adjunction

Noun

1. The act of joining; the thing joined or added.
2. The joining of personal property owned by one to that owned by another.
3. 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 a,b and y to the ring R may be denoted R#91;a,b,y#93;.
   • The field adjunction #92;mathbb#123;Q#125;(#92;pi) can be obtained from #92;mathbb#123;Q#125; by adjoining #92;pi to #92;mathbb#123;Q#125;.
4. 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

5. 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 𝒞 and 𝒟 and (covariant) functors F:𝒞→𝒟 and G:𝒟→𝒞) A natural isomorphism Φ: operatorname Hom_( mathcal )C(G·,·)→ operatorname Hom_( mathcal )D(·,F·) (where the hom-functors are understood as bifunctors from 𝒟^( operatorname )op×𝒞 to mathbf Set). See Adjoint functors on Wikipedia.Wikipedia .

      Meronyms: adjoint left adjoint right adjoint