Editing Karoubi envelope (section)

==Examples==

* If '''C''' has products, then given an [[isomorphism]] <math>f: A \rightarrow B</math> the mapping <math>f \times f^{-1}: A \times B \rightarrow B \times A</math>, composed with the canonical map <math>\gamma:B \times A \rightarrow A \times B</math> of symmetry, is a partial [[Involution (mathematics)|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]].<ref>{{Harvard citations| last1=Balmer | last2=Schlichting | year=2001 | nb=yes}}</ref>
*The Karoubi envelope is used in the construction of several categories of [[motive (algebraic geometry)|motives]].
*The Karoubi envelope construction takes semi-adjunctions to [[adjoint functors|adjunction]]s.<ref>{{cite journal
  | author = Susumu Hayashi  | title = Adjunction of Semifunctors: Categorical Structures in Non-extensional Lambda Calculus
  | journal = Theoretical Computer Science
  | volume = 41
  | pages = 95–104
  | year = 1985
  | doi=10.1016/0304-3975(85)90062-3| doi-access = 
  }}</ref> 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.<ref>{{cite journal | author = C.P.J. Koymans  | title = Models of the lambda calculus
  | journal = Information and Control
  | volume = 52
  | pages = 306–332
  | year = 1982
 | doi=10.1016/s0019-9958(82)90796-3| doi-access = free
  }}</ref><ref>{{cite conference | author= DS Scott
  | author-link = Dana Scott
  | title = Relating theories of the lambda calculus
  | book-title = To HB Curry: Essays in Combinatory Logic
  | year = 1980 }}
</ref>
* The category of [[projective module]]s over any ring is the Karoubi envelope of its full subcategory of free modules.
* The category of [[vector bundle]]s over any [[paracompact space]] is the Karoubi envelope of its full subcategory of trivial bundles. This is in fact a special case of the previous example by the [[Serre–Swan theorem]] and conversely this theorem can be proved by first proving both these facts, the observation that the [[global section]]s functor is an equivalence between trivial vector bundles over <math>X</math> and free modules over <math>C(X)</math> and then using the [[universal property]] of the Karoubi envelope.