The ℓ p -metrization of functors with finite supports

Abstract

AbstractLet p ∈ [1, ∞] 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 ∈ ℕ and a ∈ Fn the map Xn → FX, f → Ff(a), is non-expanding with respect to the ℓp-metric on Xn. 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, dX) 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, dX) → (Y, dY) 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 FpX ≤ 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.Mathematics Subject Classification (2020): 54B3054E3554F45Key words: FunctordistancemonoidHausdorff distancefinite supportdimension