Category metrics and valuations of concrete categories

Abstract

We will introduce the notion of a category metric ϱ on some category S , as a system of metrics (ϱ XY; X,Y ∈ 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 ϱ C of C in the sense of Zlatoš (Fuzzy Sets and Systems 82 (1996) 73–96), such that, for any S -morphism f : UA → UB, ϱ 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 Zlatoš (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 field K and K-linear maps.