Category metrics and valuations of concrete categories

We will introduce the notion of a category metric rho on some category S, as a system of metrics (rho (XY); X,Y is an element of Ob S) on the sets of morphisms S(X, Y), compatible, in some precise sense, with the composition of S-morphisms. Given a category metric on a category S and a locally closed concrete category (C, U) over S, one can construct a fairly natural valuation rho (C) of C in the sense of Zlatos (Fuzzy Sets and Systems 82 (1996) 73-96), such that, for any S-morphism f:UA --> UB, rho (C)(f) is the distance of f from the set C(A, B), embedded into S(UA, UB) via the faithful forgetful functor U: C --> S. It turns out that most of the valuations constructed in Zlatos (Fuzzy Sets and Systems 82 (1996) 73-96) arise in this way from the same category metric on the category Set(fin) of all finite sets and mappings, or from an analogous metric on the category Vect(fin)(K) of all finite-dimensional vector spaces over some held K and K-linear maps. (C) 2001 Elsevier Science B.V. All rights reserved.