nLab biadjunction

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.

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.

Related concepts

• adjoint functor, adjoint triple, adjoint quadruple

• proadjoint, Hopf adjunction

• 2-adjunction

  biadjunction, lax 2-adjunction, pseudoadjunction

• pseudoadjunction

• biequivalence, biadjoint biequivalence?

• 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

• Nick Gurski, Biequivalences in tricategories, Theory and applications of categories 2012, journal web site, arxiv

• Emily Riehl and Dominic Verity, Homotopy coherent adjunctions and the formal theory of monads, 2015, arxiv

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

• Emily Riehl and Mike Shulman, A type theory for synthetic ∞-categories, 2017, arxiv