Source: https://researchportal.bath.ac.uk/en/publications/enriched-and-internal-categories-an-extensive-relationship
Timestamp: 2019-04-21 23:16:41+00:00

Document:
We consider an extant infinitary variant of Lawvere’s 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.
Enriched and internal categories: an extensive relationship. / Power, Anthony; Cottrell, Thomas; Fujii, Soichiro.
In: Tbilisi Mathematical Journal, Vol. 10, No. 3, 01.06.2017, p. 239-254.
Power, Anthony ; Cottrell, Thomas ; Fujii, Soichiro. / Enriched and internal categories: an extensive relationship. In: Tbilisi Mathematical Journal. 2017 ; Vol. 10, No. 3. pp. 239-254.
N2 - We consider an extant infinitary variant of Lawvere’s 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.
AB - We consider an extant infinitary variant of Lawvere’s 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.

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