Definition 0.1   Let us recall that as a quantum algebraic topology object, a quantum automaton is defined by the quantum triple ), where is a (locally compact) quantum groupoid, are the unitary representations of on rigged Hilbert spaces of quantum states and quantum operators on the Hilbert space , and is the transformation, or automorphism groupoid of quantum transitions that represents all flip-flop quantum transitions of one cubit each between the permitted quantum states of the quantum automaton.

Definition 0.2   The category of quantum automata is defined as an algebraic category whose objects are triples (where is either a Hilbert space or a rigged Hilbert space of quantum states and operators acting on , and is a measure related to the quantum logic, , and (quantum) transition probabilities of this quantum system), and whose morphisms are defined between such triples by homomorphisms of Hilbert spaces, , naturally compatible with the operators , and by homomorphisms between the associated Haar measure systems.

Definition 0.3   A quantum algebraic topology definition of the category of quantum algebraic automata involves the objects specified above in Definition 0.1 as quantum automaton triples , and quantum automata homomorphisms defined between such triples; these morphisms are defined by groupoid homomorphisms and , together with unitarity preserving mappings between unitary representations of on rigged Hilbert spaces (or Hilbert space bundles).