Homotopy covariant functors (Q1059891)

In this paper we prove a theorem on the representability of convariant functors formulated by \textit{E. H. Brown} jun. [Am. Math., II. Ser. 75, 467-484 (1962; Zbl 0101.406); correction ibid. 78, 201 (1963; Zbl 0113.169)]. We denote by PW' the category whose objects are cellular spaces with distinguished point, and whose morphisms are the homotopy classes of mappings. The full subcategory of PW' generated by simply connected spaces which have only a finite number of cells in each dimension is denoted by C. We denote by PEns the category of sets with distinguished element, and the morphisms preserve the distinguished elements. We consider the funtors \(\pi_ X=[X, ]\) for an arbitrary X of the category PW'. Theorem: For a covariant functor F: PW'\(\to PEns\) (which has three natural properties), there exists in the category C a space X, which is unique up to homotopy equivalence, such that for any Y of C the set F(Y) is naturally isomorphic with the set [X,Y].