The Homotopy Theory of $A_\infty$Categories

TL;DR

This paper develops the homotopy theory of $A_ abla$ categories by introducing semi-free $A_ abla$ categories and resolutions, advancing the understanding of their homotopical properties.

Contribution

It introduces semi-free $A_ abla$ categories and resolutions for non-unital $A_ abla$ and DG categories, providing new tools for their homotopy theory.

Findings

Defined semi-free $A_ abla$ categories as cofibrations.

Proved existence of resolutions for non-unital $A_ abla$ and DG categories.

Established the homotopy category structure for $A_ abla$ categories.

Abstract

In this paper we describe the homotopy category of the $A_\infty$categories. To do that we introduce the notion of semi-free $A_\infty$category, which plays the role of standard cofibration. Moreover, we define the non unital $A_\infty$ (resp. DG)categories with cofibrant morphisms and we prove that any non unital $A_\infty$ (resp. DG)category has a resolution of this kind.