Extension theories for categories (preliminary report)

Let C and A be categories and P : C → A a functor which is bijective on objects and surjective on arrows; then C is an extension of A. This notion of extension is studied here from the point of view of classifying and synthesizing extensions, by generalizing methods used in studying extensions of groups and semigroups. (Functors which merge objects don’t behave so much like homomorphisms and probably require intrinsically categorical methods to study them.) More specifically, in this paper I describe how to study certain extensions of small categories by a method which is essentially the Eilenberg-Mac Lane theory of group extensions in a more general setting. In Section 2 I describe how to generalize certain well-known concepts in semigroup theory to small categories. In Section 3 I define the particular type of extension to be considered (extensions by a right semifunctor). Theorem 3.1 constructs such extensions by factor sets, generalizing Schreier’s theory for groups. Theorem 3.2 says that split-extensions correspond to trivial factor sets. Theorems 3.3 and 3.4 show that (under a certain restriction) split extensions are group objects in a certain comma category and extensions are simple transitive actions by such group objects. This generalizes the perception about group extensions of Beck [67]. In Section 4 I show how extensions by right semifunctors arise by a natural process (Theorem 4.1) and make a rather weak connection between categories of posets