Enriched and internal categories: an extensive relationshipTh. Cottrell, S. Fujii, J. PowerWe consider an extant infinitary variant of Lawvere's finitary definition of extensivity of a category V. In the presence of cartesian closedness and finite limits in V, we give two characterisations of the condition in terms of a biequivalence between the bicategory of matrices over V and the bicategory of spans over discrete objects in V. Using the condition, we prove that V-Cat and the category Catd(V) of internal categories in V with a discrete object of objects are equivalent. Our leading example has V=Cat, making V-Cat the category of all small 2-categories and Catd(V) the category of small double categories with discrete category of objects. We further show that if V is extensive, then so are V-Cat and Cat(V), allowing the process to iterate.Tbilisi Mathematical Journal, Vol. 10(3) (2017), pp. 239-254 |