Faithfulness of a functor of Quillen

TL;DR

This paper investigates the faithfulness of a canonical functor from the category of fibrant objects modulo cylinder homotopy to the homotopy category, establishing conditions for faithfulness and showing it does not always hold.

Contribution

It identifies specific conditions under which the canonical functor is faithful and demonstrates that faithfulness does not hold universally.

Findings

The functor is faithful under certain conditions.

Faithfulness of the functor is not guaranteed in general.

Provides criteria for when the functor is faithful.

Abstract

There exists a canonical functor from the category of fibrant objects of a model category modulo cylinder homotopy to its homotopy category. We show that this functor is faithful under certain conditions, but not in general.