Notes on enriched categories with colimits of some class (completed version)

Abstract

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 {\em $Φ$-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 {\em $Φ$-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 {\em saturated} (that is, what was called {\em closed} in the terminology of \cite{AK88}). We prove that for the class $\p$ of {\em all} weights, the classes $\p^+$ and $\p^-$ both coincide with the class $\Q$ of {\em 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 ${(\Q(\A^{op}))}^{op}$ 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.