adjunction

English

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adjunction

Wikipedia

Etymology

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

Noun

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.
6. (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 ${\displaystyle Gg\circ \eta _{A}=f}$. (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

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