nLab biadjunction


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



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.


Incoherent versions

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

By the Yoneda lemma for bicategories, this is equivalent to giving pseudonatural transformations η:Id AUF\eta : Id_A \to U F and ε:FUId C\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(Fa,c)A(a,Uc)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

Last revised on July 26, 2021 at 13:18:14. See the history of this page for a list of all contributions to it.