Enriched and internal categories: an extensive relationship

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We consider an extant infinitary variant of Lawvere's finitary definition of extensivity of a category Ʋ. In the presence of cartesian closedness and finite limits in Ʋ, we give two char- acterisations of the condition in terms of a biequivalence between the bicategory of matrices over Ʋ and the bicategory of spans over discrete objects in Ʋ. Using the condition, we prove that Ʋ-Cat and the category Catd(Ʋ) of internal categories in V with a discrete object of objects are equivalent. Our leading example has Ʋ = Cat, making Ʋ-Cat the category of all small 2-categories and Catd(Ʋ) the category of small double categories with discrete category of objects. We further show that if V is extensive, then so are Ʋ-Cat and Cat(Ʋ), allowing the process to iterate.


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