Flatness for the \(E_\infty\)-tensor product (Q2752878)

One starts with the Kriz-May work [\textit{I. Kriz} and \textit{J. P. May}, ``Operads, algebras, modules and motives'', Asterisque 233 (1995; Zbl 0840.18001)] in homological algebra which describes an \(E_\infty\)-operad \(C\) of differential graded modules and an \(E_\infty\)-tensor product such that \(C\)-algebras are commutative monoids for it. NEWLINENEWLINENEWLINEThe present paper studies the homotopy theory of \(E_\infty\)-algebras and the interaction with the tensor product. The here defined cell \(C\)-algebras (analogous to a free resolution giving a derived functor of the tensor product) are shown to be flat. This leads to a proof that the category of \(C\)-algebras is a proper closed model category (in the sense of Quillen), that weak equivalences being quasi-isomorphisms, fibrations being surjections and cofibrations being maps which are retracts of relative cell inclusions \(A\to B\) (i.e., describing \(B\) as a cell \(A\)-algebra).NEWLINENEWLINEFor the entire collection see [Zbl 0964.00013].