adjunction

adjunction

English

Etymology

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

Noun

adjunction (countable and uncountable, plural adjunctions)

1. The act of joining; the thing joined or added.
2. (law) The joining of personal property owned by one to that owned by another.
3. (category theory) Given a pair of categories ${\\displaystyle {\\mathcal {C}}}$ and ${\\displaystyle {\\mathcal {D}}}$: an anti-parallel pair of functors ${\\displaystyle F:{\\mathcal {C}}\\rightarrow {\\mathcal {D}}}$ and ${\\displaystyle G:{\\mathcal {D}}\\rightarrow {\\mathcal {C}}}$ and a natural transformation ${\\displaystyle \\eta$ :{\\mbox{id}}_{C}\\rightarrow GF} called “unit” such that for any object ${\\displaystyle A\\in {\\mathcal {C}}}$, for any object ${\\displaystyle B\\in {\\mathcal {D}}}$, and for any morphism ${\\displaystyle f:A\\rightarrow GB}$, there is a unique morphism ${\\displaystyle g:FA\\rightarrow B}$ such that ${\\displaystyle Gg\\circ \\eta _{A}=f}$.[1] (Note: there is another natural transformation called “counit” as well but its existence may be derived by theorem.) The pair of functors express a similarity between the pair of categories which is weaker than that of an equivalence of categories.

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

   Meronyms: adjoint, left adjoint, right adjoint

Translations

the act of joining

Polish: adiunkcja f Portuguese: adjunção f

the thing joined or added

a form of similarity between a pair of categories mapped to each other by dual morphisms

Polish: adiunkcja f

The translations below need to be checked and inserted above into the appropriate translation tables. See instructions at Wiktionary:Entry layout § Translations.

Translations to be checked

Bulgarian: (please verify) съединение (bg) (sǎedinenie)

References

1. Michael Barr; Charles Wells (1995) Category Theory for Computing Science, second edition, University Press, Cambridge, Great Britain: Prentice Hall, →ISBN, §9.2, page 258