2020/04/04 by Iryna Banakh, T. Banakh, V. Brydun +5
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #Combinatorics #Computer science #Discrete mathematics #Hyperspace #Mathematical Dynamics and Fractals #Mathematics #Metric (unit) #Metric space #Pure mathematics #Space (punctuation) #math.CT #math.GN #math.MG #msc:54B30 #msc:54E35 #msc:54F45
paper · pdf · doi:10.4064/cm8226-11-2020
published as Colloq. Math. 166 (2021) , 251-266 · 79 pages
openalex publication_date 2020/04/04 · openalex created_date 2021/10/25 · arxiv created 2021/12/13 · arxiv updated 2022/02/08 · openalex updated_date 2026/08/05
Let p∈[1,∞] and F:Set\toSet be a functor with finite supports in the category Set of sets. Given a non-empty metric space (X,dX), we introduce the distance dpFX on the functor-space FX as the largest distance such that for every n∈\mathbb N and a∈ Fn the map Xn→ FX, f↦ Ff(a), is non-expanding with respect to the ℓp-metric dpXn on Xn. We prove that the distance dpFX is a pseudometric if and only if the functor F preserves singletons; dpFX 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 Ff:(FX,dpFX)→ (FY,dpFY) 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\ledeg(F)⋅dim X. Using injective envelopes, we introduce a modification \check dpFX of the distance dpFX and prove that the functor \check Fp:Dist\toDist, \check Fp:(X,dX)↦ (FX,\check dpFX), in the category Dist of distance spaces preserves Lipschitz maps and isometries between metric spaces.