Reference. Network Models from Petri Nets with Catalysts

John C. Baez, John Foley, Joe Moeller · Compositionality · 2019 · petri-nets · DOI · arXiv
Petri networks and network models are two frameworks for the compositional design of systems of interacting entities. Here we show how to combine them using the concept of a ‘catalyst’: an entity that is neither destroyed nor created by any process it engages in. In a Petri net, a place is a catalyst if its in-degree equals its out-degree for every transition. We show how a Petri net with a chosen set of catalysts gives a network model. This network model maps any list of catalysts from the chosen set to the category whose morphisms are all the processes enabled by this list of catalysts. Applying the Grothendieck construction, we obtain a category fibered over the category whose objects are lists of catalysts. This category has as morphisms all processes enabled by some list of catalysts. While this category has a symmetric monoidal structure that describes doing processes in parallel, its fibers also have premonoidal structures that describe doing one process and then another while reusing the catalysts.

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Cite as @baez-2019-network (helia, typst) · \cite{baez-2019-network} (LaTeX)
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@article{baez-2019-network, title={Network Models from Petri Nets with Catalysts}, volume={1}, ISSN={2631-4444}, url={http://dx.doi.org/10.32408/compositionality-1-4}, DOI={10.32408/compositionality-1-4}, journal={Compositionality}, publisher={Centre pour la Communication Scientifique Directe (CCSD)}, author={Baez, John C. and Foley, John and Moeller, Joe}, year={2019}, month=Dec, pages={4} }
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baez-2019-network:
  type: article
  title: Network Models from Petri Nets with Catalysts
  author:
  - Baez, John C.
  - Foley, John
  - Moeller, Joe
  date: 2019-12
  page-range: '4'
  url: http://dx.doi.org/10.32408/compositionality-1-4
  serial-number:
    doi: 10.32408/compositionality-1-4
    issn: 2631-4444
  parent:
    type: periodical
    title: Compositionality
    publisher: Centre pour la Communication Scientifique Directe (CCSD)
    volume: 1
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Cited by (3)

Colored Petri Nets are Monoidal Double Functors master-2025-coloredarXiv · 2025
We give a characterization of colored Petri nets as monoidal double functors. Framing colored Petri nets in terms of category theory allows for canonical definitions of various well-known constructions on colored Petri nets. In particular, we show how morphisms of colored Petri nets may be understood as natural transformations. The displayed category construction explains how lax double functors are equivalent to functors with codomain their former domain. We use this result to characterize the unfolding of colored Petri nets in terms of free symmetric monoidal categories.
arXiv

The Grothendieck Construction in Categorical Network Theory moeller-2021-thearXiv · 2021
In this thesis, we present a flexible framework for specifying and constructing operads which are suited to reasoning about network construction. The data used to present these operads is called a network model, a monoidal variant of Joyal’s combinatorial species. The construction of the operad required that we develop a monoidal lift of the Grothendieck construction. We then demonstrate how concepts like priority and dependency can be represented in this framework. For the former, we generalize Green’s graph products of groups to the context of universal algebra. For the latter, we examine the emergence of monoidal fibrations from the presence of catalysts in Petri nets.
arXiv

Network Models baez-2017-networkarXiv · 2017
Networks can be combined in various ways, such as overlaying one on top of another or setting two side by side. We introduce “network models” to encode these ways of combining networks. Different network models describe different kinds of networks. We show that each network model gives rise to an operad, whose operations are ways of assembling a network of the given kind from smaller parts. Such operads, and their algebras, can serve as tools for designing networks. Technically, a network model is a lax symmetric monoidal functor from the free symmetric monoidal category on some set to 𝐂𝐚𝐭, and the construction of the corresponding operad proceeds via a symmetric monoidal version of the Grothendieck construction.
arXiv
Cites 25 works (3 here)
With notes (3)

Monoidal Grothendieck construction moeller_vasilakopoulou_2020TAC · 2020
We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely lax monoidal pseudofunctors to the 2-category of categories. Furthermore, we investigate the relation between this ‘global’ monoidal version where the total category is monoidal and the fibration strictly preserves the structure, and a ‘fibrewise’ one where the fibres are monoidal and the reindexing functors strongly preserve the structure, first hinted by Shulman. In particular, when the domain is cocartesian monoidal, we show how lax monoidal structures on a pseudofunctor to Cat bijectively correspond to lifts of the pseudofunctor to MonCat. Finally, we give some examples where this correspondence appears, spanning from the fundamental and family fibrations to network models and systems.
Web · arXiv

Noncommutative network models moeller-2019-noncommutativeMSCS · 2019
Network models, which abstractly are given by lax symmetric monoidal functors, are used to construct operads for modeling and designing complex networks. Many common types of networks can be modeled with simple graphs with edges weighted by a monoid. A feature of the ordinary construction of network models is that it imposes commutativity relations between all edge components. Because of this, it cannot be used to model networks with bounded degree. In this paper, we construct the free network model on a given monoid, which can model networks with bounded degree. To do this, we generalize Green’s graph products of groups to pointed categories which are finitely complete and cocomplete.
DOI · arXiv

Network Models baez-2017-networkarXiv · 2017
Networks can be combined in various ways, such as overlaying one on top of another or setting two side by side. We introduce “network models” to encode these ways of combining networks. Different network models describe different kinds of networks. We show that each network model gives rise to an operad, whose operations are ways of assembling a network of the given kind from smaller parts. Such operads, and their algebras, can serve as tools for designing networks. Technically, a network model is a lax symmetric monoidal functor from the free symmetric monoidal category on some set to 𝐂𝐚𝐭, and the construction of the corresponding operad proceeds via a symmetric monoidal version of the Grothendieck construction.
arXiv
External (22)
baez-2019-network reference entries/refs/baez-2019-network/baez-2019-network.hel