The l^p -metrization of functors with finite supports

Let p is an element of [1, infinity] and F : Set -> Set be a functor with finite supports in the category Set of sets. Given a non-empty metric space (X, dX), we introduce the distance on the functor-space FX as the largest distance such that for every n is an element of N and a is an element of Fn the map X-n -> FX, f -> Ff(a), is non-expanding with respect to the l(p) -metric on X-n . We prove that the distance is a pseudometric if and only if the functor F preserves singletons; is a metric if F preserves singletons and one of the following conditions holds: (1) the metric space (X, d(X) ) is Lipschitz disconnected, (2) p = 1, (3) the functor F has finite degree, (4) F preserves supports. We prove that for any Lipschitz map f : (X, d(X) ) -> (Y, d(Y) ) between metric spaces the map is Lipschitz with Lipschitz constant Lip(Ff) <= Lip(f). If the functor F is finitary, has finite degree (and preserves supports), then F preserves uniformly continuous function, coarse functions, coarse equivalences, asymptotically Lipschitz functions, quasi-isometries (and continuous functions). For many dimension functions we prove the formula dim (FX)-X-p <= deg(F) dim X. Using injective envelopes, we introduce a modification of the distance and prove that the functor Dist -> Dist, , in the category Dist of distance spaces preserves Lipschitz maps and isometries between metric spaces.