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

## TL;DR

This paper establishes a model structure on the category of weakly unital dg categories, extending known structures and proving a Quillen equivalence with strictly unital dg categories, using operad theory.

## Contribution

It constructs a cofibrantly generated Quillen model structure on weakly unital dg categories and shows its equivalence to the strict case via operad quasi-isomorphism.

## Key findings

- Model structure on weakly unital dg categories established
- Quillen equivalence with strictly unital dg categories proved
- Operad governing weakly unital dg categories is quasi-isomorphic to associative operad

## 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 $\mathrm{Cat}_{\mathrm{dgwu}}(\Bbbk)$ of small weakly unital dg categories over a field $\Bbbk$. Our model structure can be thought of as an extension of the model structure on the category $\mathrm{Cat}_{\mathrm{dg}}(\Bbbk)$ of (strictly unital) small dg categories over $\Bbbk$, due to Tabuada [Tab]. More precisely, we show that the imbedding of $\mathrm{Cat}_{\mathrm{dg}}(\Bbbk)$ to $\mathrm{Cat}_{\mathrm{dgwu}}(\Bbbk)$ is a right adjoint of a Quillen pair of functors. We prove that this 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.