## 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

### Keywords:

set endofunctor; set axiom
Full Text:

### References:

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