nLab
biadjunction

Idea

The notion of 2-adjunction of biadjunction is the higher generalization of the notion of adjunction from category theory to 2-category theory.

Definition

Given (possibly weak) 2-categories, $A$ and $C$, and (possibly weak) 2-functors $F:A\to C$ and $U:C\to A$, a biadjunction is given by specifying for each object $a$ in $A$ and each object $c$ in $C$ an equivalence of categories $C(Fa,c)\cong A(a,Uc)$, which is pseudonatural both in $a$ and in $c$.

There are several other characterizations of biadjointness.

Properties

If there is a biadjunction in this sense, it can be replaced by a biadjunction for which this equivalence of categories is an adjoint equivalence.

Related concepts

• adjoint functor, adjoint triple, adjoint quadruple

• proadjoint, Hopf adjunction

• 2-adjunction

  biadjunction, lax 2-adjunction, pseudoadjunction

• adjoint (infinity,1)-functor

References

• John Gray?, Formal category theory: Adjointness for 2-categories, Lecture Notes in Mathematics 391, Springer, Berlin, 1974.

• Thomas M. Fiore, Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory, Memoirs of the American Mathematical Society 182 (2006), no. 860. 171 pages, MR2007f:18006, math.CT/0408298