论文标题
$ \ ell^p $ - 具有有限支持的函子的函子
The $\ell^p$-metrization of functors with finite supports
论文作者
论文摘要
令$ p \在[1，\ infty] $和$ f：\ mathbf {set} \ to \ mathbf {set} $中是一个有限支持的函数$ \ mathbf {set set} $。给定一个非空的度量空间$（x，d_x）$，我们在函数空间$ fx $上介绍距离$ d^p_ {fx} $，是最大的距离，以至于每$ n \ in \ mathbb n $和$ n $和$ a \ in fn $ in fn $ in fn $ in Map $ x^n \ to fx $ x^n \ to fx $ f x $ f x $ f \ f \ r.ap ff \ ryap ff \ ryap ff（amap ff ff \ ryap ff（a MapSto ff（and Map） $ \ ell^p $ -metric $ d^p_ {x^n} $ on $ x^n $。我们证明，当且仅当函数$ f $保留单例时，距离$ d^p_ {fx} $是一个伪计； $ d^p_ {fx} $如果$ f $保留单例，并且以下条件之一保留：（1）公制空间$（x，d_x）$ is lipschitz newnected，（2）$ p = 1 $，（3）函数$ f $ f $具有有限f $ f $ f $ f $ preserves Supports支持。我们证明，对于任何Lipschitz地图$ f：（x，d_x）\ to（y，d_y）$之间的$ ff：（fx，d^p_ {fx}）\ to（fy，d^p_ {fy}） \ Mathrm {Lip}（f）$。如果函子$ f $是一个有限度（并且保留支持），则$ f $保留均匀连续的功能，粗糙的功能，粗糙的等价，渐近lipschitz函数，准等法（和连续功能）。对于许多维函数，我们证明公式$ \ dim f^px \ le \ mathrm {deg}（f）（f）\ cdot \ dim x $。使用Injextive信封，我们引入了一个修改$ \ check d^p_ {fx} $的距离$ d^p_ {fx} $的$ d^p_ {fx}）$，在类别中$ \ mathbf {dist} $的距离空间保留Lipschitz地图和度量空间之间的异构体。
Let $p\in[1,\infty]$ and $F:\mathbf{Set}\to\mathbf{Set}$ be a functor with finite supports in the category $\mathbf{Set}$ of sets. Given a non-empty metric space $(X,d_X)$, we introduce the distance $d^p_{FX}$ on the functor-space $FX$ as the largest distance such that for every $n\in\mathbb N$ and $a\in Fn$ the map $X^n\to FX$, $f\mapsto Ff(a)$, is non-expanding with respect to the $\ell^p$-metric $d^p_{X^n}$ on $X^n$. We prove that the distance $d^p_{FX}$ is a pseudometric if and only if the functor $F$ preserves singletons; $d^p_{FX}$ 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 $f:(X,d_X)\to (Y,d_Y)$ between metric spaces the map $Ff:(FX,d^p_{FX})\to (FY,d^p_{FY})$ is Lipschitz with Lipschitz constant $\mathrm{Lip}(Ff)\le \mathrm{Lip}(f)$. 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 $\dim F^pX\le\mathrm{deg}(F)\cdot\dim X$. Using injective envelopes, we introduce a modification $\check d^p_{FX}$ of the distance $d^p_{FX}$ and prove that the functor $\check F^p:\mathbf{Dist}\to\mathbf{Dist}$, $\check F^p:(X,d_X)\mapsto (FX,\check d^p_{FX})$, in the category $\mathbf{Dist}$ of distance spaces preserves Lipschitz maps and isometries between metric spaces.
