# Directed homotopy in the Cayley graph of a monoid

There is a the notion of the Cayley graph $C(G)$ of a group $G$ (which depends on a given presentation $G \cong \mathcal F(S) / \sigma$ where $\mathcal F$ is the free group functor and $\sigma$ some congruence). This graph has several nice properties like that the fundamental group $\pi_1(C(G), e)$ ($e$ the identity of $G$) is the normal subgroup of $\mathcal F(S)$ corresponding to $\sigma$, or the subgroup of zero-sum reduced words with entries in the generators $S$ and their inverses.

One can easily imagine the analagous construction for monoids. A monoid presentation $M=\mathcal F(S) / \sigma$ defines a Cayley graph $C(M)$. The fundamental monoid $\pi_1(C(M),e)$ is the monoid of zero-sum sequences of generators $S$. Also, considering the free category generated by this graph $F(C(M))$, we have $\mathsf{Hom}_{F(C(M))}(e,x) \cong \mathsf Z_M(x)$ where $\mathsf Z_M(x)$ is the factorization set of $x$ with respect to the generators $S$. (Because these are just sets, this is just a structure-less bijection, but they both correspond to strings with entries in the generators which get evaluate to $x$ under the operation of $M$.)

Has this connection been explored? I don't think I saw it mentioned in "Directed Algebraic Topology" (but I haven't done more than skim it).

edit: I have found a variant of the basic construction I had in mind by N.D. Gilbert (Monoid presentations and associated groupoids), but this construction is nondirected and so only describes non-directed homotopy (fundamental group $\cong$ Grothendieck group of $M$, rather than fundamental monoid $\cong M$).