Many categories of interest in mathematics are not cocomplete at all (for example, the homotopy category and the category of fields). A category A is cocomplete when every functor \(f: K\to A\), with K small, has a colimit. The categories in nature which are cocomplete generally satisfy a stronger cocompleteness property: they are total. This means that every functor \(f: K\to A\), whose associated discrete fibration has small fibres, has a colimit.
Equivalently, A is total when its Yoneda embedding \(y: A\to PA\) into the category PA of presheaves on A has a left adjoint z. This concept goes back to R. Walters and the reviewer [J. Algebra 50, 350-379 (1978; Zbl 0401.18004)] in a setting which includes not only categories, but ordered sets, enriched categories, finitely complete categories, and so on. Later papers by various authors have mainly examined totality for ordinary categories and finitely complete categories (the latter because of its connection with Grothendieck toposes).
The present paper looks in detail at totality for categories enriched over a good base monoidal category V. It brings together the known results at this level, and, lifts and extends results previously known only for ordinary categories (especially concerning compactness and hypercompleteness). Furthermore, the relationship between totality of an enriched category and totality of its underlying ordinary category is examined.