 Cauchy-complete category

ID: cauchy-complete-category

Codex  0 Created 2026-10-05 Updated 2026-10-07

A category is Cauchy-complete when every idempotent morphism splits: for

e : A \to A

with

e^{2} = e

, there are

r : A \to B

and

i : B \to A

with

i r = e

and

r i = 1_{B}

. Its Karoubi envelope freely supplies these splittings. For a small category, this condition makes every retract of a representable functor representable.

 New to topics? Read the docs here!