TY - JOUR
ID - 50747
TI - On lifting of biadjoints and lax algebras
JO - Categories and General Algebraic Structures with Applications
JA - CGASA
LA - en
SN - 2345-5853
AU - Lucatelli Nunes, Fernando
AD - CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal.
Y1 - 2018
PY - 2018
VL - 9
IS - 1
SP - 29
EP - 58
KW - Lax algebras
KW - pseudomonads
KW - biadjunctions
KW - adjoint triangles
KW - lax descent objects
KW - descent categories
KW - weighted bi(co)limits
DO - 10.29252/cgasa.9.1.29
N2 - 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 giveĀ 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} \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 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.
UR - https://cgasa.sbu.ac.ir/article_50747.html
L1 - https://cgasa.sbu.ac.ir/article_50747_e7751692a69d525e49259ebe2763142f.pdf
ER -