Notes on enriched categories with colimits of some class
G. M. Kelly, V. Schmitt
备注：
22 pages. This work is on going. We intend to add further sections to this work in the near future. Results presented here are issued mainly from unpublished notes of the first author and contains those in the notes math.CT/0309209 and math.CT/0403164 of the second author. The latter is also the only responsible for the typos, spelling mistakes and other errors that could occur in the present paper!
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 -> [B^op,V] and show how it restricts to an equivalence (P^-(B^op))^op ~ P^-(B) between subcategories of small projectives.
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