Contents

This page is about the generalisation of an adjunction to bicategories. For a functor $L$ which is both a left and a right adjoint to a functor $R$, see ambidextrous adjunction.

### Context

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

# Contents

## Idea

A biadjunction is the “maximally weak” kind of 2-adjunction: a higher generalization of the notion of adjunction from category theory to 2-category theory, and specifically to bicategories (or more generally internally in a tricategory. See 2-adjunction for other kinds of 2-adjunction.

## Definition

### Incoherent versions

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 $C(F a,c)\cong A(a,U c)$, which is pseudonatural both in $a$ and in $c$.

By the Yoneda lemma for bicategories, this is equivalent to giving pseudonatural transformations $\eta : Id_A \to U F$ and $\varepsilon : F U \to Id_C$ satisfying the triangle identities up to invertible modifications. This latter definition can be internalized in any (weak) 3-category, such as a Gray-category or a tricategory.

### Coherent versions

Both the “equivalence of hom-categories” definition and the “unit and counit” definition have stronger “coherent” versions: We can ask the equivalences $C(F a,c)\cong A(a,U c)$ to be adjoint equivalences, or for the “triangulator” modifications to satisfy the swallowtail equations. These two conditions are roughly equivalent, and any “incoherent” biadjunction can be improved to a coherent one by altering one of the triangulators; see Gurski, Riehl-Verity, Pstrągowski, Riehl-Shulman, and this proof in Globular.

• Piotr Pstrągowski, On dualizable objects in monoidal bicategories, framed surfaces and the Cobordism Hypothesis, 2014 arxiv