Marek Antoni Bednarczyk,
Andrzej Borzyszkowski, Wieslaw Pawlowski

Epimorphic functors

864

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 subsequently to introduce the notion of a kernel of a functor, quotient category and quotient functor.

Then extremal epimorphisms in $\cat$ are characterized, up to isomorphism, as quotient functors. It turns out that $\cat$ has an extremal-epi--mono factorization structure.

A class of regular congruences plays a special r\^ole. We show that, essentially, the standard congruences are all regular. Moreover, regular epimorphisms are identified as those extremal epimorphisms which have regular kernels.

The construction of coproducts in $\cat$ is elementary. Here, an elementary construction of coequalizers is described as a step toward characterization of regular epimorphisms. Thus, arbitrary colimits in $\cat$ can now be constructed by elementary means. This provides an elementary proof of the cocompleteness of $\cat$.

Theory of concurrent processes seems to be a natural place to look for applications of the notions and results presented here in the area of computer science. As an example we show how a construction that leads the notion of trace introduced by Mazurkiewicz can be explained within our framework.

Key words: category of (small) categories, generalized congruences, regular and extremal epimorphisms, transition systems, processes.