2011/12/11 by Boris Shoikhet, Shoikhet, Boris
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.1112.2360
openalex publication_date 2011/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be a field of any characteristic. In this paper, we construct a functorial cofibrant resolution \mathfrakR(A) for the ℤ≤ 0-graded dg algebras A over k, such that the functor A\rightsquigarrow \mathfrakR(A) is colax-monoidal with quasi-isomorphisms as the colax maps. More precisely, there are maps of bifunctors \mathfrakR(A⊗ B)→ \mathfrakR(A)⊗ \mathfrakR(B), compatible with the projections to A⊗ 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.