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Notes on enriched categories with colimits of some class

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G.M. Kelly and V. Schmitt

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.

http://www.tac.mta.ca/tac/volumes/14/17/14-17.dvi

http://www.tac.mta.ca/tac/volumes/14/17/14-17.ps

http://www.tac.mta.ca/tac/volumes/14/17/14-17.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/14/17/14-17.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/14/17/14-17.ps

Revised 2006-04-29. Original version at

http://www.tac.mta.ca/tac/volumes/14/17/14-17a.dvi

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