Concerning universal categories

Abstract

The present pper contains a definition of universal cate gory and some results concerning this notion.In the 1 we give the definition and two simple criteria which enables us to show that some frequently discussed categories are not universal.At the end of the paragraph we prove that the category of commutative semi-groups and their homomorphisms is not universal, which result is not an immediate consequence of the criteria* The category 7t of sets with binary relations and their compatible mapping is shown to be universal in the 2 A proof crf the fact was sketched in [2] using the constructions introduced in [1] Here it is don in a more detailed way* The main results of the pper are given in the 3 .We examine some categories similar to Ti, and stat which of them are universal.Every transformation cf t X -> X may be considered as a particular ase of a relation on X , nmely {{cf{x) t x) I |x e Xf Another particular kind of relations form multivalued mappings AJU t X-*X f i.e. such relations, that for every xe X there is at least one y with y (ti i , Roughly speaking, the de finition of fa contains two kinds of relations: nmely, relati ons in objects and morphisms, which are, in fact, a particular Mnd f compatible generalized relations.To sug^st it, we writc ^ * ^(r, f) .We may replace r by f , or r and the