𝑆-fibrations and Calculus Left Fractions

Abstract

Let 𝒞 be any small 𝒰-category, where 𝒰 is a fixed Grothendeick universe. Let 𝑆 be a set of morphisms in the category 𝒞. Let 𝒞[𝑆−1] be the category of fractions of 𝑆 and 𝐹𝑆∶ 𝒞 → 𝒞[𝑆−1] be the canonical functor. For convenience we write 𝐹𝑆=𝐹. Bauer and Dugundji [2] have introduced the concept of 𝑆-fibration, weak 𝑆-fibration, 𝑆-cofibration and weak 𝑆-cofibration in the category 𝒞 and have explored the properties of these concepts. There are some other advantages over the assumption that the set of morphisms 𝑆 admits a calculus of left (right) fractions [4, 6]. In this note we study some cases showing how the assumption that 𝑆 admits a calculus of left (right) fractions helps us to prove that weak 𝑆-fibration implies 𝑆-fibration and weak 𝑆-cofibration implies 𝑆-cofibration.