Categorical Homotopy and Fibrations

Abstract

Let X be an arbitrary category, and let 90c be any family of its morphisms. It is known [3] that there is a category X/W (the Gabriel-Zisman "category of fractions of X by 30c ") having the same objects as X, and a covariant functor 7j:Jf-> X/'OJl which is the identity on objects, such that r¡(f) is invertible in X/Wl for each/e 9JÎ.We will use each class 9JÍ to determine a notion of homotopy in X, by defining two morphisms/, g to be 9JÎ-homotopic if -n(f) = rj(g). This notion has the usual properties expected of a homotopy notion; moreover, in the category Top of topological spaces and continuous maps, suitable choices of 9Ji (for example, TO = all homotopy equivalences) reveal 9Jc-homotopy equivalent to the usual notion of homotopy.Each class 501 determines a notion of fibration in X In the category Top, a suitable choice of 9Jt determines both the usual homotopy and the Hurewicz fibrations; however, different classes 9JI may yield the usual notion of homotopy and distinct notions of fibration. More generally, in an arbitrary category X, it is the given class 9JÎ itself, rather than the homotopy notion induced by 9JÏ, that determines the concept of fibration; from this viewpoint, it turns out, surprisingly, that the notion of a Hurewicz fibration is not a homotopy notion. By "reversing arrows," 9JI determines also a concept of cofibration; and again there is a splitting: in fact, two classes may determine the same notion of homotopy but distinct notions of cofibration.In the last section, we introduce the concept of a weak 9Jl-fibration. This notion does not, in general, possess all the advantages of the previous one; however, it reduces to the previous notion under suitable restrictions on the class 9JÎ. Moreover, there are classes, 9JÎ, 9? in the category Top such that {weak W-fibrations} = {Hurewicz fibrations} and {weak 9c-fibrations} = {Dold fibrations}.Each covariant functor O: X -> if determines a O-homotopy in X, by choosing 9J?={/|<p(/) is invertible}. Most of the homotopy notions encountered in various categories turn out to stem from this general construction. For example: if tlz is the category of graded groups and degree zero homomorphisms, and if ir(X) is the total homotopy of a space X, then in the category X of spaces dominated by CW-complexes the functor it: X ->► tlz determines the usual notion of homotopy;

Categories of fuzzy sets

“…We will follow the construction of Bauer-Dugundji, which is described in some detail in [2]. In this construction, the objects of C=M are the same as the objects of C, and the maps from X to Y in the category C=M are equivalence classes of diagrams of the form…”

“…Definition 11 (Dugundji-Bauer [2]). Let C be any category, and let M be any family of its morphisms.…”

Esqueleto Homoto-Homológico en la Categoría de los Grupos Abelianos

“…En [2] se define una relación de homotopía en una categoría A por medio de la categoría de cocientes (A/M, κ) (o llamada también en [5] como categoría de fracciones de A por M), donde A/M es una categoría con los mismos objetos de A y κ : A → A/M es un funtor covariante que preserva los objetos de A y satisface las siguientes dos condiciones:…”

Gray-categories model algebraic tricategories

“…We denote by W the collection of weak equivalences in GrayCat. The category GrayCat[W −1 ] is the localization of GrayCat at the weak equivalences, which is determined (up to unique isomorphism) by the following universal property [BD69]:…”