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

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

2005-09-05

arxiv.org/abs/math/0509102v1

Mathematics

Category Theory

This is a completion of CT/0501383. Results presented here are mainly from unpublished notes of the first author and contains

Scientific paper

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.

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