Monoidal cofibrant resolutions of dg algebras

Abstract

Let $k$ be a field of any characteristic. In this paper, we construct a functorial cofibrant resolution $\mathfrak{R}(A)$ for the ${\mathbb{Z}}_{\le 0}$-graded dg algebras $A$ over $k$, such that the functor $A⇝\mathfrak{R}(A)$ is colax-monoidal with quasi-isomorphisms as the colax maps. More precisely, there are maps of bifunctors $\mathfrak{R}(A\otimes B)\to \mathfrak{R}(A)\otimes \mathfrak{R}(B)$, compatible with the projections to $A\otimes B$, and obeying the colax-monoidal axiom. The main application of such resolutions (which we consider in our next paper) is the existence of a colax-monoidal dg localization of pre-triangulated dg categories, such that the localization is a genuine dg category, whose image in the homotopy category of dg categories is isomorphic to the Toën's dg localization.

v3 69 pages (vs v2 39 pages) thoroughly improved and corrected

References in corpus (2)

• Algebras and modules in monoidal model categories
• Enriched model categories and an application to additive endomorphism spectra