Notes on enriched categories with colimits of some class

By G. M. Kelly, Victoria Schmitt. Theory and Applications of Categories, 2005

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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 Pboth coincide with the class Q of absolute weights. For any class Φ and any category A, we have the free Φ-cocompletion Φ(A )o fA; and we recognize Q(A) as the Cauchy-completion of A. We study the equivalence between (Q(A op )) op and Q(A), which we exhibit as the restriction of the Isbell adjunction between (A, V) op and (A op , 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.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

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