A model structure on the category of A$_\infty$-categories with strict morphisms

TL;DR

This paper establishes a cofibrantly generated model structure on the category of strictly unital A$_ finite$-categories over a commutative ring, showing that semi-free A$_ finite$-categories are cofibrant objects, facilitating homotopical analysis.

Contribution

It introduces a new model structure on A$_ finite$-categories with strict morphisms, proving all objects are fibrant and identifying semi-free categories as cofibrant.

Findings

The category of A$_ finite$-categories admits a cofibrantly generated model structure.

All objects in this category are fibrant.

Semi-free A$_ finite$-categories are cofibrant objects.

Abstract

We prove that the category of (strictly unital) A$_\infty$-categories, linear over a commutative ring $R$, with strict A$_\infty$-morphisms has a cofibrantly generated model structure. In this model structure every object is fibrant and the cofibrant objects have cofibrant morphisms. As a consequence we prove that the semi-free A$_\infty$-categories (resp. resolutions) are cofibrant objects (resp. resolution) in this model structure.