On Evrard’s Homotopy Fibrant Replacement of a Functor

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. Introduction The problem treated here is how to replace a functor f : C→ D between small categories with a “homotopy fibration of categories”. By latter we mean a functor fulfilling the assumption of Corollary to Quillen Theorem B [Q, Section 1], see Proposition 1.18 for several equivalent formulations. Such a replacement of a functor f : C → D between small categories by a homotopy fibration is a commutative diagram