View source for Adjunction

==English==
{{wikipedia}}

===Etymology===
{{root|en|ine-pro|*yewg-}}
From {{lbor|en|la|adjunctio|notext=1}}, from {{m|la|adjungere}}: compare {{cog|fr|adjonction}}, and see {{m|en|adjunct}}.

===Pronunciation===
* {{audio|en|LL-Q1860 (eng)-Flame, not lame-adjunction.wav|a=US}}

===Noun===
{{en-noun|~}}

# {{senseid|en|joining}} The act of [[join]]ing; the thing joined or added.
# {{senseid|en|legal}} {{lb|en|legal}} The joining of [[personal property]] owned by one to that owned by another.
# {{lb|en|mathematics|chiefly|algebra|and|number theory}} The process of [[adjoin]]ing elements to an [[algebraic structure]] (usually a [[ring]] or [[field]]); the result of such a process. 
#: {{ux|en|The ring obtained after the '''adjunction''' of the elements <math>a,b</math> and <math>y</math> to the ring <math>R</math> may be denoted <math>R[a, b, y]</math>.}}
#: {{ux|en|The field '''adjunction''' <math>\mathbb{Q}(\pi)</math> can be obtained from <math>\mathbb{Q}</math> by adjoining <math>\pi</math> to <math>\mathbb{Q}</math>.}}
# {{lb|en|category theory|loosely}} A relationship between a pair of [[category|categories]] that makes the pair, in a weak sense, equivalent.
#: {{hyponyms|en|equivalence of categories|isomorphism of categories|Galois connection}}
# {{senseid|en|category}} {{lb|en|category theory|strictly}} A [[natural isomorphism]] between a pair of [[functor]]s satisfying certain conditions, whose existence implies a close relationship between the functors and between their ([[codomain|co]])[[domain]]s; the natural isomorphism, functors, and their (co)domains thought of as a single [[object]]. 
## (''formally, given two categories <math>\mathcal{C}</math> and <math>\mathcal{D}</math> and ([[covariant]]) [[functor]]s <math>F:\mathcal{C}\rightarrow \mathcal{D}</math> and <math>G:\mathcal{D}\rightarrow \mathcal{C}</math>'') A [[natural isomorphism]] <math>\Phi : \operatorname{Hom}_\mathcal{C}(G\cdot, \cdot) \to \operatorname{Hom}_\mathcal{D}(\cdot, F\cdot)</math> (''where the hom-functors are understood as [[bifunctor]]s from <math>\mathcal{D}^\operatorname{op} \times \mathcal{C}</math> to <math>\mathbf{Set}</math>''). ''See {{pedia|Adjoint functors}}''.
##: {{meronyms|en|adjoint|left adjoint|right adjoint}}

====Derived terms====
{{col|en|biadjunction}}

====Translations====
{{trans-top|the act of joining}}
* Polish: {{t|pl|adiunkcja|f}}
* Portuguese: {{t|pt|adjunção|f}}
{{trans-bottom}}

{{trans-top|the thing joined or added}}
{{trans-bottom}}

{{trans-top|a form of similarity between a pair of categories mapped to each other by dual morphisms}}
* Polish: {{t|pl|adiunkcja|f}}
{{trans-bottom}}

{{checktrans-top}}
* Bulgarian: {{t+check|bg|съединение|sc=Cyrl}}
{{trans-bottom}}

====References====
<references />
<!--#: &nbsp;&nbsp;&nbsp;&nbsp;Given a pair of categories <math>\mathcal{C}</math> and <math>\mathcal{D}</math> and an anti-parallel pair of functors between them (<math>F:\mathcal{C}\rightarrow\mathcal{D}</math> and <math>G:\mathcal{D}\rightarrow\mathcal{C}</math>) which form an '''adjunction''' then the compositions ''GF'' and ''FG'' are endofunctors which are projections: <math>(FG)^2 \cong FG</math> and <math>(GF)^2 \cong GF</math>. The fact that they are projections can be derived from the triangle identities, which amount to saying that <math>FGF \cong F</math> and <math>GFG \cong G</math>. Then the restrictions of functors ''F'' and ''G'' to <math>GF(\mathcal{C})</math> and <math>FG(\mathcal{D})</math> yields an equivalence of categories.-->
<!--#: &nbsp;&nbsp;&nbsp;&nbsp;If functors <math>F:\mathcal{C} \rightarrow \mathcal{D}</math> and <math>G:\mathcal{D}\rightarrow \mathcal{C}</math> form an '''adjunction''', then there is an isomorphism between the homsets <math>\mbox{Hom}_\mathcal{C}(X, G(Y))</math> and <math>\mbox{Hom}_\mathcal{D}(F(X), Y)</math> which is natural in ''X'' and in ''Y''. This means that the functor <math>\mbox{Hom}_\mathcal{C}(-, G(Y)):\mathcal{C}\rightarrow \mathbf{Set}</math> is naturally isomorphic to the functor <math>\mbox{Hom}_\mathcal{D}(F(-), Y):\mathcal{C}\rightarrow \mathbf{Set}</math> and that the functor <math>\mbox{Hom}_\mathcal{C}(X, G(-)):\mathcal{D}\rightarrow \mathbf{Set}</math> is naturally isomorphic to the functor <math>\mbox{Hom}_\mathcal{D}(F(X), -):\mathcal{D}\rightarrow \mathbf{Set}</math>.-->
<!--#: &nbsp;&nbsp;&nbsp;&nbsp;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 <math>\neg</math> (negation) which when applied to a formula <math>X</math> yields its negation <math>\neg X</math>, and which when applied to an entailment <math>X \vdash Y</math> yields its contrapositive <math>\neg Y \vdash \neg X</math>. Then <math>\neg</math> is adjoint to itself, ''i.e.'' it is self-adjoint: <math>\neg \dashv \neg</math>, because <math>\neg \neg \neg X \dashv \vdash \neg X</math>. This is true both in classical logic and in intuitionistic logic. However, in classical logic it is also true that <math>\neg \neg X \dashv \vdash X</math>, so that <math>\neg</math> is self-inverse (up to isomorphism), which means that <math>\neg</math> is classically a self-equivalence of the “theory” category; this is stronger than the intuitionistic self-'''adjunction''' of the said theory.-->