  Henriques and Kamnitzer have defined a commutor for the category of crystals
of a finite-dimensional complex reductive Lie algebra that gives it the
structure of a coboundary category (somewhat analogous to a braided monoidal
category). Kamnitzer and Tingley then gave an alternative definition of the
crystal commutor, using Kashiwara's involution on Verma crystals, that
generalizes to the setting of symmetrizable Kac-Moody algebras. In the current
paper, we give a geometric interpretation of the crystal commutor using quiver
varieties. Equipped with this interpretation we show that the commutor endows
the category of crystals of a symmetrizable Kac-Moody algebra with the
structure of a coboundary category, answering in the affirmative a question of
Kamnitzer and Tingley.
