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

The paper is in essence a survey of categories having $\varphi$-weighted colimits for all the weights $\varphi$ in some class $\mathrm{\Phi }$. We introduce the class ${\mathrm{\Phi }}^{+}$ of {\em $\mathrm{\Phi }$-flat} weights which are those $\psi$ for which $\psi$-colimits commute in the base $\text{V}$ with limits having weights in $\mathrm{\Phi }$; and the class ${\mathrm{\Phi }}^{-}$ of {\em $\mathrm{\Phi }$-atomic} weights, which are those $\psi$ for which $\psi$-limits commute in the base $\text{V}$ with colimits having weights in $\mathrm{\Phi }$. 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 $\text{p}$ of {\em all} weights, the classes ${\text{p}}^{+}$ and ${\text{p}}^{-}$ both coincide with the class $\text{Q}$ of {\em absolute} weights. For any class $\mathrm{\Phi }$ and any category $\text{A}$, we have the free $\mathrm{\Phi }$-cocompletion $\mathrm{\Phi }(\text{A})$ of $\text{A}$; and we recognize $\text{Q}(\text{A})$ as the Cauchy-completion of $\text{A}$. We study the equivalence between ${(\text{Q}({\text{A}}^{op}))}^{op}$ and $\text{Q}(\text{A})$, which we exhibit as the restriction of the Isbell adjunction between ${[\text{A},\text{V}]}^{op}$ and $[{\text{A}}^{op},\text{V}]$ when $\text{A}$ is small; and we give a new Morita theorem for any class $\mathrm{\Phi }$ containing $\text{Q}$. We end with the study of $\mathrm{\Phi }$-continuous weights and their relation to the $\mathrm{\Phi }$-flat weights.