Research Article

The ℓ^p -metrization of functors with finite supports

Published in: Quaestiones Mathematicae
Volume 46 , issue S1 : SPECIAL ISSUE DEDICATED TO THE MEMORY OF PROFESSOR HORST HERRLICH , 2023 , pages: 53–144

DOI: 10.2989/16073606.2023.2247240

Author(s): Taras Banakh Ivan Franko National University of Lviv, Ukraine , Viktoria Brydun Ivan Franko National University of Lviv, , Lesia Karchevska , Ukraine , Mykhailo Zarichnyi Ivan Franko National University of Lviv, Ukraine

Keywords: 54B30 , 54E35 , 54F45 , Functor , distance , monoid , Hausdorff distance , finite support , dimension

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Abstract

Let 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 ℓ^p -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, 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 F^pX ≤ 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.

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