Date of Award

5-1973

Degree Type

Thesis

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Supervisor

Professor B. Banaschewski

Abstract

This thesis represents an attempt to marry two distinct directions of research in Modern Algebra. On the one hand we have the theory of topoi which has been undergoing vigorous development within the last three years. This area grew out of careful consideration of the tools of Modern Algebraic Geometry as promoted by the French School under the leadership of Alexandre Grothendieck. Lawvere and Tierney have shown how the theory of topoi may be axiomatized conveniently and have established that much of Mathematics may be carried out in the environment of a topos, in which scheme the topos of definition replaces the category of sets. The second direction referred to is what is called Universal Algebra. By this we mean internal Universal Algebra, that is to say the study of equations and their solutions in mathematical structures. In the present work we undertake the study of the behaviour of universal algebras modelled in a topos.

In fact the topoi we choose to study are those arising as categories of sheaves of sets on a suitable parametrizing object (Grothendieck site). In this framework we introduce the notion of equation and solution, now for sheaves of algebras. We establish as major results that a compact Hausdorff sheaf of algebras is equationally (algebraically) compact and that for sheaves of modules, homological purity is equivalent to a form of equational purity. On the path towards the proof of these results we establish new facts about certain topoi, and some new facts regarding "external" universal algebra, that is, the purely categorical aspect of universal algebra. For example we characterize the points of the category of double negation sheaves on a topological space, we give a useful characterization of those continuous maps whose inverse image functor on the associated sheaf categories is cotripleable, we establish a Birkhoff-type subdirect representation theorem for sheaves of algebras and we exhibit sufficient conditions for injectivity to be well-behaved in the category of sheaves of universal algebras associated with a specified theory.

What we have carried out here represents only a beginning in terms of the possibilities inherent in studying universal algebras modelled in topoi. The old theorems of universal algebra take on a more dynamic and geometrical flavour in their new environment, which we hope will lead to their application in Combinatorics and Algebraic Geometry.

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