Let us consider a 1-category $C$. For any commutative and unital ring $k$ the the free $k$-module generated by the morphisms of $C$ can be equipped with an algebra structure by setting $fg$ to be either 0 or their composition.
Now consider the differential graded $k$-module associated to an $\omega$-category. (In degree $m$ a basis is given by $m$-morphisms and the boundary is given by the difference of the $(m-1)$-target and the $(m-1)$-source map). The above construction endows the $k$-module in each degree with an algebra structures.
Is there a way of unifying all these individual algebra structures into a natural chain complex structure?