On the Evrard fibrant replacement of a functor

TL;DR

This paper refines Evrard's categorical cocylinder factorization, providing a more economical approach to factor any functor between small categories into a homotopy equivalence followed by a (co)fibred functor satisfying Quillen's Theorem B conditions.

Contribution

It introduces a more efficient version of Evrard's factorization, enhancing the understanding of functor decompositions in category theory.

Findings

Provides a refined Evrard fibrant replacement construction.

Shows any functor can be factored into a homotopy equivalence and a (co)fibred functor.

Ensures the (dual) assumption of Quillen's Theorem B is satisfied.

Abstract

We provide a more economical refined version of Evrard's categorical cocylinder factorization of a functor [Ev1,2]. We show that any functor between small categories can be factored into a homotopy equivalence followed by a (co)fibred functor which satisfies the (dual) assumption of Quillen's Theorem B.