A categorical axiomatic system is an axiomatic system that has a model and for which any two models are isomorphic. The names of the objects may differ, but the isomorphism must preserve the relations and operations specified by the axioms.
An example of an axiomatic system which isn't categorical is a geometry described by the following four axioms (Smart):
1. There exist five points.
2. Each line is a subset of those five points.
3. There exist two lines.
4. Each line contains at least two points.
This axiomatic system is non-categorical because the axioms admit two models that are not isomorphic, for example,
1. Two disjoint lines each containing two points plus a separate point not on either line.
2. Two lines containing three points each which intersect in one of the points.
The presence of an intersection in one model and not the other implies that the models are fundamentally different and hence are inequivalent.
