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 $\Phi$-flat weights which are those $\psi$ for which $\psi$-colimits commute in the base $\cal V$ with limits having weights in $\Phi$; and the class $\Phi^-$ of $\Phi$-atomic weights, which are those $\psi$ for which $\psi$-limits commute in the base $\cal V$ with colimits having weights in $\Phi$. We show that both these classes are saturated (that is, what was called closed in the terminology of Albert and Kelly). We prove that for the class $\cal P$ of all weights, the classes $\cal P^+$ and $\cal P^-$ both coincide with the class $\Q$ of absolute weights. For any class $\Phi$ and any category $\cal A$, we have the free $\Phi$-cocompletion $\Phi(\cal A)$ of $\cal A$; and we recognize $\cal Q(\cal A)$ as the Cauchy-completion of $\cal A$. We study the equivalence between ${(\cal Q(\cal A^{op}))}^{op}$ and $\cal Q(\cal A)$, which we exhibit as the restriction of the Isbell adjunction between ${[\cal A,\cal V]}^{op}$ and $[\cal A^{op},\cal V]$ when $\cal A$ is small; and we give a new Morita theorem for any class $\Phi$ containing $\cal Q$. We end with the study of $\Phi$-continuous weights and their relation to the $\Phi$-flat weights.
Keywords: limits, colimits, flat, atomic, small presentable, Cauchy completion
2000 MSC: 18A35, 18C35, 18D20
Theory and Applications of Categories,
Vol. 14, 2005,
No. 17, pp 399-423.
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Revised 2006-04-29. Original version at
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