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NounModificar

Singular
Adjunction

Plural
Adjunctions

Adjunction (plural Adjunctions)
  1. Patrono:Category theory A form of similarity between a pair of categories \mathcal{C} and \mathcal{D} which is weaker than equivalence, which in turn is weaker than isomorphism. Given functors  F:\mathcal{C} \rightarrow \mathcal{D} and  G:\mathcal{D} \rightarrow \mathcal{C} , F is "left adjoint" of G, and G "right adjoint" of F, denoted as  F \dashv G , if
    1. there are a pair of natural transformations  \eta: \mbox{id}_\mathcal{C} \rightarrow GF and  \epsilon: FG \rightarrow \mbox{id}_\mathcal{D} satisfying the following "triangle identities":
      1.  F \eta : F \rightarrow FGF composed with  \epsilon G: FGF \rightarrow F commutes with  \mbox{id}_F: F \rightarrow F and
      2.  \eta G: G \rightarrow GFG composed with  F \epsilon: GFG \rightarrow G commutes with  \mbox{id}_G: G \rightarrow G .[1]
    2. there is a natural isomorphism  \alpha: \mathcal{D}(FX,Y) \cong \mathcal{C}(X,GY) , which is natural in the sense of being "natural in X and Y", where
      1. "natural in X" means that for every  f:X\rightarrow X', \alpha:\mathcal{D}(FX',Y) \rightarrow \mathcal{C}(X',GY) composed with \mathcal{C}(f,GY):\mathcal{C}(X',GY) \rightarrow \mathcal{C}(X,GY) commutes with \mathcal{D}(Ff,Y):\mathcal{D}(FX',Y) \rightarrow \mathcal{D}(FX,Y) composed with  \alpha:\mathcal{D}(FX,Y)\rightarrow \mathcal{C}(X,GY);
      2. "natural in Y" means that for every  f:Y \rightarrow Y',  \alpha:\mathcal{D}(FX,Y) \rightarrow \mathcal{C}(X,GY) composed with \mathcal{C}(X,Gf):\mathcal{C}(X,GY) \rightarrow \mathcal{C}(X,GY') commutes with  \mathcal{D}(FX,f):\mathcal{D}(FX,Y) \rightarrow \mathcal{D}(FX,Y') composed with  \alpha:\mathcal{D}(FX,Y') \rightarrow \mathcal{C}(X,GY').[2]

ReferencesModificar

  1. TheCatsters – Adjunctions 1
  2. TheCatsters – Adjunctions 2

TranslationsModificar


pl:adjunction ru:adjunction tr:adjunction vi:adjunction

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