engelska-finska översättning av adjunction

The act of joining; the thing joined or added
Given a pair of categories $\mathcal{C}$ and $\mathcal{D}$ : an anti-parallel pair of functors $F:\mathcal{C}\rightarrow \mathcal{D}$ and $G:\mathcal{D}\rightarrow \mathcal{C}$ and a natural transformation $\eta:\mbox{id}_C \rightarrow GF$ called “unit” such that for any object $A \in \mathcal{C}$ , for any object $B \in \mathcal{D}$ , and for any morphism $f:A\rightarrow GB$ , there is a unique morphism $g:FA\rightarrow B$ such that $Gg \circ \eta_A = f$ . The pair of functors express a similarity between the pair of categories which is weaker than that of an equivalence of categories

Example of an adjunction: this is from propositional logic. Consider a “logical theory” category whose objects are well-formed propositional formulae and whose arrows are logical entailment. (Entailment is a pre-ordering; and prosets qualify as a type of category.) Consider the contravariant endofunctor $\neg$ (negation) which when applied to a formula $X$ yields its negation $\neg X$ , and which when applied to an entailment $X \vdash Y$ yields its contrapositive $\neg Y \vdash \neg X$ . Then $\neg$ is adjoint to itself, i.e. it is self-adjoint: $\neg \dashv \neg$ , because $\neg \neg \neg X \dashv \vdash \neg X$ . This is true both in classical logic and in intuitionistic logic. However, in classical logic it is also true that $\neg \neg X \dashv \vdash X$ , so that $\neg$ is self-inverse (up to isomorphism), which means that $\neg$ is classically a self-equivalence of the “theory” category; this is stronger than the intuitionistic self-adjunction of the said theory.--

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