Idempotents and splittings

An idempotent in a category is an endomorphism that squares to itself. Idempotents are “generalized projections.”

Any section-retraction pair gives rises to a split idempotent . In general, not all idempotents can be split but a category can always be freely completed to one in which idempotents split, variously called its idempotent completion, Cauchy completion , or Karoubi envelope . These concepts generalize to enriched categories, including to Lawvere metric spaces where they recover the original notion of Cauchy completeness.

Literature

Books and surveys

  • Borceux & Dejean, 1986: Cauchy completion in category theory (pdf)
  • Borceux, 1994: Handbook of categorical algebra, Vol. 1, Ch. 6: Flat functors and Cauchy completeness, especially:
    • Sec 6.5: The splitting of idempotents
    • Sec 7.9: Cauchy completeness versus distributors
  • Reyes, Reyes, Zolfaghari, 2004: Generic figures and their glueings, Ch. 5: Generic figures
    • Sec 5.1 characterizes “continuous” presheaves as retracts of representables
    • Sec 5.2 shows that the category of continuous presheaves on a category \(C\) is equivalent to the Cauchy completion of \(C\)