Notes on Enriched Categories with Colimits of Some Class

The paper is in essence a survey of categories having φ-weighted colimits for all the weights φ in some class Φ. We introduce the class Φ of Φ-flat weights which are those ψ for which ψ-colimits commute in the base V with limits having weights in Φ; and the class Φ− of Φ-atomic weights, which are those ψ for which ψ-limits commute in the base V with colimits having weights in Φ. We show that both these classes are saturated (that is, what was called closed in the terminology of [AK88]). We prove that for the class P of all weights, the classes P+ and P− both coincide with the class Q of absolute weights. For any class Φ and any category A, we have the free Φ-cocompletion Φ(A) of A; and we recognize Q(A) as the Cauchy-completion of A. We study the equivalence between (Q(Aop)) and Q(A), which we exhibit as the restriction of the Isbell adjunction between [A,V] and [Aop,V] when A is small; and we give a new Morita theorem for any class Φ containing Q. We end with the study of Φ-continuous weights and their relation to the Φ-flat weights.