Differential graded algebras *dga-s* are fundamental objects of study in homological algebra and category theory. On the **nlab** webpage, they are defined as follows:

a

dgais a monoid in the symmetric monoidal category of (possibly unbounded) chain complexes or cochain complexes with its standard structure of a monoidal category by the tensor product of chain complexes;

I wonder if this definition makes for a general abelian category $\cal{A}$. Namely, can we define a *dga* for the category ${\cal A}$
to be a monoid object in the category of cohain complexes of ${\cal A}$, such that the differential respects the graded Leibniz rule? Is this making sense; do other people look at such things?