Editing Karoubi envelope

{{Short description|Category theory}}
In [[mathematics]] the '''Karoubi envelope''' (or '''Cauchy completion''' or '''idempotent completion''') of a [[category (mathematics)|category]] '''C''' is a classification of the [[idempotent]]s of '''C''', by means of an auxiliary category. Taking the Karoubi envelope of a [[preadditive category]] gives a [[pseudo-abelian category]], hence for additive categories, the construction is sometimes called the pseudo-abelian completion. It is named for the French mathematician [[Max Karoubi]].

Given a category '''C''', an idempotent of '''C''' is an [[endomorphism]] 

:<math>e: A \rightarrow A</math>

with 

:<math>e\circ e = e</math>.

An idempotent ''e'': ''A'' → ''A'' is said to '''split''' if there is an object ''B'' and morphisms ''f'': ''A'' → ''B'',
''g'' : ''B'' → ''A'' such that ''e'' = ''g'' ''f'' and 1<sub>''B''</sub> = ''f'' ''g''.

The '''Karoubi envelope''' of '''C''', sometimes written '''Split(C)''', is the category whose objects are pairs of the form (''A'', ''e'') where ''A'' is an object of '''C''' and <math>e : A \rightarrow A</math> is an idempotent of '''C''', and whose [[morphism]]s are the triples

: <math>(e, f, e^{\prime}): (A, e) \rightarrow (A^{\prime}, e^{\prime})</math>

where <math>f: A  \rightarrow A^{\prime}</math> is a morphism of '''C''' satisfying <math>e^{\prime} \circ f = f = f \circ e</math> (or equivalently <math>f=e'\circ f\circ e</math>).

Composition in '''Split(C)''' is as in '''C''', but the identity morphism 
on <math>(A,e)</math> in '''Split(C)''' is <math>(e,e,e)</math>, rather than
the identity on <math>A</math>.

The category '''C''' embeds fully and faithfully in '''Split(C)'''. In '''Split(C)''' every idempotent splits, and '''Split(C)''' is the universal category with this property.
The Karoubi envelope of a category '''C''' can therefore be considered as the "completion" of '''C''' which splits idempotents.

The Karoubi envelope of a category '''C''' can equivalently be defined as the [[full subcategory]] of <math>\hat{\mathbf{C}}</math> (the [[presheaf (category theory)|presheaves]] over '''C''') of retracts of [[representable functor]]s. The category of presheaves on '''C''' is equivalent to the category of presheaves on '''Split(C)'''.

== Automorphisms in the Karoubi envelope ==

An [[automorphism]] in '''Split(C)''' is of the form <math>(e, f, e): (A, e) \rightarrow (A, e)</math>, with inverse <math>(e, g, e): (A, e) \rightarrow (A, e)</math> satisfying:

: <math>g \circ f = e = f \circ g</math>
: <math>g \circ f \circ g = g</math>
: <math>f \circ g \circ f = f</math>

If the first equation is relaxed to just have <math>g \circ f = f \circ g</math>, then ''f'' is a partial automorphism (with inverse ''g''). A (partial) involution in '''Split(C)''' is a self-inverse (partial) automorphism.

==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.

==References==
<references/>

* {{Citation | last1=Balmer | first1=Paul | last2=Schlichting | first2=Marco | title=Idempotent completion of triangulated categories | url=https://www.math.ucla.edu/~balmer/research/Pubfile/IdempCompl.pdf | year=2001 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=236 | issue=2 | pages=819–834 | doi=10.1006/jabr.2000.8529| doi-access=free }}

[[Category:Category theory]]