The following is a very naive construction, and I am almost embarrassed to ask questions about it.

Suppose I have a connected simple graph $\Gamma = (\Gamma_0,\Gamma_1)$ (i.e., vertex set, edge set) with a total ordering of its vertices and a Lie group $G$. I can then form the short cochain complex $C^i(\Gamma;G) =$ the functions $\Gamma_i\to G$, with smooth coboundary operator $$ \delta:C^0(\Gamma;G) \to C^1(\Gamma;G) $$ defined by $\delta(g)(\alpha) = g(d_0(\alpha))g(d_1(\alpha))^{-1}$, where $d_0,d_1$ are given by taking faces of edges ordered by the total ordering. In fact, $\delta$ is an orbit map associated with an action of $C^0(\Gamma;G)$ on $C^1(\Gamma;G)$ defined by $$ (g\cdot f)(\alpha) := g(d_0(\alpha))g(d_1(\alpha))^{-1}f(\alpha) $$ This leads to the following definition of $H^1(\Gamma;G)$ as the orbit space of this action.

When $G$ is discrete, this is a studied notion going back to Olum in the 1950s. But I am interested in the non-discrete case.

For example, if $G = S^1$, one has $H^1(\Gamma;S^1) \cong T^n$ where $T^n$ is the $n$-torus with $n$ is the first Betti number of $\Gamma$.

**Questions:** Is the above notion to be found in the literature? If so, where? Has it been generalized to other degrees for arbitrary simplicial complexes, or spaces?

I did a google search and a mathscinet search but I could not find any references.

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