Definition 0.1   Let us recall that as a quantum algebraic topology object, a quantum automaton is defined by the quantum triple $Q_A =(\grp,\H -\Re_G, Aut(\grp)$ ), where $\grp$ is a (locally compact) quantum groupoid, $\H -\Re_G$ are the unitary representations of $\grp$ on rigged Hilbert spaces $\Re_G$ of quantum states and quantum operators on the Hilbert space $\H$ , and $Aut(\grp)$ 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 $\mathcal{\Q}_A$ is defined as an algebraic category whose objects are triples $(\H, \Delta: \H \rightarrow \H, \mu)$ (where $\H$ is either a Hilbert space or a rigged Hilbert space of quantum states and operators acting on $\H$ , and $\mu$ 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, $\O: \H \rightarrow \H$ , naturally compatible with the operators $\Delta$ , and by homomorphisms between the associated Haar measure systems.

Definition 0.3   A quantum algebraic topology definition of the category of quantum algebraic automata is in terms of 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 $h: \grp \rightarrow \grp ^*$ and $\alpha: Aut(\grp) \rightarrow Aut(\grp ^*)$ , together with unitarity preserving mappings $u$ that are defined between unitary representations of $\grp$ on rigged Hilbert spaces (or Hilbert space bundles).