# Enriched Orthogonality and Equivalences

## M. Golasinski and L. Stramaccia

In this paper, we consider an enriched orthogonality for classes of spaces, with respect to groupoids, simplicial sets and spaces themselves. This point of view allows one to characterize homotopy equivalences, shape and strong shape equivalences. We show that there exists a class of spaces, properly containing ANR-spaces, for which shape and strong shape equivalences coincide. For such a class of spaces homotopy orthogonality implies enriched orthogonality.

Keywords: ANR-space, enriched orthogonality, groupoid, homotopy structure, prospace, shape equivalence, simplicial set, ${\mathcal V}$-orthogonal object, strongly fibered space

2000 MSC: 55P10, 55P15, 55P5

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