ω-Ind Completion of Monoidal Categories and Infinite Petri Net Computations.

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Description/Abstract

There exists a KZ-doctrine on the 2-category of the locally small categories whose algebras are exactly the categories which admits all the colimits indexed by $\omega$-chains. The paper presents a wide survey of this topic. In addition, we show that this chain cocompletion KZ-doctrine lifts smoothly to KZ-doctrines on (many variations of) the 2-categories of monoidal and symmetric monoidal categories, thus yielding a universal construction of colimits of $\omega$-chains in those categories. Since the processes of Petri nets may be axiomatized in terms of symmetric monoidal categories this result provides a universal construction of the algebra of infinite processes of a Petri net.