  A Quillen model structure on the category Gray-Cat of Gray-categories is
described, for which the weak equivalences are the triequivalences. It is shown
to restrict to the full subcategory Gray-Gpd of Gray-groupoids. This is used to
provide a functorial and model-theoretic proof of the unpublished theorem of
Joyal and Tierney that Gray-groupoids model homotopy 3-types. The model
structure on Gray-Cat is conjectured to be Quillen equivalent to a model
structure on the category Tricat of tricategories and strict homomorphisms of
tricategories.
