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.

[1] Jean Bénabou, Introduction to bicategories, Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1-77. MR 0220789

[2] Renato Betti, Aurelio Carboni, Ross Street, and Robert Walters, Variation through enrichment, J. Pure Appl. Algebra 29 (1983), no. 2, 109-127. MR 707614

[3] Renato Betti and A. John Power, On local adjointness of distributive bicategories, Boll. Un. Mat. Ital. B (7) 2 (1988), no. 4, 931-947. MR MR977597 (89k:18014)

[4] Francis Borceux and George Janelidze, Galois theories, Cambridge Studies in Advanced Mathematics, vol. 72, Cambridge University Press, Cambridge, 2001.

[5] A. Carboni, G. M. Kelly, D. Verity, and R. J. Wood, A 2-categorical approach to change of base and geometric morphisms. II, Theory Appl. Categ. 4 (1998), No. 5, 82-136. MR 1633360

[6] Aurelio Carboni, Stephen Lack, and R. F. C.Walters, Introduction to extensive and distributive categories, J. Pure Appl. Algebra 84 (1993), no. 2, 145-158. MR 1201048

[7] Claudia Centazzo and Enrico M. Vitale, Sheaf theory, Categorical foundations, Encyclopedia Math. Appl., vol. 97, Cambridge University Press, Cambridge, 2004, pp. 311-357.

[8] Tobias Heindel and Pawe lSobociński, Van Kampen colimits as bicolimits in span, Algebra and coalgebra in computer science, Lecture Notes in Comput. Sci., vol. 5728, Springer, Berlin, 2009, pp. 335-349. MR 2557839

[9] Peter T. Johnstone, Sketches of an elephant: a topos theory compendium. Vol. 1, Oxford Logic Guides, vol. 43, The Clarendon Press, Oxford University Press, New York, 2002. MR 1953060

[10] G. M. Kelly, Basic concepts of enriched category theory, Repr. Theory Appl. Categ. (2005), no. 10, vi+137, Reprint of the 1982 original [Cambridge Univ. Press, Cambridge; MR0651714]. MR 2177301

[11] F. William Lawvere, Some thoughts on the future of category theory, Category theory (Como, 1990), Lecture Notes in Math., vol. 1488, Springer, Berlin, 1991, pp. 1-13. MR 1173000

[12] , Categories of space and of quantity, The space of mathematics (San Sebastiàn, 1990), Found. Comm. Cogn., de Gruyter, Berlin, 1992, pp. 14-30. MR 1214608

[13] Saunders Mac Lane, Categories for the working mathematician, second ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872

[14] Michael Shulman, Framed bicategories and monoidal fibrations, Theory Appl. Categ. 20 (2008), No. 18, 650-738. MR 2534210

[15] Dominic Verity, Enriched categories, internal categories and change of base, Repr. Theory Appl. Categ. (2011), no. 20, 1-266. MR 2844536