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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?

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    $\begingroup$ You need to assume A is symmetric monoidal to define any notion of a "monoid object". Once you have that, you can define a dga in A to be a monoid in the category of chain complexes of A (which acquires a symmetric monoidal structure). Note that the homotopy category of E_oo-algebras in the derived oo-category of A is equivalent to the category of dgas in A (in the above sense) localized at the quasi-isomorphisms. $\endgroup$
    – skd
    Commented Sep 13, 2018 at 15:57
  • $\begingroup$ Sorry, I don't see why symmetric is needed to define a monoid object. I understood it was not usual. For example $\endgroup$
    – Max Schattman
    Commented Sep 13, 2018 at 16:04
  • $\begingroup$ see ncatlab.org/nlab/show/monoid+in+a+monoidal+category $\endgroup$
    – Max Schattman
    Commented Sep 13, 2018 at 16:04
  • $\begingroup$ or indeed en.wikipedia.org/wiki/Monoid_(category_theory) $\endgroup$
    – Max Schattman
    Commented Sep 13, 2018 at 16:05
  • $\begingroup$ Sorry, that should've just read "monoidal". $\endgroup$
    – skd
    Commented Sep 13, 2018 at 18:23

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The concept you're asking about has been studied by Christensen and Hovey in Quillen model structures for relative homological algebra. 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, Barthel-May-Riehl, and others.

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  • $\begingroup$ Another place to read about this: Kashiwara/Schapira: Categories and Sheaves - However, it is not specifically concerned with monoids in these monoidal categories. But it seems the original questions is more about transitioning from 'actual' module categories to more general settings and Kashiwara/Shapira is definitely a good guide for this transition. $\endgroup$
    – Gerrit Begher
    Commented Sep 14, 2018 at 6:39

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