Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58539120180701On lifting of biadjoints and lax algebras29585074710.29252/cgasa.9.1.29ENFernando Lucatelli NunesCMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal.Journal Article20170430Given a pseudomonad $\mathcal{T} $ on a $2$-category $\mathfrak{B} $, if a right biadjoint $\mathfrak{A}\to\mathfrak{B} $ has a lifting to the pseudoalgebras $\mathfrak{A}\to\mathsf{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 giveĀ general results on lifting of biadjoints. As a consequence, we get a <em>biadjoint triangle theorem</em> 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} \to\mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} _\ell $ the inclusion, if $R: \mathfrak{A}\to\mathfrak{B} $ is right biadjoint and has a lifting $J: \mathfrak{A}\to \mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} $, then $\ell\circ J$ is right biadjoint as well provided that $\mathfrak{A} $ has some needed weighted bicolimits. In order to prove such result, we study <em>descent objects</em> and <em>lax descent objects</em>. At the last section, we study direct consequences of our theorems in the context of the $2$-monadic approach to coherence.https://cgasa.sbu.ac.ir/article_50747_e7751692a69d525e49259ebe2763142f.pdf