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

Abstract

The paper is in essence a survey of categories having $\phi$-weighted colimits for all the weights $\phi$ in some class $\Phi$. We introduce the class $\Phi^+$ of {\em $\Phi$-flat} weights which are those $\psi$ for which $\psi$-colimits commute in the base $\V$ with limits having weights in $\Phi$; and the class $\Phi^-$ of {\em $\Phi$-atomic} weights, which are those $\psi$ for which $\psi$-limits commute in the base $\V$ with colimits having weights in $\Phi$. 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 $\p$ of {\em all} weights, the classes $\p^+$ and $\p^-$ both coincide with the class $\Q$ of {\em absolute} weights. For any class $\Phi$ and any category $\A$, we have the free $\Phi$-cocompletion $\Phi(\A)$ 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 $\Phi$ containing $\Q$. We end with the study of $\Phi$-continuous weights and their relation to the $\Phi$-flat weights.