Abstract

Abstract The menagerie of epimorphisms in the category Cat of small categories is studied. The standard notion of a congruence on a category is generalized and used to characterize the classes of regular epimorphisms and extremal epimorphisms. It is shown that the hierarchy of epimorphisms is fine, with the classes of isomorphisms, retractions, regular, extremal and general epimorphisms being properly included, one in another. Finally, functors in Cat admit extremal epi-mono factorization. 1 Introduction Co-completeness of Cat, the category of small categories, is a part of folklore. Yet, to the best of our knowledge, most of the textbooks on category theory, e.g., cf. [2, 3, 4], do not discuss the subject at all. The only place where we could,nd a hint on the direction was [1], where the reader was asked to conduct some `abstract nonsense ' type of reasoning to establish the co-compleness as an excersise. The construction of coproducts in Cat is elementary. So, a concrete presentation of coequalizers would do the job. But an elementary characterization of co-equalizers, or, more generally, regular epimorphisms, was missing so far.

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