On lifting of biadjoints and lax algebras

By the biadjoint triangle theorem, given 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 $\mathfrak{A}\to \mathsf{P}\mathsf{s}\text{-}\mathcal{T}\text{-}\mathsf{A}\mathsf{l}\mathsf{g}$ 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 $\mathit{\text{biadjoint}}\mathit{\text{triangle}}\mathit{\text{theorem}}$ which, in particular, allows us to study triangles involving the $2$-category of lax algebras, proving analogues of the result described above. More precisely, we prove that, denoting by $\ell :$\mathsf{L}\mathsf{a}\mathsf{x}$$\text{-}$$\mathcal{T}$$\text{-}$$\mathsf{A}\mathsf{l}\mathsf{g}$ \to$\mathsf{L}\mathsf{a}\mathsf{x}$$\text{-}$$\mathcal{T}$$\text{-}$$\mathsf{A}\mathsf{l}\mathsf{g}$_\ell $ the inclusion, if $R:\mathfrak{A}\to \mathfrak{B}$ is right biadjoint and has a lifting $J: \mathfrak{A} \to $\mathsf{L}\mathsf{a}\mathsf{x}$$\text{-}$$\mathcal{T}$$\text{-}$$\mathsf{A}\mathsf{l}\mathsf{g}$ $,then$\ell\circ J$isrightbiadjointaswellprovidedthat$\mathfrak{A} $ has some needed weighted bicolimits. In order to prove such theorem, we study the descent objects and the lax descent objects. At the last section, we study direct consequences of our theorems in the context of the $2$-monadic approach to coherence. In particular, we give the construction of the left $2$-adjoint to the inclusion of the strict algebras into the lax algebras.