Flatness, weakly lex colimits, and free exact completions

Research output: Contribution to journalArticlepeer-review


We capture in the context of lex colimits, introduced by Garner and Lack, the universal property of the free regular and Barr-exact completions of a weakly lex category. This is done by introducing a notion of flatness for functors F:C→E with lex codomain, and using this to describe the universal property of free Φ-exact completions in the absence of finite limits, for any given class Φ of lex weights. In particular, we shall give necessary and sufficient conditions for the existence of free lextensive and free pretopos completions in the non-lex world, and prove that the ultraproducts, in the categories of models of such completions, satisfy an universal property.

Original languageEnglish
Pages (from-to)823-856
Number of pages34
JournalAnnali di Matematica Pura ed Applicata
Issue number2
Early online date5 Oct 2023
Publication statusPublished - 1 Apr 2024


  • 18A35
  • 18B25
  • 18D20
  • 18E08
  • Enriched categories
  • Flatness
  • Free completions
  • Lex colimits
  • Regular/exact categories


Dive into the research topics of 'Flatness, weakly lex colimits, and free exact completions'. Together they form a unique fingerprint.

Cite this