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Categorical structures enriched in a quantaloid: regular presheaves, regular semicategories. (English) Zbl 1086.18005
The theory of regular modules on an \(R\)-algebra without unit for a commutative ring \(R\) was generalized to a theory of regular presheaves on a \(\mathcal V\)-enriched semicategory for a symmetric monoidal closed base category \(\mathcal V\) by E. J. Dubuc and J. G. Zilber [Cah. Topologie Géom. Différ. Catég. 35, 49–73 (1994; Zbl 0790.32009)]. In this paper the author considers presheaves on a regular \(\mathcal Q\)-semicategory \(\mathbb A\). It is shown that regular presheaves on a regular \(\mathcal Q\)-semicategory \(\mathbb A\) form a \(\mathcal Q\)-category \({\mathcal R}\mathbb A\) that is an essential (co)localisation of the category \({\mathcal P}\mathbb A\) of all presheaves on \(\mathbb A\), in which the image of the ultimate right adjoint is the \(\mathcal Q\)-category \({\mathcal Y}\mathbb A\) of Yoneda presheaves on \(\mathbb A\). He straightforwardly generalizes \(\mathcal Q\)-category theory to obtain an aspect of Morita equivalence for semicategories.

MSC:
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18D20 Enriched categories (over closed or monoidal categories)
06F07 Quantales
18B35 Preorders, orders, domains and lattices (viewed as categories)
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