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An axiomatics for bicategories of modules. (English) Zbl 0615.18006
The reviewer [Rend. Semin. Mat. Fis. Milano 51, 217-233 (1983; Zbl 0538.18005)] called a bicategory W a base when it had a small set of objects, each hom-category had small colimits and composition on either side with an arrow preserved small colimits. He gave a characterization of the bicategory W-Mod of small categories with homs enriched in W and modules between them. Bases should be compared with the sites of topos theory in the light of R. F. C. Walters [J. Pure Appl. Algebra 24, 95-102 (1982; Zbl 0497.18016)].
The present paper considers W-Mod freed from the restriction that W have a small set of objects. Then one can consider (W-Mod)-Mod which is shown to be biequivalent to W-Mod. This result compares with the fact that the category of sheaves for the canonical topology on a topos is equivalent to the original topos. Their result bears the same relationship to the reviewer’s characterization as the result of the last sentence does to Giraud’s topos-characterization theorem.
Reviewer: R.H.Street

MSC:
18D20 Enriched categories (over closed or monoidal categories)
16B50 Category-theoretic methods and results in associative algebras (except as in 16D90)
18D30 Fibered categories
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
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References:
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