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In Felix, Halperin, Thomas "Rational homotopy theory" on page 46 we find the following definition :

A cochain algebra is a dga $(R, d)$ with $R = \{R^n\}_{n\ge o}$.

I can't get the definition of a cochain algebra, I mean what is special about a cochain algebra that leads to have the distinguished name of cochain algebra among other dga's ( Differential Graded Algebra) ?

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I would say that it comes from historical reasons, cochain algebras were invented first. Furthemore, there is a difference between bounded and unbounded dga, for example in term of convergence of spectral sequences.

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