Notes on enriched categories with colimits of some class

TL;DR

This paper investigates classes of weights in enriched category theory related to colimits and limits, establishing their saturation properties and exploring enriched adjunctions and equivalences for small projectives.

Contribution

It introduces and analyzes classes of weights Phi^+ and Phi^- with respect to colimits and limits, proving their saturation and establishing an enriched adjunction and equivalence for small projectives.

Findings

Phi^+ and Phi^- are saturated classes of weights.

For the class P of all weights, P^+ = P^-.

An enriched adjunction and an equivalence between subcategories of small projectives are established.

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 -> [B^op,V] and show how it restricts to an equivalence (P^-(B^op))^op ~ P^-(B) between subcategories of small projectives.