ω-Inductive Completion of Monoidal Categories and Infinite Petri Net Computations

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 ω-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 ω-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. Introduction The idea of completing a mathematical stucture by adding to it some desirable limit ‘points’ is indeed a very natural one and it arises in many different fields of mathematics, particularly in topology and partial order theory. Since categories are a generalization of the notion of partial orders, the issue of completing categories for a given class of limits or colimits arose rather early in the development of the theory (see [21] and references therein). As far as computer science is concerned, the theory of complete partial orders and the associated completion techniques have assumed great relevance since the pioneering work on semantics by Scott [33]. In the last few years, however, many computing systems have been given a semantics through the medium of category theory, the general pattern being to look at objects as representing states and at arrows as representing computations. It is therefore natural to expect that the theory of cocompletion of categories may play an interesting role in this kind of semantics. The main purpose of this paper is to illustrate how this theory fits well with the issue of infinite computations and, therefore, to make it more easily available to the computer science community. In a sense, by viewing categories as generalized posets, this view of infinite computations is very natural and indeed generalizes to categories similar constructions for adding limits to posets [28]. We motivate this further in terms of processes of Petri nets in Section 1. * Supported by EU Human Capital and Mobility grant ERBCHBGCT920005. ** Supported by Office of Naval Research Contract N00014-92-C-0518, National Science Foundation Grant CCR-9224005, and by the Information Technology Promotion Agency, Japan, as a part of the R&D of Basic Technology for Future Industries ‘New Models for Software Architecture’ sponsored by NEDO (New Energy and Industrial Technology Development Organization). *** Partially supported by the EU SCIENCE Programme, Project MASK, and by the Italian National Research Council (CNR), Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo, obiettivo Lambrusco.