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We give a construction of a commutor (natural isomophisms A x B -> B x A) for the category of crystals of a semisimple Lie algebra. This commutor is symmetric but does not satisfy the usual hexagon axiom. Instead it obeys a different axiom which makes the category of crystals into a coboundary category.

Motivated by the above construction, we investigate the structure of coboundary categories. Just as the braid group acts on repeated tensor products in braided category, the fundamental group of the moduli space of stable real genus 0 curves with n marked points acts on repeated tensor products in a coboundary category.