Embedding of categories

In this paper we generalize the notion of exact functor to an arbitrary category and show that every small category has a full embedding into a category of all set-valued functors on some small category. The notion of exact is such that this result generalizes the author's exact embedding of regular categories and, indeed, Mitchell's embedding of abelian categories. An example is given of the type of diagram-chasing argument that can be given with this embedding. Introduction. In [Barr] we proved that every small regular category has an exact embedding into a set-valued functor category (C, 5) where C is some small category. In this paper we show that the same result can be proved for any small category when the definition of exact functor is slightly extended. This new definition will agree with the previous one when the category is regular. The embedding which results can be used to chase diagrams in completely arbitrary categories, much as Mitchell's theorem does for abelian categories. As an illustration of this, we derive a theorem of Grothendieck on the descent of pullbacks. 1. Universal regular epimorphisms. In any category a map is called a regular epimorphism if it is the coequalizer of some parallel pair of maps. One usually defines /: X—>Y to be a universal regular epimorphism if for every Y'-+ Y, the fibred product Y' x r X exists and the projection Y' xr X-* Y' is always a regular epimorphism. It is gradually becoming clearer that the universal regular epimorphisms are the "good" epimorphisms. In fact a good case could be made that these should be termed the quotient mappings. In a regular category, every regular epimorphism is universally so. In [Verdier] it is shown how every category X has a full embedding into a topos E where E is the category of sheaves for the so-called canonical topology on X. A brief description of E and the embedding R follows. Received by the editors May 12, 1972. AMS (A/OS) subject classifications (1970). Primary 18A25, 18B15.