Enriched and internal categories: an extensive relationship

Anthony Power, Thomas Cottrell, Soichiro Fujii

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Pages (from-to)239-254
Number of pages16
JournalTbilisi Mathematical Journal
Issue number3
Publication statusPublished - 1 Jun 2017


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

ASJC Scopus subject areas

  • Mathematics(all)


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