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Cartesian bicategories. I. (English) Zbl 0637.18003
From the authors’ introduction: “A locally posetal bicategory is Cartesian if it has a symmetric tensor product, every object is a cocommutative comonoid object, every arrow is a lax comonoid homomorphism and comultiplication and counit have right adjoints. Alternatively, a locally postal bicategory is Cartesian if the subbicategory of arrows with right-adjoints has finite biproducts, each hom-category has finite products and the obvious induced tensor product on arrows is functorial. After describing the first consequences of our definition, we investigate the second main notion of ‘discrete object’ in a Cartesian bicategory. Modulo a ‘functional completeness’ axiom, bicategories of relations are characterized by Cartesianness and discreteness of every object (Sections 2, 3), and these properties together with small bicoproducts, effectiveness and generators characterize bicategories of relations of a Grothendieck topos (Section 6). Bicategories of ordered objects in an exact category can be characterized as follows: they are Cartesian, closed under the Kleisli construction, and the subbicategory of discrete objects is functionally complete and generates (in a suitable sense).”
Reviewer: W.Tholen

MSC:
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
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