Flat vs. filtered colimits in the enriched context

Stephen Lack, Giacomo Tendas*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)


The importance of accessible categories has been widely recognized; they can be described as those freely generated in some precise sense by a small set of objects and, because of that, satisfy many good properties. More specifically finitely accessible categories can be characterized as: (a) free cocompletions of small categories under filtered colimits, and (b) categories of flat presheaves on some small category. The equivalence between (a) and (b) is what makes the theory so general and fruitful. Notions of enriched accessibility have also been considered in the literature for various bases of enrichment, such as Ab,SSet,Cat and Met. The problem in this context is that the equivalence between (a) and (b) is no longer true in general. The aim of this paper is then to: 1. give sufficient conditions on V so that (a) ⇔ (b) holds; 2. give sufficient conditions on V so that (a) ⇔ (b) holds up to Cauchy completion; 3. explore some examples not covered by (1) or (2).

Original languageEnglish
Article number108381
Pages (from-to)1-50
Number of pages50
JournalAdvances in Mathematics
Issue numberPart A
Publication statusPublished - 6 Aug 2022


  • 2-category
  • Accessible category
  • Enriched category
  • Filtered colimit
  • Flat functor
  • Simplicial category


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