-gauge symmetries -- the structure of a Hopf category, and that their associated higher $R$-matriex gives it a comonoidal {\it cobraiding} structure. This is an explicit realization of the categorical ladder proposal of Baez-Dolan, in the context of Lie group lattice 2-gauge theories. Moreover, we will also analyze the lattice 2-algebra on the graph $\Gamma$, and extract the observables of discrete 2-Chern-Simons theory from it.