Abstract. A particularly powerful representation of the fundamental fermionic state emerges when we take the gamma algebra of the Dirac equation as a 64-part double Clifford algebra, which has many isomorphic representations, including double vector, vector quaternion, complexified double quaternion, and even left- and right-multiplied octonions. A special significance for physics in general follows when we take the algebra as the tensor product of the real, complex, quaternion and vector (or complexified quaternion) algebras, for these can be taken to be the algebras which determine the physical characteristics of the fundamental parameters mass, time, charge and space, and their exact symmetry and zero totality as a Klein-4 group. They can also be seen as the opening finite section of an infinite series generated by a zero-totality universal rewrite system acting according to the kind of principles involved in computer logic. Both the Klein-4 parameter group and the universal rewrite system have natural representations in terms of category theory. These representations show clearly the connection between them and lead to some more general reflections about fundamental aspects of category theory itself.