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

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