### Categorical Homotopy and Fibrations

F.-W. Bauer, J. Dugundji
1969 Transactions of the American Mathematical Society
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
more » ... on 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?={/|► tlz determines the usual notion of homotopy; if Jf is the category of Kan-complexes, then ir-homotopy is the same as Kanhomotopy; and if 77: Jf->-'S7, is the total homology functor on the category of chain-complexes, then 77-homotopy is the usual chain-homotopy. In the Appendix, we give a construction of Jf/SJl that differs slightly from that of Gabriel-Zisman, and contains slightly more detail; one advantage of this procedure is that some results in [3] are seen to hold without the additional requirement that "2JI admits a calculus of left fractions".