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Large systems of independent objects in concrete categories. I, II. (English) Zbl 0606.18002

A classical problem by Ulam solved by Z. Hedrlin and A. Pultr [Commentat. Math. Univ. Carol. 7, 357-400 (1966; Zbl 0143.029)] asks whether there exists \(2^{\aleph_ 0}\) countable graphs such that there is no morphism from one to another. This leads to the question whether in a category there is a subset of objects with the same cardinality as the category such that there is no morphism from one to another.
Here such questions are studied for an important class of concrete categories S(F) determined by set functors F. The objects of S(F) are the pairs (X,R) where X is a set and \(R\subseteq FX\), and the morphisms from (X,R) to (Y,S) are the mappings f fulfilling Ff(R)\(\subseteq S\), (Ff(S)\(\subseteq R)\) if F is covariant (contravariant). Many day-life categories fit into this framework. In the first paper the case of covariant set functors F, in the second paper the case of contravariant set functors F is investigated.
Reviewer: W.Deuber

MSC:

18B05 Categories of sets, characterizations
08A35 Automorphisms and endomorphisms of algebraic structures
18A20 Epimorphisms, monomorphisms, special classes of morphisms, null morphisms

Citations:

Zbl 0143.029
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References:

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