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Change of base, Cauchy completeness and reversibility

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Anna Labella and Vincent Schmitt

We investigate the effect on Cauchy complete objects of the change of base
2-functor ${\cal V}-Cat \rightarrow {\cal W}-Cat$ induced by a two-sided
enrichment
${\cal V} \rightarrow {\cal W}$. We restrict our study to the case of
locally partially ordered bases. The reversibility notion introduced
by Walters is extended to two-sided enrichments and Cauchy completion.
We show that a reversible left adjoint two-sided enrichment $F: {\cal V}
\rightarrow {\cal W}$ between locally partially ordered reversible
bicategories induces an adjunction $F_{\sim} \dashv F^{\sim}: \VSkCRcCat
\rightharpoonup \WSkCRcCat$ between sub-categories of skeletal and
Cauchy-reversible complete enrichments. We give two applications: sheaves
over locales and group actions.

Keywords: Enriched categories, two-sided enrichments, change of base, reversibility,
Cauchy completion, sheaves.

2000 MSC: 18D20,18D99.

*Theory and Applications of Categories*, Vol. 10, 2002, No. 10, pp 187-219.

http://www.tac.mta.ca/tac/volumes/10/10/10-10.dvi

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