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 dga is 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 will wonder if this definition makes for a general abelian category $\cal{A}$. Namely, if we can well-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?

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 dga is 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?