Start with a category $C$. Form a monoid $M$ whose elements are lists of morphisms in the category $C$ subject to commuting diagrams in $C$. Is there a name for this construction or a better way to categorially understand this?



The monoid of lists of morphisms in a category subject to commuting diagrams
Do you mean take the monoid with the following presentation? Take the arrows of C as generators and add the relations that f.g = fg if f and g are composable and that each identity of C be equivalent to 1? I would call this the universal monoid U(C) of C. It has the universal property that there is a functor $F:C\to U(C)$ (the obvious one) such that any functor from C to a monoid M, viewed as a one-object category, factors through $U(C)$. So U is the left adjoint to the inclusion of the category of monoids into the category of small categories.


I myself have used the group version of this many times.


