An introduction to totally cocomplete categories

Abstract

A total category is defined as a locally small category whose Yoneda embedding, Y, has a left adj oint, L. Totality implies cocompleteness (and completeness) . The converse is not true. However, many familiar cocomplete categories are total. In fact , total categories enjoy good closure properties. In the total setting, arguments are more conceptual than for merely cocomplete categories; often expressed in terms of adjointness situations. For example, one may specialize total categories by considering lex total categories, total categories whose L is lex. Such categories are closely related to topoi. Two interesting conj ectures are. introduced. Attempts to characterize set A 0" (for small A) and set , via adj oints left of Yoneda, are made. vi