A functional S-dual in a strong shape category

In the S-category P (with compact-open strong shape mappings, cf. §1, instead of continuous mappings, and arbitrary finite-dimensional separable metrizable spaces instead of finite polyhedra) there exists according to [1], [2] an S-duality. The S-dual DX, X = (X,n) ∈ P, turns out to be of the same weak homotopy type as an appropriately defined functional dual (S0)X (Corollary 4.9). Sometimes the functional object XY is of the same weak homotopy type as the “real” function space X (§5). 0. Introduction. S-duality occurs in the S-category P, having pairs X = (X,n), X a separable metrizable finite-dimensional space, n ∈ Z, as objects, with stable homotopy classes of so-called coss-morphisms (= strong shape morphisms with compact-open carrier, cf. [1, §2], [2, §1]) as morphisms. The main result is the existence of an S-dual DX, displaying all properties to be expected from classical S-duality (cf. [1, Theorem 4.3], [2, Theorem 2.1]). One of the basic achievements of classical S-duality lies in the possibility of exhibiting DX, up to weak homotopy equivalence, as a functional object F (X,S0) ([4, Theorem 3.8] and [7]). In the present strong-shape-theoretic approach to S-duality (which can be administered to any X ⊂ S, not only to finite polyhedra) we have, in order to detect functional objects, to follow the same lines which led to “virtual spaces” X ∧ Y in [2, §1] (which are only under additional assumptions equivalent to the ∧-product X ∧ Y ) and establish “virtual function spaces” XY (§3). It turns out (§5) that they are sometimes of the same weak homotopy type as the function spaces X . In general they are, like X ∧ Y in [2, §1], new objects, which are defined by introducing new coss-morphisms Z → XY (§2, §3). 1991 Mathematics Subject Classification: Primary 55P25, 55P55; Secondary 55N20, 55M05, 55C40.