Definition 0.1. Let us recall that as a Quantum Algebraic Topology object, a quantum automaton is defined by the quantum triple Q_A = (𝒢,−ℜ_G,Aut(𝒢)), where 𝒢 is a (locally compact) quantum groupoid, −ℜ_G are the unitary representations of 𝒢 on rigged Hilbert spaces ℜ_G of quantum states and quantum operators on the Hilbert space , and Aut(𝒢) 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 𝒬_A is defined as an algebraic category whose objects are triples (,Δ : →,μ) (where is either a Hilbert space or a rigged Hilbert space of quantum states and operators acting on , and μ is a measure related to the quantum logic, LM, and (quantum) transition probabilities of this quantum system), and whose morphisms are defined between such triples by homomorphisms of Hilbert spaces, Ø : →, naturally compatible with the operators Δ, 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 (Q_A), and quantum automata homomorphisms defined between such triples; these Q_A morphisms are defined by groupoid homomorphisms h : 𝒢 → 𝒢^∗ and α : Aut(𝒢) → Aut(𝒢^∗), together with unitarity preserving mappings u between unitary representations of 𝒢 on rigged Hilbert spaces (or Hilbert space bundles).