Examples
- If C has products, then given an isomorphism the mapping, composed with the canonical map of symmetry, is a partial involution.
- If C is a triangulated category, the Karoubi envelope Split(C) can be endowed with the structure of a triangulated category such that the canonical functor C → Split(C) becomes a triangulated functor.
- The Karoubi envelope is used in the construction of several categories of motives.
- The Karoubi envelope construction takes semi-adjunctions to adjunctions. For this reason the Karoubi envelope is used in the study of models of the untyped lambda calculus. The Karoubi envelope of an extensional lambda model (a monoid, considered as a category) is cartesian closed.
Read more about this topic: Karoubi Envelope
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