Abstract homotopy theory

Abstract

i) 1. Introduction. Let "^ be a category and let Sf be the category of sets and set maps. If X and Y are objects in <, \X, Y~] will denote the set of morphisms from X to Y. Let H : %> -> f be a contravariant functor. In the case where # is the category of CW complexes with base point and homotopy classes of maps, conditions on H were given in [1] which implied that H was naturally equivalent to [ ,Y\ for some CW complex YH. Furthermore, the proof of this result used, for the most part, abstract category arguments. The aim of this paper is to formalize this latter fact. That is, we wish to give conditions on an abstract category < and conditions on H from which we can deduce that there is a natural equivalence T: [ , Y]-*H. Furthermore, we want the category of CW complexes and homotopy classes of maps to satisfy our conditions on c.