Coherence, Homotopy and 2-Theories

2-Theories are a canonical way of describing categories with extra structure. 2-theory-morphisms are used when discussing how one structure can be replaced with another structure. This is central to categorical coherence theory. We place a Quillen model category structure on the category of 2-theories and 2-theory-morphisms where the weak equivalences are biequivalences of 2-theories. A biequivalence of 2-theories induces a biequivalence of 2-categories of algebras. This model category structure allows one to talk of the homotopy of 2-theories and discuss the universal properties of coherence.