Notes on enriched categories with colimits of some class
G. M. Kelly, V. Schmitt
Abstract
Given a class Phi of weights, we study the following classes: Phi+ of Phi-flat weights which are the psi for which psi-colimits commute in the base V with limits with weights in Phi; and Phi-, dually defined, of weights psi for which psi-limits commute in the base V with colimits with weights in Phi. We show that both these classes are saturated (i.e. closed under the terminology of Albert-Kelly or Betti's coverings). We prove that for the class P of all weights P+ = P-. For any small B, we defined an enriched adjunction a` la Isbell [B,V]op -> [Bop,V] and show how it restricts to an equivalence (P-(Bop))op ~ P-(B) between subcategories of small projectives.
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