Editing Automata theory (section)

==Category-theoretic models==
One can define several distinct [[category (mathematics)|categories]] of automata<ref>Jirí Adámek and [[Věra Trnková]]. 1990. ''Automata and Algebras in Categories''. Kluwer Academic Publishers:Dordrecht and Prague</ref> following the automata classification into different types described in the previous section. The mathematical category of deterministic automata, [[sequential machine]]s or ''sequential automata'', and Turing machines with ''automata homomorphisms'' defining the arrows between automata is a [[Cartesian closed category]],<ref>{{cite book|first = Saunders|last = Mac Lane|author-link = Saunders Mac Lane|title = Categories for the Working Mathematician|publisher = Springer|location= New York |date =1971|isbn = 978-0-387-90036-0}}</ref> it has both categorical [[limit (category theory)|limit]]s and [[colimit]]s. An automata homomorphism maps a quintuple of an automaton ''A''<sub>''i''</sub> onto the quintuple of another automaton 
'' A''<sub>''j''</sub>. Automata homomorphisms can also be considered as ''automata transformations'' or as [[semigroup]] homomorphisms, when the state space, '''''S''''', of the automaton is defined as a semigroup '''S'''<sub>g</sub>. [[Monoid]]s are also considered as a suitable setting for automata in [[monoidal category|monoidal categories]].<ref>http://www.math.cornell.edu/~worthing/asl2010.pdf James Worthington.2010.Determinizing, Forgetting, and Automata in Monoidal Categories. ASL North American Annual Meeting, 17 March 2010</ref><ref>Aguiar, M. and Mahajan, S.2010. ''"Monoidal Functors, Species, and Hopf Algebras"''.</ref><ref>Meseguer, J., Montanari, U.: 1990 Petri nets are monoids. ''Information and Computation'' '''88''':105–155</ref>

;Categories of variable automata
One could also define a ''variable automaton'', in the sense of [[Norbert Wiener]] in his book on ''[[The Human Use of Human Beings]]'' ''via'' the endomorphisms <math>A_{i}\to A_{i}</math>. Then one can show that such variable automata homomorphisms form a mathematical group. In the case of non-deterministic, or other complex kinds of automata, the latter set of endomorphisms may become, however, a ''variable automaton [[groupoid]]''. Therefore, in the most general case, categories of variable automata of any kind are [[categories of groupoids]] or [[groupoid category|groupoid categories]]. Moreover, the category of reversible automata is then a 
[[2-category]], and also a subcategory of the 2-category of groupoids, or the groupoid category.