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. • • John C. Baez:

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. Network ModelsOne way to combine two graphs is to identify the vertices of one with some of the vertices of the other in a one-to-one way, then gluing the two graphs together at the identified vertices. We can decompose such a combination into a sequence of simpler operations of three types: disjoint union of any two graphs, gluing two graphs together which have the same vertex set, which we call overlay, and permutation of vertices. In previous work, network models were introduced to formally encode these operations [2]. The algebras of a network operad can serve as tools for designing complex multi-agent networks. Network operads are constructed from network models, which are certain symmetric lax monoidal functors. There is a functorial construction of a network model from a monoid, which we call the ordinary network model for weighted graphs. In this paper, we provide a different construction in order to realize a larger class of networks as algebras of network operads, which we call the free varietal network model for weighted graphs. In Section 4, we give an example of a family of networks which cannot form an algebra for any ordinary network model for weighted graphs, but does for a varietal one.The reader is assumed to be familiar with basic notions from category theory [11], especially symmetric monoidal categories and lax symmetric monoidal functors [7]. Let S be the symmetric groupoid, i.e. the category with objects n = {1, . . . , n} (including the empty set for 0) and bijections for morphisms. Let Mon denote the category of monoids. A one-colored network model is a symmetric lax monoidal functorwhere Φ is the laxator of F , i.e. a natural transformation with components Φ x,y : F n × F m → F (n + m).We call the monoids F (n) the constituent monoids of the network model F . There is a more general notion of network model which replaces the category S with a free symmetric monoidal category. We do not consider this generalization here, so we always mean a one-colored network model when we say network model. Let NetMod denote the category of network models with monoidal natural transformations as morphisms.Essentially, a network model is a family of monoids {M n } n∈N each with a group action of the corresponding symmetric group S n , such that the product of any two embed into the one indexed by the sum of their indices e...

It is known that the Grothendieck group of the category of Schur functors is the ring of symmetric functions. This ring has a rich structure, much of which is encapsulated in the fact that it is a 'plethory': a monoid in the category of birings with its substitution monoidal structure. We show that similarly the category of Schur functors is a '2-plethory', which descends to give the plethory structure on symmetric functions. Thus, much of the structure of symmetric functions exists at a higher level in the category of Schur functors.

We define a thermostatic system to be a convex space of states together with a concave function sending each state to its entropy, which is an extended real number. This definition applies to classical thermodynamics, classical statistical mechanics, quantum statistical mechanics, and also generalized probabilistic theories of the sort studied in quantum foundations. It also allows us to treat a heat bath as a thermostatic system on an equal footing with any other. We construct an operad whose operations are convex relations from a product of convex spaces to a single convex space and prove that thermostatic systems are algebras of this operad. This gives a general, rigorous formalism for combining thermostatic systems, which captures the fact that such systems maximize entropy subject to whatever constraints are imposed upon them.

We define a thermostatic system to be a convex space of states together with a concave function sending each state to its entropy, which is an extended real number. This definition applies to classical thermodynamics, classical statistical mechanics, quantum statistical mechanics, and also generalized probabilistic theories of the sort studied in quantum foundations. It also allows us to treat a heat bath as a thermostatic system on an equal footing with any other. We construct an operad whose operations are convex relations from a product of convex spaces to a single convex space, and prove that thermostatic systems are algebras of this operad. This gives a general, rigorous formalism for combining thermostatic systems, which captures the fact that such systems maximize entropy subject to whatever constraints are imposed upon them.