Geodesic
On linear functorial operators extending pseudometrics
Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) no. 2, pp. 343-348 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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For a functor $F\supset Id$ on the category of metrizable compacta, we introduce a conception of a linear functorial operator $T=\{T_X:Pc(X)\to Pc(FX)\}$ extending (for each $X$) pseudometrics from $X$ onto $FX\supset X$ (briefly LFOEP for $F$). The main result states that the functor $SP^n_G$ of $G$-symmetric power admits a LFOEP if and only if the action of $G$ on $\{1,\dots,n\}$ has a one-point orbit. Since both the hyperspace functor $\exp$ and the probability measure functor $P$ contain $SP^2$ as a subfunctor, this implies that both $\exp$ and $P$ do not admit LFOEP.
For a functor $F\supset Id$ on the category of metrizable compacta, we introduce a conception of a linear functorial operator $T=\{T_X:Pc(X)\to Pc(FX)\}$ extending (for each $X$) pseudometrics from $X$ onto $FX\supset X$ (briefly LFOEP for $F$). The main result states that the functor $SP^n_G$ of $G$-symmetric power admits a LFOEP if and only if the action of $G$ on $\{1,\dots,n\}$ has a one-point orbit. Since both the hyperspace functor $\exp$ and the probability measure functor $P$ contain $SP^2$ as a subfunctor, this implies that both $\exp$ and $P$ do not admit LFOEP.
Classification : 46M15, 54B30, 54C20, 54E35
Keywords: linear functorial operator extending (pseudo)metrics; the functor of $G$-symmetric power 
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     author = {Banakh, T. and Pikhurko, O.},
     title = {On linear functorial operators extending pseudometrics},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {343--348},
     year = {1997},
     volume = {38},
     number = {2},
     mrnumber = {1455501},
     zbl = {0886.54010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1997_38_2_a14/}
}
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Banakh, T.; Pikhurko, O. On linear functorial operators extending pseudometrics. Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) no. 2, pp. 343-348. http://geodesic.mathdoc.fr/item/CMUC_1997_38_2_a14/