Research

Papers

  • Extensions of representation stable categories
    Accepted to New York Journal of Mathematics, 2025.
    Preprint arXiv:2209.03879, 2022.
    Representation stability began with the observation that functors out of the category FI of finite sets and injections are better behaved than arbitrary families of representations of symmetric groups. This turned out to be quite useful in deriving some results about the cohomology of configuration spaces. Since then, people have sought to find similar properties in other categories, hopefully shedding light on representations of other (families of) groups. One particularly nice example is the category FI_G, where G is your favorite finite group. The objects are still finite sets, but the morphisms are injections decorated with an element of G on each “string”. This category is so similar to FI, it essentially has all the same representation stability properties. It turns out the forgetful functor from FI_G to FI is a fibration, and so FI_G is the result of the Grothendieck construction applied to the functor FI^op -> Cat which sends each n to G^n and each injection to pulling back.
  • Colored Petri nets are monoidal double functors, with Jade Master.
    Preprint arXiv:2510.01946, 2025.
    Petri nets are used to model networks of resource-dependent processes. A Petri net can be “marked” with an assignment of a number of tokens to each resource type. Then transitions are either enabled or not depending on whether the minimum required resources are available. An “execution” is then a sequence of firings of transitions which are enabled at each step based on the movement of resources from the previous transition firings. In this way, a Petri net can be seen to present a symmetric monoidal category. Inspired by the monoidal Grothendieck construction, we show that a colored Petri net can been viewed as a certain sort of functor which assigns extra data to each component of an ordinary Petri net.
  • 2-rig extensions and the splitting principle, with John Baez and Todd Trimble
    Theory and Applications of Categories, Vol. 44, 2025, No. 31, pp 964-1019.
    The classical splitting principle in K-theory says that every vector bundle can be pulled back along some map so that it splits as a direct sum of line bundles, and this pullback behaves well with respect to K-theory. In this paper, we categorify that principle. We work with the same notion of 2-rigs as in “Schur functors and categorified plethysm”. We formulate a conjectural version of the splitting principle in this context. Roughly, we conjecture that objects in a 2-rig can be faithfully embedded into another 2-rig where they split into subline objects, and that this embedding reflects enough structure to induce an injective map on Grothendieck rings. We prove the conjecture for the free 2-rig on one generator, the category of Schur functors. Along the way, we develop some foundational theory of affine categories, meaning categories enriched in affine schemes.
  • Categorical Lyapunov theory II: stability of systems, with Aaron Ames and Sebastien Mattenet.
    Preprint arXiv:2505.22968, 2025.
    This paper continues the development of categorical Lyapunov theory, shifting focus from flows to systems modeled as coalgebras. Given an endofunctor F, we study the stability of F-coalgebras using Lyapunov-type methods. We define two kinds of categorical settings for stability: one minimal and easy to check in examples, the other more structured and theoretically robust. In both cases, we prove a Lyapunov theorem characterizing stability in terms of Lyapunov morphisms. For the richer setting, we also prove a converse theorem. This framework generalizes classical stability analysis to coalgebraic systems and lays groundwork for further categorical approaches to dynamical behavior.
  • Categorical Lyapunov theory I: stability of flows, with Aaron Ames and Paulo Tabuada.
    Preprint arXiv:2502.15276, 2025.
    Lyapunov’s theorem gives a way to prove stability of equilibria in dynamical systems using Lyapunov functions. In this paper, we recast that idea categorically. We define a general framework for “doing Lyapunov theory” in any category that satisfies a small list of axioms. These axioms allow us to define equilibria, flows, and Lyapunov morphisms in a unified setting. We prove that the existence of a Lyapunov morphism is necessary and sufficient for stability. This framework captures classical Lyapunov theory for both continuous and discrete time systems, and it extends naturally to quantale enriched categories, e.g. Lawvere metric spaces. This suggests that Lyapunov-like theorems may apply in a wide range of settings.
  • Schur functors and categorified plethysm, with John Baez and Todd Trimble, Higher Structures, Vol. 8, Iss. 1, 2024, pp 1-53.
    – Categories & Companions 2021 talk: “2-plethories” [video]
    – CT20->21 talk: “Schur functors” [video]
    It is well-known to representation theorists that the Grothendieck ring of the category of finite representations of the symmetric groupoid is isomorphic to the ring of symmetric functions, Λ. The ring Λ is rich with additional structure. In particular, Tall and Wraith point out that Λ is a plethory (so named by Borger and Wieland). In this paper, we categorify this structure. First, we define abstract Schur functors to be “natural operations” on 2-rigs (symmetric monoidal, Vect-enriched, Cauchy complete categories). Then we show that the category of abstract Schur functors (denoted by Schur) and the category of finite representations of the symmetric groupoid are equivalent to the free 2-rig on a single generator. Next, we define the categorified notions of biring and plethory, which we call “2-birig” and “2-plethory”, and show that Schur is a 2-plethory. Then we show that this 2-plethory structure on Schur induces the plethory structure on the Grothendieck ring Λ. This last step involves taking the homology of differential \mathbb Z/2 graded Schur functors.
  • Compositional Thermostatics, with John Baez and Owen Lynch
    Journal of Mathematical Physics, Vol. 64, Iss. 2, 2023.
    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.
  • Network Models from Petri Nets with Catalysts with John Baez and John Foley,
    Compositionality, Volume 1, Issue 4, 2019.
    – ACT2020: “Petri nets with Catalysts” [video]
    We explore the categorical implications of the presence of catalysts in a Petri net. The category of executions naturally decomposes into subcategories corresponding to fixed amounts of catalyst. We take advantage of this decomposition to give a partial model of the individual token philosophy. These subcategories inherit premonoidal structures from the monoidal structure of the big category. We give a novel string diagram language which can be used to keep track of the role of the catalyst in complex sequences of executions performed in the net.
  • Monoidal Grothendieck Construction, with Christina Vasilakopoulou,
    Theory and Applications of Categories, Vol. 35, 2020, No. 31, pp 1159-1207.
    – MIT Categories Seminar: “Monoidal Grothendieck construction” [video]
    We give a gold-plated version of the idea first given in Network Models: a monoidal variant of the Grothendieck construction. The classical Grothendieck construction gives an equivalence between indexed categories and fibred categories. We define two different monoidal variants of these objects: a fibre-wise one and a global one, and prove the corresponding equivalences. We also give braided and symmetric versions. We also do all of this for opindexed categories and opfibred categories. We also give conditions under which the global and fibre-wise versions are equivalent to each other. There’s a bunch of applications at the end in categorical algebra and categorical network theory.
  • Noncommutative Network Models,
    Mathematical Structures in Computer Science, Volume 30, Issue 1, pages 14-32, 2020, doi:10.1017/S0960129519000161.
    – ACT2021: “Noncommutative network models” [slides]
    I construct the free network model on a given monoid. Moreover, I construct the free varietal network model on a monoid of a given variety. To do this, I give a generalization of the graph product of groups. Writing some classical combinatorial structures in categorical language is also involved.
  • Network Models, with John Baez, John Foley, and Blake Pollard,
    Theory and Applications of Categories, Vol. 35, 2020, No. 20, pp 700-744.
    – UNAM Categories Seminar: “Network models”
    The purpose of this paper is to define a bunch of operads where the operations are networks of a certain type. The way we specify what type of network we want is with something we call a network model. This is basically a combinatorial species that has extra structure capturing the notions of union and disjoint union. The corresponding operad is then constructed via a monoidal variant of the Grothendieck construction.

Other talks