Differential graded algebras *dga-s* are fundamental objects of study in homological algebra and category theory. On the **nlab** [webpage](https://ncatlab.org/nlab/show/differential+graded+algebra), they are defined as follows:



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> 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;
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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? 



DGA for a general abelian category
The concept you're asking about has been studied by Christensen and Hovey in [Quillen model structures for relative homological algebra](https://arxiv.org/abs/math/0011216). Specifically, see Example 3.1 on page 17 of the pdf. In this paper, $\mathscr{A}$ is only required to be closed monoidal (as well as bicomplete, to have a chosen projective class $P$, with the unit $P$-projective - but these conditions are really just to do homotopy theory). If you check citeseer or Google scholar, you can find many papers that cite this work, and further develop the theory of DGAs in abelian categories, including [Schwede and Shipley](http://homepages.math.uic.edu/~bshipley/classTopFinal.pdf), [Barthel-May-Riehl](http://emis.ams.org/journals/NYJM/j/2014/20-53v.pdf), and others. 


