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Keywords:
linear functorial operator extending (pseudo)metrics; the functor of $G$-symmetric power
linear functorial operator extending (pseudo)metrics; the functor of $G$-symmetric power
Summary:
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.
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References:
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