Comparing (co)localizations across Quillen adjunctions

The talk is based on joint work with Oriol Raventós and Andy
Tonks. We show that several apparently unrelated formulas involving
localizations or cellularizations in homotopy theory arise
from comparison maps associated with pairs of adjoint functors.
Such comparison maps are used to study liftings of homotopical
(co)localizations to categories of (co)algebras over (co)monads in
suitable model categories. We discuss, with different methods, the
case of strict algebras and the case of homotopy algebras. Warnings
are issued about situations where symmetry unexpectedly fails to
hold.