adjunction

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adjunction

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From Latin adjunctio, from adjungere: compare French adjonction, and see adjunct.

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adjunction (countable and uncountable, plural adjunctions)

2. The act of joining; the thing joined or added.
4. (law) The joining of personal property owned by one to that owned by another.
5. (mathematics, chiefly algebra and number theory) The process of adjoining elements to an algebraic structure (usually a ring or field); the result of such a process.

   The ring obtained after the adjunction of the elements ${\displaystyle a,b}$ and ${\displaystyle y}$ to the ring ${\displaystyle R}$ may be denoted ${\displaystyle R}$.

   The field adjunction ${\displaystyle \mathbb {Q} (\pi )}$ can be obtained from ${\displaystyle \mathbb {Q} }$ by adjoining ${\displaystyle \pi }$ to ${\displaystyle \mathbb {Q} }$.

6. (category theory, loosely) A relationship between a pair of categories that makes the pair, in a weak sense, equivalent.

   Hyponyms: equivalence of categories, isomorphism of categories, Galois connection

8. (category theory, strictly) A natural isomorphism between a pair of functors satisfying certain conditions, whose existence implies a close relationship between the functors and between their (co)domains; the natural isomorphism, functors, and their (co)domains thought of as a single object.
   1. (formally, given two categories ${\displaystyle {\mathcal {C}}}$ and ${\displaystyle {\mathcal {D}}}$ and (covariant) functors ${\displaystyle F:{\mathcal {C}}\rightarrow {\mathcal {D}}}$ and ${\displaystyle G:{\mathcal {D}}\rightarrow {\mathcal {C}}}$) A natural isomorphism ${\displaystyle \Phi :\operatorname {Hom} _{\mathcal {C}}(G\cdot ,\cdot )\to \operatorname {Hom} _{\mathcal {D}}(\cdot ,F\cdot )}$ (where the hom-functors are understood as bifunctors from ${\displaystyle {\mathcal {D}}^{\operatorname {op} }\times {\mathcal {C}}}$ to ${\displaystyle \mathbf {Set} }$). See Adjoint functors on Wikipedia.Wikipedia .

      Meronyms: adjoint, left adjoint, right adjoint