nLab 2-category of adjunctions




For any 2-category KK, there are two 2-categories (each with several variants) that could be called “the 2-category of adjunctions in KK”.

  • The 2-category which here we call Adj(K)Adj(K) has the same objects as KK, its morphisms are the adjunctions in KK (pointing in the direction of, say, the left adjoint), and its 2-morphisms are mate-pairs of 2-morphisms between adjunctions in KK.

  • The 2-category [Adj,K][Adj,K] is the functor 2-category from the walking adjunction AdjAdj to KK. Thus its objects are the adjunctions in KK — or more precisely, triples (x,y,(f,g,η,ϵ))(x,y,(f,g,\eta,\epsilon)) where x,yx,y are objects of KK and (f,g,η,ϵ)(f,g,\eta,\epsilon) is an adjunction between xx and yy. Its morphisms are pairs of morphisms xxx\to x' and yyy\to y' such that certain squares commute (perhaps up to a transformation or isomorphism), and its 2-cells are similarly composed of cylinders.

Note that the 1-morphisms of Adj(K)Adj(K) are the objects of [Adj,K][Adj,K].



  • Stephen Schanuel and Ross Street, The free adjunction, Cahiers de topologie et géométrie différentielle catégoriques, tome 27, no 1 (1986), p. 81-83

Last revised on May 5, 2023 at 08:19:40. See the history of this page for a list of all contributions to it.