arXiv subjects

James Fairbanks

Publications and source records attributed to James Fairbanks.

At least 19 recordsLinked to original sources

arXiv2026

Time-Varying Data as Sheaves: an Invitation to Narratives

Modern science and engineering increasingly rely on time-varying data, yet the mathematical tools used to model temporal phenomena are often developed within separate disciplines, obscuring common principles and limiting the transfer of ideas across fields. This chapter presents the theory of narratives, an abstract framework for time-varying objects of any mathematical kind that supports both theoretical investigations and applications. To illustrate this perspective, the chapter develops three vignettes, each illustrating a different research direction. The first addresses a general concern: What information loss can occur when switching between different representations of temporal data? The second concerns structural and algorithmic approaches: How can we systematically decompose time-varying data into simple pieces and obtain invariants describing its structural complexity? The third is an application to control theory: How can we model multi-agent systems with switching communication topologies? More important than any individual vignette, the central message of this invitation is that a suitable abstract perspective can organize and guide research across remarkably diverse mathematical and scientific domains.

cs.AI↗

arXiv2026

A Parameterized Algorithm for Testing whether the Limit of a Diagram is Empty

A limit of a (small) diagram $d : J \to E$ in a complete category $E$ can be thought of as specifying a set of equations involving the objects of $E$. To motivate this intuitively, one can think of each object $d(j)$ as a "variable" and each morphism in $J$ as a "constraint" connecting these variables. If $E$ has an initial object, a natural question arises: does our set of equations have any solution at all? Equivalently, we can ask: is the limit of $d$ initial? In this paper we consider the computational problem that, given finite diagram $d$ in a finitely complete category $E$, asks whether its limit is empty. We construct a fast algorithm (in the sense of parameterized complexity theory) that solves this problem when $E$ is of the form $\mathbf{FinSet}^{J}$ for a finite category $J$ and $d$ is a structured co-decomposition, i.e. a diagram arising from the opposite of the Grothendieck construction of a simple graph.

math.CT↗

arXiv2025

Heterogeneous Multi-Agent Multi-Target Tracking using Cellular Sheaves

Multi-agent target tracking in the presence of nonlinear dynamics and agent heterogeneity, where state-space dimensions may differ, is a challenging problem that traditional graph Laplacian methods cannot easily address. This work leverages the framework of cellular sheaves, a mathematical generalization of graph theory, to natively model such heterogeneous systems. While existing coordination sheaf frameworks focus on cooperative problems like consensus, this work extends them to the non-cooperative target-tracking problem. The tracking of multiple, unknown targets is formulated as a harmonic extension problem on a cellular sheaf, accommodating nonlinear dynamics and external disturbances for all agents. A decentralized control law is developed using the sheaf Laplacian, and a corresponding Lyapunov-based stability analysis is provided to guarantee tracking error convergence, with results validated by simulation.

eess.SY↗

arXiv2025

Algorithmic and Extremal Obstructions Through the Language of Cohomology

We model problems as presheaves that assign sets of certificates to input instances, and we show how to use presheaf Čech cohomology to capture the precise ways in which local solutions fail to patch into global ones. Applied to problems like Vertex Cover, Cycle Cover, and Odd Cycle Transversal, our framework exposes emergent phenomena such as hidden cycles or the inflation of small, local solutions. This approach not only rephrases classical results like König's Theorem in cohomological terms, but also reveals how to systematically account for failures of compositionality. Although our main focus is on presheaves of sets, the methods generalize naturally to Abelian presheaves, suggesting a rich interplay between graph theory, cohomology, and complexity. This work represents a first step toward a systematic, sheaf-theoretic theory of algorithmic structure and related obstructions.

math.AC↗

arXiv2025

Asynchronous Nonlinear Sheaf Diffusion for Multi-Agent Coordination

Cellular sheaves and sheaf Laplacians provide a far-reaching generalization of graphs and graph Laplacians, resulting in a wide array of applications ranging from machine learning to multi-agent control. In the context of multi-agent systems, so called coordination sheaves provide a unifying formalism that models heterogeneous agents and coordination goals over undirected communication topologies, and applying sheaf diffusion drives agents to achieve their coordination goals. Existing literature on sheaf diffusion assumes that agents can communicate and compute updates synchronously, which is an unrealistic assumption in many scenarios where communication delays or heterogeneous agents with different compute capabilities cause disagreement among agents. To address these challenges, we introduce asynchronous nonlinear sheaf diffusion. Specifically, we show that under mild assumptions on the coordination sheaf and bounded delays in communication and computation, nonlinear sheaf diffusion converges to a minimizer of the Dirichlet energy of the coordination sheaf at a linear rate proportional to the delay bound. We further show that this linear convergence is attained from arbitrary initial conditions and the analysis depends on the spectrum of the sheaf Laplacian in a manner that generalizes the standard graph Laplacian case. We provide several numerical simulations to validate our theoretical results.

math.OC↗

arXiv2025

Generalized Gradient Descent is a Hypergraph Functor

Cartesian reverse derivative categories (CRDCs) provide an axiomatic generalization of the reverse derivative, which allows generalized analogues of classic optimization algorithms such as gradient descent to be applied to a broad class of problems. In this paper, we show that generalized gradient descent with respect to a given CRDC induces a hypergraph functor from a hypergraph category of optimization problems to a hypergraph category of dynamical systems. The domain of this functor consists of objective functions that are 1) general in the sense that they are defined with respect to an arbitrary CRDC, and 2) open in that they are decorated spans that can be composed with other such objective functions via variable sharing. The codomain is specified analogously as a category of general and open dynamical systems for the underlying CRDC. We describe how the hypergraph functor induces a distributed optimization algorithm for arbitrary composite problems specified in the domain. To illustrate the kinds of problems our framework can model, we show that parameter sharing models in multitask learning, a prevalent machine learning paradigm, yield a composite optimization problem for a given choice of CRDC. We then apply the gradient descent functor to this composite problem and describe the resulting distributed gradient descent algorithm for training parameter sharing models.

math.CT↗

arXiv2025

Porous Convection in the Discrete Exterior Calculus with Geometric Multigrid

The discrete exterior calculus (DEC) defines a family of discretized differential operators which preserve certain desirable properties from the exterior calculus. We formulate and solve the porous convection equations in the DEC via the Decapodes.jl embedded domain-specific language (eDSL) for multiphysics problems discretized via CombinatorialSpaces.jl. CombinatorialSpaces.jl is an open-source Julia library which implements the DEC over simplicial complexes, and now offers a geometric multigrid solver over maps between subdivided simplicial complexes. We demonstrate numerical results of multigrid solvers for the Poisson problem and porous convection problem, both as a standalone solver and as a preconditioner for open-source Julia iterative methods libraries.

cs.CE↗

arXiv2025

Lassos: Pushing Tree Decompositions Forward Along Homomorphisms

It is folklore that tree-width is monotone under taking subgraphs (i.e. injective graph homomorphisms) and contractions (certain kinds of surjective graph homomorphisms). However, although tree-width is obviously not monotone under any surjective graph homomorphism, it is not clear whether contractions are canonically the only class of surjections with respect to which it is monotone. Under the requirement that the decomposition shape must be preserved, we prove that this is indeed the case. Our results provide a framework for answering questions of this sort for many other kinds of combinatorial data structures (such as directed multigraphs, hypergraphs, Petri nets, circular port graphs, half-edge graphs, databases, simplicial sets etc.) for which natural analogues of tree decompositions can be defined. Furthermore and of independent interest, we prove these results by introducing the notion of a lasso, a generalization of contractions of graphs to arbitrary categories with pushouts of monomorphisms.

math.CO↗

arXiv2025

Distributed Multi-agent Coordination over Cellular Sheaves

Techniques for coordination of multi-agent systems are vast and varied, often utilizing purpose-built solvers or controllers with tight coupling to the types of systems involved or the coordination goal. In this paper, we introduce a general unified framework for heterogeneous multi-agent coordination using the language of cellular sheaves and nonlinear sheaf Laplacians, which are generalizations of graphs and graph Laplacians. Specifically, we introduce the concept of a nonlinear homological program encompassing a choice of cellular sheaf on an undirected graph, nonlinear edge potential functions, and constrained convex node objectives, which constitutes a standard form for a wide class of coordination problems. We use the alternating direction method of multipliers to derive a distributed optimization algorithm for solving these nonlinear homological programs. To demonstrate the applicability of this framework, we show how heterogeneous coordination goals including combinations of consensus, formation, and flocking can be formulated as nonlinear homological programs and provide numerical simulations showing the efficacy of our distributed solution algorithm.

math.OC↗

arXiv2025

Towards a Unified Theory of Time-Varying Data

What is a time-varying graph, a time-varying topological space, or, more generally, a mathematical structure that evolves over time? In this work, we lay the foundations for a general theory of temporal data by introducing categories of narratives. These are sheaves on posets of time intervals that encode snapshots of a temporal object along with the relationships between them. This theory satisfies five desiderata distilled from the burgeoning field of time-varying graphs: (D1) it defines both time-varying objects and their morphisms; (D2) it distinguishes between cumulative and persistent interpretations and provides principled methods for transitioning between them; (D3) it systematically lifts static notions to their temporal analogues; (D4) it is object agnostic; (D5) it integrates with theories of dynamical systems. To achieve this, we build upon existing categorical and sheaf-theoretic approaches to temporal graph theory, generalizing them to any category with limits and colimits. We also formalize tacit intuitions that, while present, often remain implicit in temporal graph theory. Beyond synthesizing and reformulating existing ideas in categorical language, we introduce sheaf-theoretic constructions and prove results that, to our knowledge, have not appeared in the temporal data literature - such as the adjunction between persistent and cumulative narratives. More importantly, we integrate these existing and novel elements into a consistent and coherent framework, setting the stage for a unified theory of time-varying data.

math.CT↗

arXiv2024

GATlab: Modeling and Programming with Generalized Algebraic Theories

Categories and categorical structures are increasingly recognized as useful abstractions for modeling in science and engineering. To uniformly implement category-theoretic mathematical models in software, we introduce GATlab, a domain-specific language for algebraic specification embedded in a technical programming language. GATlab is based on generalized algebraic theories (GATs), a logical system extending algebraic theories with dependent types so as to encompass category theory. Using GATlab, the programmer can specify generalized algebraic theories and their models, including both free models, based on symbolic expressions, and computational models, defined by arbitrary code in the host language. Moreover, the programmer can define maps between theories and use them to declaratively migrate models of one theory to models of another. In short, GATlab aims to provide a unified environment for both computer algebra and software interface design with generalized algebraic theories. In this paper, we describe the design, implementation, and applications of GATlab.

cs.LO↗

arXiv2024

A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks

Regulatory networks depict promoting or inhibiting interactions between molecules in a biochemical system. We introduce a category-theoretic formalism for regulatory networks, using signed graphs to model the networks and signed functors to describe occurrences of one network in another, especially occurrences of network motifs. With this foundation, we establish functorial mappings between regulatory networks and other mathematical models in biochemistry. We construct a functor from reaction networks, modeled as Petri nets with signed links, to regulatory networks, enabling us to precisely define when a reaction network could be a physical mechanism underlying a regulatory network. Turning to quantitative models, we associate a regulatory network with a Lotka-Volterra system of differential equations, defining a functor from the category of signed graphs to a category of parameterized dynamical systems. We extend this result from closed to open systems, demonstrating that Lotka-Volterra dynamics respects not only inclusions and collapsings of regulatory networks, but also the process of building up complex regulatory networks by gluing together simpler pieces. Formally, we use the theory of structured cospans to produce a lax double functor from the double category of open signed graphs to that of open parameterized dynamical systems. Throughout the paper, we ground the categorical formalism in examples inspired by systems biology.

q-bio.MN↗

arXiv2024

Modeling Model Predictive Control: A Category Theoretic Framework for Multistage Control Problems

Model predictive control (MPC) is an optimal control technique which involves solving a sequence of constrained optimization problems across a given time horizon. In this paper, we introduce a category theoretic framework for constructing complex MPC problem formulations by composing subproblems. Specifically, we construct a monoidal category - called Para(Conv) - whose objects are Euclidean spaces and whose morphisms represent constrained convex optimization problems. We then show that the multistage structure of typical MPC problems arises from sequential composition in Para(Conv), while parallel composition can be used to model constraints across multiple stages of the prediction horizon. This framework comes equipped with a rigorous, diagrammatic syntax, allowing for easy visualization and modification of complex problems. Finally, we show how this framework allows a simple software realization in the Julia programming language by integrating with existing mathematical programming libraries to provide high-level, graphical abstractions for MPC.

math.OC↗

arXiv2024

A Compositional Framework for First-Order Optimization

Optimization decomposition methods are a fundamental tool to develop distributed solution algorithms for large scale optimization problems arising in fields such as machine learning and optimal control. In this paper, we present an algebraic framework for hierarchically composing optimization problems defined on hypergraphs and automatically generating distributed solution algorithms that respect the given hierarchical structure. The central abstractions of our framework are operads, operad algebras, and algebra morphisms, which formalize notions of syntax, semantics, and structure preserving semantic transformations respectively. These abstractions allow us to formally relate composite optimization problems to the distributed algorithms that solve them. Specifically, we show that certain classes of optimization problems form operad algebras, and a collection of first-order solution methods, namely gradient descent, Uzawa's algorithm (also called gradient ascent-descent), and their subgradient variants, yield algebra morphisms from these problem algebras to algebras of dynamical systems. Primal and dual decomposition methods are then recovered by applying these morphisms to certain classes of composite problems. Using this framework, we also derive a novel sufficient condition for when a problem defined by compositional data is solvable by a decomposition method. We show that the minimum cost network flow problem satisfies this condition, thereby allowing us to automatically derive a hierarchical dual decomposition algorithm for finding minimum cost flows on composite flow networks. We implement our operads, algebras, and algebra morphisms in a Julia package called AlgebraicOptimization.jl and use our implementation to empirically demonstrate that hierarchical dual decomposition outperforms standard dual decomposition on classes of flow networks with hierarchical structure.

math.OC↗

arXiv2024

The diagrammatic presentation of equations in categories

Lifts of categorical diagrams $D\colon\mathsf{J}\to\mathsf{X}$ against discrete opfibrations $π\colon\mathsf{E}\to\mathsf{X}$ can be interpreted as presenting solutions to systems of equations. With this interpretation in mind, it is natural to ask if there is a notion of equivalence of diagrams $D\simeq D'$ that precisely captures the idea of the two diagrams "having the same solutions''. We give such a definition, and then show how the localisation of the category of diagrams in $\mathsf{X}$ along such equivalences is isomorphic to the localisation of the slice category $\mathsf{Cat}/\mathsf{X}$ along the class of initial functors. Finally, we extend this result to the 2-categorical setting, proving the analogous statement for any locally presentable 2-category in place of $\mathsf{Cat}$.

math.CT↗

arXiv2023

A Categorical Representation Language and Computational System for Knowledge-Based Planning

Classical planning representation languages based on first-order logic have preliminarily been used to model and solve robotic task planning problems. Wider adoption of these representation languages, however, is hindered by the limitations present when managing implicit world changes with concise action models. To address this problem, we propose an alternative approach to representing and managing updates to world states during planning. Based on the category-theoretic concepts of $\mathsf{C}$-sets and double-pushout rewriting (DPO), our proposed representation can effectively handle structured knowledge about world states that support domain abstractions at all levels. It formalizes the semantics of predicates according to a user-provided ontology and preserves the semantics when transitioning between world states. This method provides a formal semantics for using knowledge graphs and relational databases to model world states and updates in planning. In this paper, we conceptually compare our category-theoretic representation with the classical planning representation. We show that our proposed representation has advantages over the classical representation in terms of handling implicit preconditions and effects, and provides a more structured framework in which to model and solve planning problems.

cs.AI↗

arXiv2023

Compositional Algorithms on Compositional Data: Deciding Sheaves on Presheaves

Algorithmicists are well-aware that fast dynamic programming algorithms are very often the correct choice when computing on compositional (or even recursive) graphs. Here we initiate the study of how to generalize this folklore intuition to mathematical structures writ large. We achieve this horizontal generality by adopting a categorial perspective which allows us to show that: (1) structured decompositions (a recent, abstract generalization of many graph decompositions) define Grothendieck topologies on categories of data (adhesive categories) and that (2) any computational problem which can be represented as a sheaf with respect to these topologies can be decided in linear time on classes of inputs which admit decompositions of bounded width and whose decomposition shapes have bounded feedback vertex number. This immediately leads to algorithms on objects of any C-set category; these include -- to name but a few examples -- structures such as: symmetric graphs, directed graphs, directed multigraphs, hypergraphs, directed hypergraphs, databases, simplicial complexes, circular port graphs and half-edge graphs. Thus we initiate the bridging of tools from sheaf theory, structural graph theory and parameterized complexity theory; we believe this to be a very fruitful approach for a general, algebraic theory of dynamic programming algorithms. Finally we pair our theoretical results with concrete implementations of our main algorithmic contribution in the AlgebraicJulia ecosystem.

cs.CC↗

arXiv2023

Characterizing Compositionality of LQR from the Categorical Perspective

Composing systems is a fundamental concept in modern control systems, yet it remains challenging to formally analyze how controllers designed for individual subsystems can differ from controllers designed for the composition of those subsystems. To address this challenge, we propose a novel approach to composing control systems based on resource sharing machines, a concept from applied category theory. We use resource sharing machines to investigate the differences between (i) the linear-quadratic regulator (LQR) designed directly for a composite system and (ii) the LQR that is attained through the composition of LQRs designed for each subsystem. We first establish novel formalisms to compose LQR control designs using resource sharing machines. Then we develop new sufficient conditions to guarantee that the LQR designed for a composite system is equal to the LQR attained through composition of LQRs for its subsystems. In addition, we reduce the developed condition to that of checking the controllability and observability of a certain linear, time-invariant system, which provides a simple, computationally efficient procedure for evaluating the equivalence of controllers for composed systems.

eess.SY↗