Finite background groupoid and cocycle-weighted gauging data

Let $K$ be a finite ordered simplicial complex. A finite background field groupoid $\mathcal{F}(K)$ is a small category of declared fields and invertible gauge equivalences, with finitely many isomorphism classes and finite automorphism groups. Its automorphisms retain the multiplicities declared by the field model. A cocycle/action weight is a function $W_K:\mathrm{Ob}\mathcal{F}(K)\to U(1)$ constant on every gauge arrow. Its normalized gauging sum is

$$Z_K(W_K)=\sum _{[x]\in {\pi }_0\mathcal{F}(K)}\frac{W_K(x)}{\mathrm{\mid }{\mathrm{Aut}}_{\mathcal{F}(K)}(x)\mathrm{\mid }}\mathrm{.}$$

For the finite $2$-group specialization, fix finite $(G,A,\rho ,\beta )$ and let $\mathcal{B}(K)$ be the explicit background gauge $2$-groupoid: objects are flat $G$-edge cochains and $A$-face cochains satisfying ${\delta }_{a}B=a^{\ast }\beta$; $1$-arrows retain raw vertex/edge parameters $(g,\lambda )$, and $2$-arrows are vertex cochains $t$ with ${\lambda }^{\mathrm{\prime }}-\lambda ={\delta }_{a^g}t$, for fixed $g$. The raw $1$-composition is the displayed $\tau$-corrected composition; its associator is $\beta (k_i,h_i,g_i)$. Write ${\pi }_0$ for object equivalence classes, ${\pi }_1(\mathcal{B}(K),x)$ for $2$-isomorphism classes of $1$-automorphisms, and ${\pi }_2(\mathcal{B}(K),x)$ for $2$-automorphisms of the identity $1$-arrow. For a phase weight $W_K$ constant on all $1$-arrows, define the higher normalized gauging sum

$$Z_K^{(2)}(W_K)=\sum _{[x]\in {\pi }_0\mathcal{B}(K)}{W}_K(x)\frac{\mathrm{\mid }{\pi }_2(\mathcal{B}(K),x)\mathrm{\mid }}{\mathrm{\mid }{\pi }_1(\mathcal{B}(K),x)\mathrm{\mid }}\mathrm{.}$$

Distinct raw gauge labels are not identified by equality of their effects on fields. The higher sum retains ${\pi }_2$; it is not ordinary groupoid integration of the $1$-truncation. In particular a point has higher weight $\mathrm{\mid }A\mathrm{\mid }\mathrm{/}\mathrm{\mid }G\mathrm{\mid }$ for unit action, whereas an ordinary field groupoid $BG$ has weight $1\mathrm{/}\mathrm{\mid }G\mathrm{\mid }$.

For a subcomplex $L\subseteq K$ and fixed field $s$ on $L$, relative fields and arrows use exact restriction to $s$, identity vertex gauge labels on $L$, zero edge gauge labels on $L$, and zero vertex $2$-gauge labels on $L$. Their higher sum uses the same ${\pi }_2\mathrm{/}{\pi }_1$ prescription. Ordinary groupoids are included as $2$-groupoids with only identity $2$-arrows, so this prescription reduces to the preceding ordinary sum. A higher cocycle supplies an admissible scalar action only after its evaluated weight is proved invariant; nontrivial boundary line-bundle actions require their own transgression data and are not silently treated as scalar functions.

Finite 2-group background cochains

Finite background gauge 2-groupoid

Coherence and stabilizers of finite background gauge 2-groupoids

Finite groupoid cardinality and homotopy-fiber integration