Change of base, Cauchy completeness and reversibility

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.

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