We show that the generic symmetric monoidal category with a commutative separable algebra which has a $\Sigma$-family of actions is the category of cospans of finite $\Sigma$-labelled graphs restricted to finite sets as objects, thus providing a syntax for automata on the alphabet $\Sigma$. We use this result to produce semantic functors for $\Sigma$-automata.
Keywords: separable algebra, cospan category
2000 MSC: 18B20, 18D10, 68Q05, 68Q85
Theory and Applications of Categories,
Vol. 15, CT2004,
No. 6, pp 164-177.
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