  In this expository paper, we discuss and compare the notions of braided and
coboundary monoidal categories. Coboundary monoidal categories are analogues of
braided monoidal categories in which the role of the braid group is replaced by
the cactus group. We focus on the categories of representations of quantum
groups and crystals and explain how while the former is a braided monoidal
category, this structure does not pass to the crystal limit. However, the
categories of representations of quantum groups of finite type also possess the
structure of a coboundary category which does behave well in the crystal limit.
We explain this construction and also a recent interpretation of the coboundary
structure using quiver varieties. This geometric viewpoint allows one to show
that the category of crystals is in fact a coboundary monoidal category for
arbitrary symmetrizable Kac-Moody type.

Quantum Algebra;Representation Theory