On lifting of biadjoints and lax algebras

TL;DR

This paper extends the biadjoint triangle theorem to lax algebras, providing conditions under which liftings of biadjoints are preserved, and explores applications in 2-monadic coherence theory.

Contribution

It generalizes the biadjoint triangle theorem to lax algebra contexts and establishes conditions for liftings to be right biadjoint, including the construction of left 2-adjoints for strict and lax algebras.

Findings

Established a biadjoint triangle theorem for lax algebras.

Proved that certain liftings preserve biadjointness under weighted bicolimits.

Constructed the left 2-adjoint to the inclusion of strict into lax algebras.

Abstract

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{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 \textit{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. More precisely, we prove that, 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 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.