nLab adjoint

In linear algebra

The term adjoint originates in linear algebra, where it may refer, in increasing generality, to:

• adjoint matrix

• Hermitian adjoint

• adjoint operator

In category theory

In category theory, the term “adjoint” foremost refers to “adjoint functors”, a terminology which is probably inspired by the similarity between the characteristic hom-isomorphism of such functors and the defining equality $\big\langle A v ,\, w \big\rangle \,=\, \big\langle v ,\, A^\dagger w \big\rangle$ for Hermitian adjoint linear operators.

Proceeding from here, there are various variants of adjointness in category theory:

• adjunction, adjoint morphism

• adjoint functor

  • strong adjoint functor

  • adjoint triple, adjoint quadruple, recollement

• proadjoint, Hopf adjunction

• 2-adjunction

  biadjunction, lax 2-adjunction, pseudoadjunction

• adjoint (infinity,1)-functor

• (∞,n)-category with adjoints