Karoubi envelope Automorphisms in the Karoubi envelope Examples References Navigation menu10.1016/0304-3975(85)90062-310.1016/s0019-9958(82)90796-3"Idempotent completion of triangulated categories"10.1006/jabr.2000.85290021-8693

Category theory


mathematicscategoryidempotentspreadditive categorypseudo-abelian categoryMax Karoubiendomorphismmorphismsfull subcategorypresheavesrepresentable functorsautomorphism




In mathematics the Karoubi envelope (or Cauchy completion or idempotent completion) of a category C is a classification of the idempotents of C, by means of an auxiliary category. Taking the Karoubi envelope of a preadditive category gives a pseudo-abelian category, hence 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


e:A→Adisplaystyle e:Arightarrow A

with



e∘e=edisplaystyle ecirc e=e.

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 1B = 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 e:A→Adisplaystyle e:Arightarrow A is an idempotent of C, and whose morphisms are the triples


(e,f,e′):(A,e)→(A′,e′)displaystyle (e,f,e^prime ):(A,e)rightarrow (A^prime ,e^prime )

where f:A→A′displaystyle f:Arightarrow A^prime is a morphism of C satisfying e′∘f=f=f∘edisplaystyle e^prime circ f=f=fcirc e (or equivalently f=e′∘f∘edisplaystyle f=e'circ fcirc e).


Composition in Split(C) is as in C, but the identity morphism
on (A,e)displaystyle (A,e) in Split(C) is (e,e,e)displaystyle (e,e,e), rather than
the identity on Adisplaystyle A.


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 C^displaystyle hat mathbf C (the presheaves over C) of retracts of representable functors. 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 (e,f,e):(A,e)→(A,e)displaystyle (e,f,e):(A,e)rightarrow (A,e), with inverse (e,g,e):(A,e)→(A,e)displaystyle (e,g,e):(A,e)rightarrow (A,e) satisfying:


g∘f=e=f∘gdisplaystyle gcirc f=e=fcirc g

g∘f∘g=gdisplaystyle gcirc fcirc g=g

f∘g∘f=fdisplaystyle fcirc gcirc f=f

If the first equation is relaxed to just have g∘f=f∘gdisplaystyle gcirc f=fcirc g, 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 f:A→Bdisplaystyle f:Arightarrow B the mapping f×f−1:A×B→B×Adisplaystyle ftimes f^-1:Atimes Brightarrow Btimes A, composed with the canonical map γ:B×A→A×Bdisplaystyle gamma :Btimes Arightarrow Atimes B 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.[1]

  • The Karoubi envelope is used in the construction of several categories of motives.

  • The Karoubi envelope construction takes semi-adjunctions to adjunctions.[2] 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.[3][4]

  • The category of projective modules over any ring is the Karoubi envelope of its full subcategory of free modules.

  • The category of vector bundles 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 sections functor is an equivalence between trivial vector bundles over Xdisplaystyle X and projective modules over C(X)displaystyle C(X) and then using the universal property of the Karoubi envelope.


References



  1. ^ Balmer & Schlichting 2001


  2. ^ Susumu Hayashi (1985). "Adjunction of Semifunctors: Categorical Structures in Non-extensional Lambda Calculus". Theoretical Computer Science. 41: 95–104. doi:10.1016/0304-3975(85)90062-3..mw-parser-output cite.citationfont-style:inherit.mw-parser-output .citation qquotes:"""""""'""'".mw-parser-output .citation .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-ws-icon abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center.mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-maintdisplay:none;color:#33aa33;margin-left:0.3em.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em


  3. ^ C.P.J. Koymans (1982). "Models of the lambda calculus". Information and Control. 52: 306–332. doi:10.1016/s0019-9958(82)90796-3.


  4. ^ DS Scott (1980). "Relating theories of the lambda calculus". To HB Curry: Essays in Combinatory Logic.



  • Balmer, Paul; Schlichting, Marco (2001), "Idempotent completion of triangulated categories" (PDF), Journal of Algebra, 236 (2): 819–834, doi:10.1006/jabr.2000.8529, ISSN 0021-8693

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