2025 JMM — AMS Special Session on Applied Category Theory

At the 2025 Joint Mathematics Meetings (JMM), there was an American Mathematical Society (AMS) Special Session on Applied Category Theory.

Program

9–9:20am: A categorical approach to Lyapunov stability

Presented by Joe Moeller, California Institute of Technology

Joint work with Aaron D. Ames, California Institute of Technology

Abstract: Lyapunov analysis forms the foundations for modern nonlinear control as Lyapunov functions characterize stability for dynamical systems. We present a categorical framework for Lyapunov theory, generalizing stability analysis through Lyapunov functions. Core to our approach is the axioms underlying a setting for stability, which give the necessary ingredients for “doing Lyapunov theory” in a category of interest. With these minimal assumptions we define stability, formulate Lyapunov morphisms, and prove a generalized Lyapunov theorem, demonstrating that the existence of Lyapunov morphisms imply stability. Examples include F-coalgebras, labeled transition systems, and quantale-enriched categories.

9:30–9:50am: Some Results on Non-Hermitian Ribbon Fusion Categories

Presented by Khyathi Komalan, California Institute of Technology

Abstract: Several open problems have been asked and are being solved on the nature of dagger structures that can be defined on Fusion Categories, and many results have been shown for Hermitian (positive dagger) Fusion Categories. However, Non-Hermitian Fusion Categories do exist, the most prominent example of them being the Yang-Lee Category. Using properties of Fusion Categories, previous results that has been found about Fusion Categories with a Hermitian Dagger Structure, and properties that arise from the existence of a Ribbon Structure and the ability to define a braiding, we now shift our focus on investigating Ribbon Fusion Categories that have a Non-Hermitian Dagger structure on them. Utilizing the rich and complicated structure of these categories, we discuss the outline of proofs of theorems on the Muger center, braiding, and spherical structure of such a fusion category, and discuss a potential construction of a TQFT using these results.

10–10:20am: Applying Stochastic Attributed C-Set Rewriting in Agent-based Modeling in Public Health & Beyond

Presented by Xiaoyan Li, University of Saskatchewan (Slides)

Joint work with:

Abstract: Agent-based modeling (ABM) is a computational method used to simulate complex systems by analyzing the emergent behavior arising from individually characterized agents interacting with their environment and each other. Traditional ABM approaches, often built using object-oriented programming techniques, typically struggle to express and capture intricate model structures, and commonly require software engineering that obscures model logic and stymies the interdisciplinary critique and feedback on which impactful modeling depends. To address these limitations, we introduce a novel stochastic attributed C-Set rewriting framework grounded in Category Theory for both ABMs and models hybridizing agent-based logic with stock and flow modeling. This categorical framework enables the structured and mathematically rigorous representation of various model elements, such as schemas, state charts, and networks, while employing multiple Category Theory methods, including declarative, attribute variable-enriched graphical rewriting, data migration, composition, etc. Different rewriting semantics can be achieved via selection of double-, single-, and sesqui-pushout-based rewriting. Using an example from Public Health, we will demonstrate how this framework enhances model representation and computation. Furthermore, as a promising future direction, the categorical foundation of this approach has the potential to support composition, stratification, separation of governing from observer processes, and seamless interaction and transformation across different model architectures—offering a more expressive and adaptable modeling paradigm that we plan to further explore.

10:30–10:50am: CatColab: A category-theoretic environment for collaborative modeling

Presented by Evan Patterson, Topos Institute (Slides)

Abstract: Ideas from applied category theory promise to make modeling more formal, interoperable, and conceptual, yet the technical nature of these ideas creates a significant barrier to entry for potential users outside the mathematical community. CatColab is a new software project that aims to make category-theoretic modeling accessible to a broader range of users. CatColab provides an intuitive user interface to specify, analyze, and critique formal models in a real-time collaborative setting. In this talk, we demonstrate the CatColab software and give a brief introduction to its category-theoretic underpinnings.

11–11:20am: Enriched Grothendieck topologies under change of base

Presented by Ariel Rosenfield, University of California Irvine (Slides)

Abstract: In the presence of a monoidal adjunction between locally finitely presentable Bénabou cosmoi U and V, we examine the behavior of enriched coverages on a V-enriched category U, and that of their constituent covering sieves, under the change of enriching category induced by the right adjoint G : V → U of the pair. We exhibit a construction of a U-Grothendieck topology on C given a V-Grothendieck topology, and prove in particular that when G is faithful and conservative, any upward-directed V-coverage on C corresponds uniquely to an upward-directed U-coverage on G_* C. We show that when G is fully faithful, base change commutes with enriched sheafification in the sense of Borceux-Quinteiro.

11:30–11:50am: (Higher) Categorical Galois Theory for the Working Mathematician

Presented by Joseph Rennie, Colorado College (Slides)

Abstract: Grothendieck’s groundbreaking approach to Galois theory (through the lens of toposes) led to myriad generalizations and adaptations of the overall idea from Galois theory into other areas. In this talk we first present a formal framework (due to Borceux and Janelidze) unifying the above (along with recent related work due to Townsend). We then present a recent higher-categorical generalization of this framework and demonstrate how use cases derive automatically from reasonable Algebro-Geometric duality results. This talk assumes no familiarity with the technicalities of higher categories or categorical Galois theory, and is intended to leave the audience with a sense of how this framework might prove relevant in their own area, along with a sense of the minimum necessary amount of work to establish such applications.

1–2:20pm: Panel discussion—Roads to learning category theory

Moderated by Kristine Bauer, University of Calgary

Panelists:

2:30–2:50pm: Category theory applied to inferentialist philosophy of language

Presented by Kristopher Brown, Topos Institute (Slides)

Abstract: There are remarkable similarities between applied category theory and inferentialist semantics in the philosophy of language: a focus on making pre-existing structure explicit, sense-making without rigid foundations, characterizing content in terms of external structure rather than internal structure, emphasis on open systems, and putting syntax and semantics in the same playing field. I will present some formalizations of inferentialism (a generalization of Girard’s phase semantics) due to Hlobil and Brandom and show some progress towards understanding what is taking place, categorically.

3–3:20pm: Analogical Plan Transfer in Robotics using Functorial Data Migrations

Presented by Angeline Aguinaldo, University of Maryland, College Park (Slides)

Abstract: Analogical transfer represents an advanced method of task plan reuse in robotics, where a task plan π from a source planning domain D_S is adapted to a new target planning domain D_T by leveraging formalized analogies between the two domains. Planning domains and their actions are represented using C-sets and double-pushout rewrite rules. Analogical transfer is achieved through functorial data migration—specifically delta data migrations and conjunctive query migrations, which make use of direct and structural analogies, respectively. This talk will discuss the benefits of this approach to applications in robotics, such as providing a general and well-defined method of plan transfer, the ability to automatically generate actions in the target planning domain, and the ability to simplify planning in complex domains by leveraging simpler ones.

3:30–3:50pm: Drazin Inverses in Categories

Presented by Priyaa Varshinee Srinivasan, Tallinn University of Technology

Joint work with:

Abstract: In this talk, I will introduce Drazin inverses from a categorical perspective. Drazin inverses are a fundamental algebraic structure which have been extensively deployed in semigroup theory and ring theory. Drazin inverses can also be defined for endomorphisms in any category. In this talk, I will introduce Drazin categories, in which every endomorphism has a Drazin inverse, and provide various examples including the category of matrices over a field, and explore various properties of these inverses.

4–4:20pm: Applications of Category Theory to Advanced Air Mobility Architecture

Presented by Nelson Niu, University of Washington (Slides)

Abstract: Advanced Air Mobility (AAM) is a rapidly-emerging sector of the aerospace industry which aims to safely and efficiently integrate highly automated aircraft into the National Airspace System (NAS). AAM is not a single technology, but a collection of new and emerging technologies being applied to the aviation transportation system, particularly in new aircraft types. These AAM technologies must integrate into one of the most complex and simultaneously safest transportation systems in the world. The evolution of the NAS poses a daunting problem: what new systems engineering architectures are needed to enable the AAM use cases, and how do we get there from here?

NASA is conducting research into advanced techniques for defining, encoding, and maintaining an open architecture for AAM and leveraging an ever-increasing database of knowledge to mature and validate requirements for the future aviation system. Category theory offers a number of promising applications that may help manage the complexity of an evolving system architecture. We will describe three examples: functorial data migration for transforming information presented one way to another; monoidal categories and wiring diagrams for formalizing activity diagrams; and polynomial functors and lenses for studying sequence diagrams.

4:30–4:50pm: Endomorphisms of Integer Valued Neural Networks with ReLU_t

Presented by Eric Dolores-Cuenca, Pusan National University (Slides)

Joint work with:

Abstract: In order theory, the lexicographic sum construction associates to every finite poset with n points an endomorphism of posets, that is Posets → End_Posets. The language of operads allows us to study other objects A with the property Posets ⊂ End_A, where End_A is the endomorphism operad of a set A. Examples of possible sets A are the Stanley order polynomials, zeta values, and order polytopes.

The paper “Tropical Geometry of Deep Neural Networks” by L. Zhang et al. introduces an equivalence between integer valued neural networks (IVNNs) with ReLU_t activation and tropical rational functions, which come with a map to polytopes. Here, IVNN refers to a network with integer weights but real biases, and ReLU_t is defined as ReLU_t(x) = max(x, t) for t ∈ R ∪ {−∞}. The authors of the above-mentioned paper ask about the consequences we can infer from connecting the study of IVNN with with the theory of polytopes. We lift the structure of an algebra over an operad of posets from order polytopes to IVNNs with . This implies that we have a (algebra) set of neural networks indexed by arbitrary finite posets, and that we have a family of associative operations indexed by posets.

We then explain how the neural networks associated to the N poset ({x < y > w < z}) can be interpreted as a 2-by-2 convolutional filter. When adapted to an implementation of the shallow ConvNet quaternion neural network classifier introduced by X. Zhu et al. in the paper “Quaternion convolutional neural networks”, we reduce the number of trainable parameters from 2,032,650 to 656,394, while also improving the performance of the neural network from a testing accuracy of 78.1% to 78.9%. We will also report ongoing experiments with state-of-the-art quaternion convolutional neural networks, as well as experiments with general convolutional neural networks and the study of the downsampling properties of poset neural networks.