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-cells are mate-pairs of 2-cells 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 morphisms of Adj(K)Adj(K) are the objects of [Adj,K][Adj,K].


  • The morphisms in Adj(Adj(K))Adj(Adj(K)) are the adjoint triples in KK.

  • The inclusion of MndMnd, the free monad, in AdjAdj induces a 2-functor from [Adj,K][Adj,K] to [Mnd,K][Mnd,K], the 2-category of monads in KK. The adjoints to this 2-functor are the Kleisli and Eilenberg-Moore constructions on monads in KK.


  • 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 July 7, 2021 at 18:46:44. See the history of this page for a list of all contributions to it.