Some properties of the A$_{\infty}$-nerve

TL;DR

This paper proves that the A$_{ abla}$-nerve construction preserves weak equivalences between quasi-equivalent A$_{ abla}$-categories and shows that the nerve of a pretriangulated A$_{ abla}$-category forms a stable $ abla$-category.

Contribution

It establishes the weak equivalence of A$_{ abla}$-nerves for quasi-equivalent categories and demonstrates the stability of the nerve of pretriangulated A$_{ abla}$-categories.

Findings

A$_{ abla}$-nerves of quasi-equivalent categories are weakly equivalent.

The nerve of a pretriangulated A$_{ abla}$-category is a stable $ abla$-category.

Provides foundational results linking A$_{ abla}$-categories and $ abla$-categories in the Joyal model structure.

Abstract

The aim of this paper is to prove that the A$_{\infty}$-nerve of two quasi-equivalent A$_{\infty}$-categories (linear over a commutative ring) are weak-equivalent in the Joyal model structure. As a consequence we prove that the A$_{\infty}$-nerve of a pretriangulated A$_{\infty}$-category is a stable $\infty$-category.