Sketches Let us transfer these ideas from groups to categories, starting with generators and laws. We write S for the set of sorts. A free unary theory has no laws. This basic tool has many different names, and it would be instructive to read through the remainder of this section several times, substituting each of the following points of view in turn. In particular, if the types are propositions , the operation-symbols are deduction steps and the terms are proofs.
Terms are paths.
The types are called states and the operation-symbols are actions. It is deterministic because the action labelled r has a unique target. Now terms are acceptable words or behaviour traces , and may be described by a regular grammar . The types are objects, the operation-symbols are arrows or generating morphisms, and the laws are called commutative polygons. Each raw term is given by an oriented path, but the faces generate an equivalence relation, as follows.