Representations for dicategories

Abstract

This thesis is concerned with functions and group homomorphisms. The tool system employed is the dicategory, an algebra of mappings with operation that of composition and in which decomposition into composites of mappings onto, isomorphisms into, identities into, and so on, is possible. The dicategory axioms are abstractions of certain properties common to functions, group and ring homomorphisms, continuous functions between topological spaces, and so on. The problems solved are those of faithful representations of abstract dicategories by particular dicategories. Chapter I reviews the notion of category and defines the dicategory. By addition of one further axiom a representation by classes and functions is obtained. The connections between this representation and two well-known ones, one for groups and one for partially ordered sets, are noted. Chapter II presents axioms for a system which is shown to be representable as a dicategory of abelian semigroup homomorphisms. Chapter III exhibits axioms for an abelian dicategory, and shows that each such dicategory is isomorphic to a dicategory of abelian group homomorphisms. The availability of a second representation and its connection with that of Chapter II are noted. Chapter IV studies homomorphisms of arbitrary groups. After developing a theorem on associative operations in groups, axioms are presented which allow representation for certain dicategories by particular ones consisting of group homomorphisms. The representation is not faithful but a remedy which will achieve faithfulness is indicated.