  In the present work we describe the category $\mathsf{WC}_2$ of weighted
2-complexes and its subcategory $\mathsf{WC}_1$ of weighted graphs. Since a
Coxeter group is defined by its Coxeter graph, the construction of Coxeter
groups defines a functor from $\mathsf{WC}_1$ to the category of groups. We
generalize the notion of a Coxeter group by extending the domain of the functor
to the category $\mathsf{WC}_2$. It appears that the resulting functor
generalizes the construction of Coxeter groups, Gauss pure braid groups $GVP_n$
(introduced by V. Bardakov, P. Bellingeri, and C. Damiani in 2015), $k$-free
braid groups on $n$ strands $G_n^k$ (introduced by V. Manturov in 2015), and
other quotients of Coxeter groups.
