An examination of some concepts in the theory of topoi

Abstract

This thesis is about the definition and some of the properties of topoi-cartesian-closed categories with a subobject classifier. Starting with the concept of a cartesian-closed category I develop some of its properties and examine a Heyting Algebra as an important example. Then I define a topos and show that for any small category A the category of contravariant functors from A to Set, the category of sets, is a topos. Following this there is an examination of various concepts, and their properties, available in a topos. These include partial maps, relations, the factorization of a map into an epi map followed by a monic map and Heyting Algebra objects. These sections are basically an exposition, of the first half of P. Freyd's paper "Aspects of Topoi" (for which I have supplied the detail which is often absent or only indicated), together with additional material some of which is drawn from Kock and Wraith's "Elementary Toposes". The final section, based on R. Paré's paper "Colimits in Topoi", shows that the existence of finite colimits is presupposed by the rest of the definition of a topos and thus does not have to be included in the definition, and, introduces an alternative definition of a topos.