# $\omega$-categorical algebra

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?