A remark on fibrancy of ($\mbox{A$_{\infty}$Cat}$,$W^{\tiny\mbox{A}_{\infty}}_{\tiny\mbox{qe}}$)

TL;DR

This paper demonstrates the existence of certain pullbacks in the category of strictly unital A$_{ty}$categories and confirms that this category with a specific class of weak equivalences is fibrant in the model category of relative categories.

Contribution

It proves the existence of pullbacks of A$_{ty}$functors satisfying property F1 and affirms the fibrancy of (A$_{ty}$Cat, W^{A$_{ty}$}_{qe}) in RelCat.

Findings

Existence of pullbacks in A$_{ty}$categories for functors with property F1.

Confirmation that (A$_{ty}$Cat, W^{A$_{ty}$}_{qe}) is fibrant in RelCat.

Positive answer to Pascaleff's question on fibrancy.

Abstract

In this note we prove the existence, in the category of (strictly unital) A$_{\infty}$categories, of the pullback of a (strictly unital) A$_{\infty}$functor, satisfying a particular property (denoted by F1), along any A$_{\infty}$functor. As a consequence we provide a positive answer to Pascaleff's question whether (A$_{\infty}$Cat,$W^{\tiny\mbox{A}_{\infty}}_{\tiny\mbox{qe}}$) is a fibrant object in RelCat.