Classically, the splitting principle says how to pull back a vector bundle in such a way that it splits into line bundles and the pullback map induces an injection on $𝐾$-theory. Here we categorify the splitting principle and generalize it to the context of 2-rigs. A 2-rig is a kind of categorified “ring without negatives”, such as a category of vector bundles with $\oplus$ as addition and $\otimes$ as multiplication. Technically, we define a 2-rig to be a Cauchy complete $𝑘$-linear symmetric monoidal category where $𝑘$ has characteristic zero. We conjecture that for any suitably finite-dimensional object $𝑟$ of a 2-rig $\text{𝖱}$, there is a 2-rig map $𝐸:\text{𝖱}\to \text{𝖱}\prime$ such that $𝐸(𝑟)$ splits as a direct sum of finitely many “subline objects” and $𝐸$ has various good properties: it is faithful, conservative, essentially injective, and the induced map of Grothendieck rings $𝐾(𝐸):𝐾(\text{𝖱})\to 𝐾(\text{𝖱}\prime )$ is injective. We prove this conjecture for the free 2-rig on one object, namely the category of Schur functors, whose Grothendieck ring is the free $𝜆$-ring on one generator, also known as the ring of symmetric functions. We use this task as an excuse to develop the representation theory of affine categories - that is, categories enriched in affine schemes - using the theory of 2-rigs.

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 present a unified framework for Petri nets and various variants, such as pre-nets and Kock’s whole-grain Petri nets. Our framework is based on a less well-studied notion that we call $\mathrm{\Sigma }$-nets, which allow finer control over whether tokens are treated using the collective or individual token philosophy. We describe three forms of execution semantics in which pre-nets generate strict monoidal categories, $\mathrm{\Sigma }$-nets (including whole-grain Petri nets) generate symmetric strict monoidal categories, and Petri nets generate commutative monoidal categories, all by left adjoint functors. We also construct adjunctions relating these categories of nets to each other, in particular showing that all kinds of net can be embedded in the unifying category of $\mathrm{\Sigma }$-nets, in a way that commutes coherently with their execution semantics.

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.

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.

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 $\text{𝐂𝐚𝐭}$, and the construction of the corresponding operad proceeds via a symmetric monoidal version of the Grothendieck construction.