Monoidal cofibrant resolutions of dg algebras

TL;DR

This paper constructs a functorial, colax-monoidal cofibrant resolution for dg algebras over any characteristic field, facilitating the development of dg localizations compatible with Toën's framework.

Contribution

It introduces a novel functorial cofibrant resolution for dg algebras that is colax-monoidal, enabling new applications in dg category localization.

Findings

Constructed a colax-monoidal cofibrant resolution functor for dg algebras.

Established compatibility of the resolution with tensor products via colax maps.

Facilitated the existence of a colax-monoidal dg localization matching Toën's localization in homotopy.

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 $A\rightsquigarrow \mathfrak{R}(A)$ 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 To\"{e}n's dg localization.