Enriched Orthogonality and Equivalences

A topological strong shape equivalence is a map inducing an isomorphism in the strong shape category sSh(Top,ANR). It turns out that $f : X \to Y$ is such a map if it gives, by composition, an equivalence $f_Z : Gpd(Y,Z) \to Gpd(X,Z)$ between fundamental groupoids for all $Z \in ANR$. In other words, one may say that strong shape equivalences form the orthogonal class of $ANR \subset Top$. Here, the notion of orthogonality is intended in an enriched sense, over the category Gpd of groupoids. Since the category Top of compactly generated spaces, besides Gpd, can be enriched as well over itself and over the category Sets of simplicial sets, Section 1 starts from the study of some interesting relations between different notions of enriched orthogonality. Every class of spaces has the same orthogonal with respect to Top and Sets (= simplicial sets), which is properly contained in its orthogonal with respect to Gpd. In Theorem 1.4, we obtain characterizations of homotopy equivalences, strong shape equivalences and shape equivalences. In particular, we are led to a general denition of a class of strong shape equivalences for any pair of categories (C,K), enriched over a monoidal closed model category V. Enriched orthogonality over a monoidal model category always implies orthogonality in the homotopy category, in the usual sense, but the converse is false in general. In Section 2, we deal with a class of spaces containing properly ANR-spaces, called strongly bered. For such a class of spaces, shape and strong shape equivalences coincide, so that, in such a case homotopy orthogonality implies enriched orthogonality.