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Zbl 0606.18002
Koubek, Václav
Large systems of independent objects in concrete categories. I, II. (English)
[J] Czech. Math. J. 34(109), 506-527, 528-540 (1984). ISSN 0011-4642; ISSN 1572-9141/e

A classical problem by Ulam solved by {\it Z. Hedrlin} and {\it A. Pultr} [Commentat. Math. Univ. Carol. 7, 357-400 (1966; Zbl 0143.029)] asks whether there exists $2\sp{\aleph\sb 0}$ 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. \par 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.
[W.Deuber]

MSC 2000:: *18B05 Category of sets
08A35 Endomorphisms on general algebraic systems
18A20 Special classes of morphisms

Keywords: systems of independent objects in categories; concrete categories; set functors

Citations: Zbl 0143.029

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