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There are two equivalent definitions of differential graded algebras with different point of view. The first one is that it is a sequence $A=(A^n)_{n\in \mathbb{Z}}$ of vector spaces together with a differential $d:A\to A$ of degree $1$ and a multiplication $\mu:A\otimes A\to A$ satisfying certain conditions. The second one is that it is an algebra $A$ together with a decomposition $A=\bigoplus_{n\in \mathbb{Z}}A^n$ and a linear map $d:A\to A$ satisfying certain conditions.

My question is which one is more natural. Thanks very much!

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    $\begingroup$ I am not sure about "more natural", but thinking of a dg-algebra as a complex immediately suggests to also consider $H^n$ or to look at homotopies of maps between dg-algebras. $\endgroup$
    – M.G.
    Commented Dec 10, 2021 at 2:42
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    $\begingroup$ And thinking of it as an algebra suggests considering modules and so on. $\endgroup$
    – Zach Teitler
    Commented Dec 10, 2021 at 2:55
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    $\begingroup$ The first one is more natural because it describes it as "an associative monoid in the category of complexes". This notion is easy to extrapolate to other situations (like operads, endofuncfors and monads, etc). When you talk about direct sums, I think about weight gradings, not homological ones, so for me the second definition is a bit misleading (and will confuse people when you define weight graded dgas, which is why some people speak about "Adams gradings" instead.) $\endgroup$
    – Pedro
    Commented Dec 25, 2021 at 22:25

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The definition as a sequence of components immediately generalizes to settings like arbitrary abelian categories without countable coproducts. The definition with all components fused together cannot be used in such a generality.

The underlying algebra functor sends a differential graded algebra $A$ to $\bigoplus_n A_n$ equipped with the induced multiplication. This is precisely the underlying object of the second definition, which may explain its popularity.

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