Definition 0.1   A classical automaton, or sequential machine, is defined as a quintuple of sets and set-theoretical mappings, . (it can also be defined as a commutative square diagram with the same components as above, in a rather obvious way.)

With the above definition one can now define morphisms between automata and their composition.

Definition 0.2   A homomorphism of automata is a morphism of automata quintuples that preserves commutativity of the set-theoretical mapping compositions of both the transition function and the output function .

With the above two definitions now we have sufficient data to define the category of automata and automaton homomorphisms.

Definition 0.3   The category of automata is a category of automata quintuples and automata homomorphisms that commute with both the transition and the output functions of the automata.

Remarks:

1. Automata homomorphisms can be considered also as automata transformations or as semigroup homomorphisms, when the state space, , of the automaton is defined as a semigroup.
2. Abstract automata have numerous realizations in the real world as : machines, robots, devices, computers, supercomputers, always considered as discrete state space sequential machines.
   
3. Fuzzy or analog devices are not included as standard automata.
   
4. Similarly, variable (transition function) automata are not included, but Universal Turing machines are.

Definition 0.4   An alternative definition of an automaton is also in use: as a five-tuple , where is a non-empty set of symbols such that one can define a configuration of the automaton as a couple of a state and a symbol . Then defines a “next-state relation, or a transition relation" which associates to each configuration a subset of S- the state space of the automaton. With this formal automaton definition, the category of abstract automata can be defined by defining automata homomorphisms in terms of the morphisms between five-tuples representing such abstract automata.

Example: A special case of automaton is when all its transitions are reversible; then its state space is a groupoid. The category of reversible automata is then a 2-category, and also a subcategory of the 2-category of groupoids, or the groupoid category.