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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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