On the Category of Petri Net Computations In M emory and D edication to my Beloved mother L iana

We introduce the notion of strongly concatenable process as a refinement of concatenable processes [3] which can be expressed axiomatically via a functor Q[ ] from the category of Petri nets to an appropriate category of symmetric strict monoidal categories, in the precise sense that, for each net N , the strongly concatenable processes of N are isomorphic to the arrows of Q[N ]. In addition, we identify a coreflection right adjoint to Q[ ] and characterize its replete image, thus yielding an axiomatization of the category of net computations. Introduction Petri nets, introduced by C.A. Petri [8] (see also [10]), are unanimously considered among the most representative models for concurrency, since they are a fairly simple and natural model of concurrent and distributed computations. However, Petri nets are, in our opinion, not yet completely understood. Among the semantics proposed for Petri nets, a relevant role is played by the various notions of process [9, 4, 1], whose merit is to provide a faithful account of computations involving many different transitions and of the causal connections between the events occurring in a computation. However, process models, at least in their standard forms, fail to bring to the foreground the algebraic structure of nets and their computations. Since such a structure is relevant to the understanding of nets, they fail, in our view, to give a comprehensive account of net behaviours. The idea of looking at nets as algebraic structures [10, 7, 13, 14, 2] has been given an original interpretation by considering monoidal categories as a suitable framework [6]. In fact, in [6, 3] the authors have shown that the semantics of Petri nets can be understood in terms of symmetric monoidal categories— where objects are states, arrows processes, and the tensor product and the arrow composition model, respectively, the operations of parallel and sequential composition of processes. In particular, [3] introduced concatenable processes—the slightest variation of Goltz-Reisig processes [4] on which sequential composition can be defined—and structured the concatenable processes of a Petri net N as the arrows of the symmetric strict monoidal category P [N ]. This yields an axiomatization of the causal behaviour of a net as an essentially algebraic theory and thus provides a unification of the process and the algebraic view of net computations. However, also this construction is somehow unsatisfactory, since it is not functorial. More strongly, given a morphism between two nets, i.e., a simulation between them, it may not be possible to identify a corresponding monoidal functor between the respective categories of computations. This fact, besides * Basic Research in Computer Science, Centre of the Danish National Research Foundation. ⋄ Supported by EU Human Capital and Mobility grant ERBCHBGCT920005.