A Quillen model structure on the category of Kontsevich-Soibelman weakly unital dg categories

Abstract

In this paper, we study weakly unital dg categories as they were defined by Kontsevich and Soibelman [KS, Sect.4]. We construct a cofibrantly generated Quillen model structure on the category ${\mathrm{Cat}}_{\mathrm{dgwu}}(\mathbb{k})$ of small weakly unital dg categories over a field $\mathbb{k}$. Our model structure can be thought of as an extension of the model structure on the category ${\mathrm{Cat}}_{\mathrm{dg}}(\mathbb{k})$ of (strictly unital) small dg categories over $\mathbb{k}$, due to Tabuada [Tab]. More precisely, we show that the imbedding of ${\mathrm{Cat}}_{\mathrm{dg}}(\mathbb{k})$ to ${\mathrm{Cat}}_{\mathrm{dgwu}}(\mathbb{k})$ is a right adjoint of a Quillen pair of functors. We prove that this Quillen pair is, in turn, a Quillen equivalence. In course of the proof, we study a non-symmetric dg operad $\mathcal{O}$, governing the weakly unital dg categories, which is encoded in the Kontsevich-Soibelman definition. We prove that this dg operad is quasi-isomorphic to the operad ${\mathrm{Assoc}}_{+}$ of unital associative algebras.

37 pages, 1 figure