Finite flat connection groupoid on a cell complex

Let $K$ be a finite connected CW complex and $G$ a finite group. Choose an orientation of each 1-cell and, for each 2-cell, its based attaching word in the finite edge presentation of ${\pi }_1(K)$; the choice of base paths is part of that presentation. A flat edge assignment is a label $g_e\in G$ for every oriented 1-cell with $g_{\bar{e}}=g_e^{-1}$, such that each attaching word has holonomy $1$. If a word traverses edges $e_1,\dots ,e_m$ in that order with exponents ${ϵ}_j\in \{1,-1\}$, its holonomy is $g_{e_m}^{{ϵ}_m}\cdots g_{e_1}^{{ϵ}_1}$.

Write $\mathrm{Flat}(K,G)$ for the finite set of flat assignments and $K_0$ for the vertices. The finite vertex gauge group $G^{K_0}$ acts by

$(h\cdot g{)}_e=h_{t(e)}{g}_e{h}_{s(e)}^{-1},$

where $s(e)$ and $t(e)$ are the source and target vertices. The finite flat connection groupoid ${\mathcal{C}}_G(K)$ is the action groupoid $\mathrm{Flat}(K,G)\mathrm{/}\mathrm{/}{G}^{K_0}$: its objects are flat assignments and its arrows $g\to g^{\mathrm{\prime }}$ are the $h\in G^{K_0}$ with $h\cdot g=g^{\mathrm{\prime }}$. Its untwisted finite BF partition function is the groupoid cardinality

$Z_G(K):=\mathrm{\mid }{\mathcal{C}}_G(K)\mathrm{\mid }={\sum }_{[g]\in \mathrm{Flat}(K,G)\mathrm{/}{G}^{K_0}}\frac{1}{\mathrm{\mid }\mathrm{Stab}(g)\mathrm{\mid }}=\frac{\mathrm{\mid }\mathrm{Flat}(K,G)\mathrm{\mid }}{\mathrm{\mid }G{\mathrm{\mid }}^{\mathrm{\mid }{K}_0\mathrm{\mid }}}\mathrm{.}$

This is a finite combinatorial model; it has no continuous Lie-algebra-valued connection integral or additional cocycle phase.

For a finite disconnected CW complex, define ${\mathcal{C}}_G(K)$ to be the product of the groupoids of its connected components; for the empty complex use the terminal groupoid.

Finite groupoid cardinality and homotopy-fiber integration

A finite connected CW complex has the edge-and-2-cell presentation