In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
the Karoubi envelope (or Cauchy completion or idempotent completion) of a
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
C is a classification of the
idempotent
Idempotence (, ) is the property of certain operation (mathematics), operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence ...
s of C, by means of an auxiliary category. Taking the Karoubi envelope of a
preadditive category
In mathematics, specifically in category theory, a preadditive category is
another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab.
That is, an Ab-category C is a category such that
every hom ...
gives a
pseudo-abelian category In mathematics, specifically in category theory, a pseudo-abelian category is a category (mathematics), category that is preadditive category, preadditive and is such that every idempotent has a kernel (category theory), kernel. Recall that an idemp ...
, hence the construction is sometimes called the pseudo-abelian completion. It is named for the French mathematician
Max Karoubi
__NOTOC__
Max Karoubi () is a French mathematician, topologist, who works on K-theory, cyclic homology and noncommutative geometry and who founded the first European Congress of Mathematics.
In 1967, he received his Ph.D. in mathematics (Doct ...
.
Given a category C, an idempotent of C is an
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
:
with
:
.
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
''B'' = ''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
is an idempotent of C, and whose
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s are the triples
:
where
is a morphism of C satisfying
(or equivalently
).
Composition in Split(C) is as in C, but the identity morphism
on
in Split(C) is
, rather than
the identity on
.
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
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...
of
(the
presheaves over C) of retracts of
representable functor In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and ...
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 mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
in Split(C) is of the form
, with inverse
satisfying:
:
:
:
If the first equation is relaxed to just have
, 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
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
the mapping
, composed with the canonical map
of symmetry, is a partial
involution
Involution may refer to:
* Involute, a construction in the differential geometry of curves
* '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
.
* If C is a
triangulated category In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy categ ...
, 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
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation t ...
. The Karoubi envelope of an extensional lambda model (a monoid, considered as a category) is cartesian closed.
[
]
* The category of
projective module
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizati ...
s over any ring is the Karoubi envelope of its full subcategory of free modules.
* The category of
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
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
In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout ...
and conversely this theorem can be proved by first proving both these facts, the observation that the
global section
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
s functor is an equivalence between trivial vector bundles over
and free modules over
and then using the universal property of the Karoubi envelope.
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
''Journal of Algebra'' (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier. ''Journal of Algebra'' was founded by Graham Higman, who was its editor from 1964 to 1 ...
, issn=0021-8693 , volume=236 , issue=2 , pages=819–834 , doi=10.1006/jabr.2000.8529, doi-access=free
Category theory