On lifting of biadjoints and lax algebras

Abstract

Given a pseudomonad $mathcal{T} $ on a $2$-category $mathfrak{B} $, if a right biadjoint $mathfrak{A}tomathfrak{B} $ has a lifting to the pseudoalgebras $mathfrak{A}tomathsf{Ps}textrm{-}mathcal{T}textrm{-}mathsf{Alg} $ then this lifting is also right biadjoint provided that $mathfrak{A} $ has codescent objects. In this paper, we give general results on lifting of biadjoints. As a consequence, we get a biadjoint triangle theorem which, in particular, allows us to study triangles involving the $2$-category of lax algebras, proving analogues of the result described above. In the context of lax algebras, denoting by $ell :mathsf{Lax}textrm{-}mathcal{T}textrm{-}mathsf{Alg} tomathsf{Lax}textrm{-}mathcal{T}textrm{-}mathsf{Alg} _ell $ the inclusion, if $R: mathfrak{A}tomathfrak{B} $ is right biadjoint and has a lifting $J: mathfrak{A}to mathsf{Lax}textrm{-}mathcal{T}textrm{-}mathsf{Alg} $, then $ellcirc J$ is right biadjoint as well provided that $mathfrak{A} $ has some needed weighted bicolimits. In order to prove such result, we study descent objects and lax descent objects. At the last section, we study direct consequences of our theorems in the context of the $2$-monadic approach to coherence.