We provide a more economical refined version of Evrard's categorical cocylinder factorization of a functor. 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.
Keywords: Homotopy fibrant replacement of a functor; Quillen theorem B; Evrard's theorem
2010 MSC: 18D30, 18G30, 18G55
Theory and Applications of Categories, Vol. 31, 2016, No. 34, pp 989-1015.