Understanding the small object argument

Richard Garner*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

55 Citations (Scopus)


The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an "algebraic" refinement of the small object argument, cast in terms of Grandis and Tholen's natural weak factorisation systems, which rectifies each of these three deficiencies.

Original languageEnglish
Pages (from-to)247-285
Number of pages39
JournalApplied Categorical Structures
Issue number3
Publication statusPublished - Jun 2009
Externally publishedYes


  • Small object argument
  • Weak factorisation system


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