Finite background gauge 2-groupoid

Fix a finite ordered simplicial complex $K$, a finite group $G$, a finite abelian group $A$ in additive notation, a left action $\rho :G\to \mathrm{Aut}(A)$, and a normalized inhomogeneous $3$-cocycle $\beta :G^3\to A$. A background $x=(a,B)$ has flat edge labels $a_{ij}{a}_{jk}=a_{ik}$ and satisfies ${\delta }_{a}B=a^{\ast }\beta$, with coefficients at the first vertex. Here $({\delta }_{a}t{)}_{ij}={\rho }_{a_{ij}}{t}_j-t_i$ for a vertex cochain, $({\delta }_a\lambda {)}_{ijk}={\rho }_{a_{ij}}{\lambda }_{jk}-{\lambda }_{ik}+{\lambda }_{ij}$ for an edge cochain, and $({\delta }_{a}B{)}_{ijkl}={\rho }_{a_{ij}}{B}_{jkl}-B_{ikl}+B_{ijl}-B_{ijk}$. The cocycle convention is

$$D(s,t,r,q):={\rho }_s\beta (t,r,q)-\beta (st,r,q)+\beta (s,tr,q)-\beta (s,t,rq)+\beta (s,t,r)=0.$$

The background gauge 2-groupoid $\mathcal{B}(K)$ has these backgrounds as objects. A $1$-arrow $f:x\to y$ is a raw pair $(g,\lambda )$ of vertex $G$-labels and edge $A$-labels with target $y=(b,C)$ given by

$$b_{ij}=g_i{a}_{ij}{g}_j^{-1},{\zeta }_{ijk}^g(a)=-\beta (g_i,a_{ij},a_{jk})+\beta (b_{ij},g_j,a_{jk})-\beta (b_{ij},b_{jk},g_k),C={\rho }_{g}B+{\zeta }^g(a)+{\delta }_b\lambda \mathrm{.}$$

Raw pairs remain distinct even when their induced functions on backgrounds coincide. For fixed $g$ and target edge labels $b$, a $2$-arrow $t:(g,\lambda )\Rightarrow (g,{\lambda }^{\mathrm{\prime }})$ is a vertex $A$-cochain satisfying ${\lambda }^{\mathrm{\prime }}-\lambda ={\delta }_{b}t$. There are no $2$-arrows between different vertex labels $g$. Vertical composition adds vertex cochains; the identity $2$-arrow is zero.

For consecutive $1$-arrows $(g,\lambda ):x\to y$ and $(h,\mu ):y\to z$, write $a,b,c$ for the successive edge labels. Their composite is $(hg,\nu )$, where

$${\nu }_{ij}={\rho }_{h_i}{\lambda }_{ij}+{\mu }_{ij}+\tau (h,g;a{)}_{ij},\tau (h,g;a{)}_{ij}=\beta (h_i,g_i,a_{ij})-\beta (h_i,b_{ij},g_j)+\beta (c_{ij},h_j,g_j)\mathrm{.}$$

Horizontal composition of $t$ on the first arrow and $s$ on the second is $s+{\rho }_{h}t$. Identity $1$-arrows have all vertex labels $e$ and edge labels zero. For three consecutive arrows $g,h,k$, the associator from $((k\circ h)\circ g)$ to $k\circ (h\circ g)$ is the vertex cochain ${\sigma }_i=\beta (k_i,h_i,g_i)$. Unit constraints are identity $2$-arrows.

A $2$-groupoid here means these data form a bicategory: each pair of objects has a category of $1$-arrows and $2$-arrows; horizontal composition preserves identities and vertical composition; associators are natural and obey the pentagon; units obey the triangle identity; all $2$-arrows are invertible and every $1$-arrow has a reverse up to invertible $2$-arrows. Finiteness means all three sets of labels are finite. Coherence of the displayed data is a separate theorem.

Write ${\pi }_0\mathcal{B}(K)$ for the equivalence classes of objects under $1$-arrows. For an object $x$, let ${\pi }_1(\mathcal{B}(K),x)$ be the group of $2$-isomorphism classes of $1$-automorphisms of $x$, and let ${\pi }_2(\mathcal{B}(K),x)$ be the abelian group of $2$-automorphisms of its identity $1$-arrow. In particular, ${\pi }_2$ retains vertex cochains with ${\delta }_{a}t=0$ rather than discarding them. For a subcomplex $L\subseteq K$ and fixed background $s$ on $L$, the relative background 2-groupoid has objects restricting exactly to $s$, $1$-arrows with $g=e$ on vertices of $L$ and $\lambda =0$ on edges of $L$, and $2$-arrows with $t=0$ on vertices of $L$.

Finite 2-group background cochains

Finite 2-group background gauge consistency

Baez and Lauda, §8.3

Benini, Cordova and Hsin, §2.3