A strong shape theory with S-duality

If in the classical S-category P, 1) continuous mappings are replaced by compact-open strong shape (= coss) morphisms (cf. §1 or [1], §2), and 2) ∧-products are properly reinterpreted, then an S-duality theorem for arbitrary subsets X ⊂ S (rather than for compact polyhedra) holds (Theorem 2.1). 0. Introduction. In a previous paper [1] we introduced the concept of coss-shape (compact-open strong shape or strong shape with compact support) and established for each X = (X,m), with m ∈ Z and finitedimensional X ∈Met (= category of separable metric spaces), i.e. for any object in an S-category P (cf. [1, §4]), an S-dual DX ∈ P which is, up to isomorphism in P, uniquely determined and natural, and satisfies D2X ≈ X and P(X,Y) ≈ P(DY, DX). In the present paper we introduce in §1 and §3 new, additional cossmorphisms X → Y ∧ Z and X ∧ Y → Z (Y ∧ Z and X ∧ Y are pairs of objects in P) which for special choices of X, Y, Z can be interpreted as coss-morphisms between X and Y ∧ Z ∈ P (cf. §3) resp. X ∧Y and Z, or sometimes even as strong shape morphisms (1.2 and §5). The main objective of this paper is the verification of Theorem 2.1 asserting the existence of a natural isomorphism (1) {X ∧Y,Z}c ≈ {X, DY ∧ Z}c ({. . . , . . .}c denoting the respective sets of coss-morphisms in the S-category, cf. §3). Among the corollaries in §2 we recover the main theorem of [1] and two other S-duality theorems for special choices of X, Y, Z. In particular, classical S-duality (cf. [9], [10]) turns out to be a corollary of (1) (cf. §2(8)). 1991 Mathematics Subject Classification: Primary 55P25, 55P55; Secondary 55N20, 55M05.