Enriched and internal categories: an extensive relationship

Research output: Contribution to journalArticle

Abstract

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.
LanguageEnglish
Pages239-254
Number of pages16
JournalTbilisi Mathematical Journal
Volume10
Issue number3
DOIs
StatusPublished - 2017

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Internal
Bicategory
Relationships
Cartesian
Iterate
Object

Keywords

  • enrichment
  • internal category
  • extensivity
  • bicategories
  • spans
  • matrices

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Enriched and internal categories: an extensive relationship. / Power, Anthony; Cottrell, Thomas; Fujii, Soichiro.

In: Tbilisi Mathematical Journal, Vol. 10, No. 3, 2017, p. 239-254.

Research output: Contribution to journalArticle

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