词条 | Karoubi envelope |
释义 |
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 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 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 is an idempotent of C, and whose morphisms 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 of (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 envelopeAn automorphism 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
References1. ^{{Harvard citations| last1=Balmer | last2=Schlichting | year=2001 | nb=yes}} 2. ^{{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}} 3. ^{{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}} 4. ^{{cite conference | author= DS Scott | authorlink = Dana Scott | title = Relating theories of the lambda calculus | booktitle = To HB Curry: Essays in Combinatory Logic | year = 1980 }}
1 : Category theory |
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