  For finite semidistributive lattices the map $\kappa$ gives a bijection
between the sets of completely join-irreducible elements and completely
meet-irreducible elements.
  Here we study the $\kappa$-map in the context of torsion classes. It is
well-known that the lattice of torsion classes for an artin algebra is
semidistributive, but in general it is far from finite. We show the
$\kappa$-map is well-defined on the set of completely join-irreducible
elements, even when the lattice of torsion classes is infinite. We then extend
$\kappa$ to a map on torsion classes which have canonical join representations
given by the special torsion classes associated to the minimal extending
modules introduced by the first and third authors and A. Carroll.
  For hereditary algebras, we show that the extended $\kappa$-map on torsion
classes is essentially the same as Ringel's $\epsilon$-map on wide
subcategories. Also in hereditary case, we relate the square of $\kappa$ to the
Auslander-Reiten translation.
