How strict is strictification?

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The subject of this paper is the higher structure of the strictification adjunction, which relates the two fundamental bases of three-dimensional category theory: the Gray-category of 2-categories and the tricategory of bicategories. We show that – far from requiring the full weakness provided by the definitions of tricategory theory – this adjunction can be strictly enriched over the symmetric closed multicategory of bicategories defined by Verity. Moreover, we show that this adjunction underlies an adjunction of bicategory-enriched symmetric multicategories. An appendix introduces the symmetric closed multicategory of pseudo double categories, into which Verity's symmetric multicategory of bicategories embeds fully.
Original languageEnglish
Pages (from-to)2948-2976
Number of pages29
JournalJournal of Pure and Applied Algebra
Issue number7
Publication statusPublished - Jul 2019


  • strictification
  • bicategory
  • 2-category
  • adjunction
  • multicategory
  • double category
  • Gray-category
  • bicategory-enriched
  • three-dimensional category theory


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