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Endofunctors of Set and cardinalities. (English) Zbl 1042.18001

We say that a set functor \(F:\mathcal {S}et\to\mathcal {S}et\) is a DVO-functor if \(F\) is naturally equivalent to any set functor \(G:\mathcal {S}et \to\mathcal {S}et\) such that for any set \(X\), cardinalities of \(FX\) and \(GX\) are the same. For a set functor \(F\), let us denote \(W_F\) the class of all cardinals \(\gamma\) such that the cardinality of \(FX\) is \(\gamma\) for any set \(X\) with cardinality \(\gamma\). The paper contains a characterization of non-faithful DVO-functors and the proof that if \(W\) is a class of cardinals such that any set functors \(F\) and \(G\) are naturally equivalent whenever \(W=W_F=W_G\), then \(W\) is the class of all cardinals (and \(F\) is the identity functor). Under special set axioms it is shown: a characterization of classes \(W\) of cardinals such that \(W=W_F\) for some set functor \(F\) and that any DVO-functor is finitary.

MSC:

18B05 Categories of sets, characterizations
03E99 Set theory
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References:

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