The monoid of lists of morphisms in a category subject to commuting diagrams 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?
 A: 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.
A: For every $n$ we have the set $N_n(C)$ of $n$-chains $X_0 \to \dotsc \to X_n$ of morphisms in $X$. This set is actually a category, the morphisms are commutative diagrams. A short definition is $N_n(C) = \mathrm{Hom}([n],C)$, where the preorder $[n] = \{0,\dotsc,n\}$ is considered as a category. In fact, $\{N_n(C)\}_n$ is a simplicial set, called the nerve of $C$. It's geometric realization is also called the geometric realization $|C|$ of $C$.
In your question, you obviously vary $n$ and want to get a single monoid. It's not clear to me how you want to achieve this. Perhaps you really mean the nerve, or do you want to consider $\coprod_n N_n(C)$?
