Sheaves of Dynamical Systems

• Sheaves of dynamical systems are mathematical constructs that integrate local state data into a unified global framework using locality and gluing axioms.
• They employ various topological frameworks, such as Alexandrov and interval topologies, to model interconnected dynamics and time-dependent behaviors.
• Applications include system verification, bifurcation analysis through sheaf cohomology, and symmetry reduction in composite and arithmetic systems.

A sheaf of dynamical systems is a mathematical structure that encodes families of dynamical systems, their interconnections, and the gluing of local behaviors into coherent global dynamics. This concept arises in multiple contexts: from the compositional modeling of hybrid, discrete, or continuous systems, to the construction of algebraic and cohomological invariants in the paper of parametric families and arithmetic schemes. Fundamental to the framework is the sheaf property: local data (e.g., states, flows, equations, or measurements on subsystems) can be uniquely and consistently assembled into global sections that represent entire system behaviors. The sheaf-theoretic perspective enables the rigorous analysis of system integration, bifurcation, uncertainty quantification, and invariance under symmetry or group action.

1. Foundational Definitions

A presheaf of dynamical systems on a topological space or site $X$ assigns to each open subset $U$ a set (or category) $F(U)$ of "sections over $U$"—typically state spaces, trajectories, or solution structures—and to each inclusion $V\subseteq U$ a restriction map $\rho_{U,V}:F(U)\to F(V)$ satisfying $\rho_{U,U} = \mathrm{id}$ and $\rho_{V,W}\circ\rho_{U,V} = \rho_{U,W}$. The presheaf is a sheaf if it satisfies the locality and gluing axioms: any family of compatible local data on an open cover $\{U_i\}$ of $U$ can be uniquely glued to a global section on $U$ (Robinson, 2016, 0810.2877).

When each $F(U)$ is itself equipped with a (possibly nonlinear) dynamical law $f_U:F(U)\to F(U)$, and the restrictions commute with dynamics, i.e., $\rho_{U,V}\circ f_U = f_V\circ\rho_{U,V}$, the presheaf is called a sheaf of dynamical systems (Robinson et al., 6 Nov 2025). The base topology may arise, for example, from the Alexandrov topology on a poset encoding subsystem inclusions or data flow in a network.

2. Construction and Topologies of Sheaves of Dynamical Systems

2.1 Alexandrov/Scott Topology

For composite or modular systems, the base space $X$ is often a poset (e.g., of system components, subsystems, or variables) equipped with the Alexandrov (or Scott) topology; basic opens are the sets $U_x = \{y \mid x\leq y\}$. Sections over opens correspond to assignments of compatible local states, behaviors, or equations to collections of subsystems (Robinson et al., 6 Nov 2025, 0810.2877).

2.2 Interval Topos and Time-Sheaves

To model time-dependent dynamical systems, the site of (continuous or discrete) time intervals (e.g., $[0,\ell]$) is used. A sheaf on such a site, for example the sheaf of local ODE solutions, assigns to each interval $U$ the set of trajectories $u:U\to X$ satisfying system dynamics, with restriction given by domain truncation (Schultz et al., 2016, Robinson, 2016).

2.3 Quotient Sites and Group/Semigroup Actions

When a site $X$ carries an action of a countable semigroup $\Sigma$ (e.g., iteration, shift, time translation), one may form the quotient site $[X/\Sigma]$ whose objects track both opens of $X$ and their equivariance under $\Sigma$. The category of sheaves on $[X/\Sigma]$ is equivalent to the category of sheaves on $X$ equipped with compatible $\Sigma$-action, supporting the formalism for dynamical sheaves with symmetries or parameter evolution (Garofali, 2022).

3. Sheaf Models for Composite and Parametric Systems

3.1 Netlist Construction

A central method for encoding complex interconnected systems is the "netlist" formalism: one constructs a bipartite acyclic graph with "part" nodes (mechanisms, subsystems) and "net" nodes (variables), endowing each with appropriate state spaces. The sheaf assigns to each node its local variable or joint input space, and restrictions follow connection data (e.g., functional dependencies or wiring). The global sections of the sheaf correspond exactly to solutions of the full composite system, revealing coupling and emergent behavior (Robinson et al., 6 Nov 2025).

3.2 Categorical, Span-Algebra, and Operad-Based Models

In the categorical framework, dynamical systems (discrete/continuous/hybrid) are modeled as machines with input and output ports, formalized as objects in a symmetric monoidal category or operad of wiring diagrams. Sheaf-valued algebras over these operads organize the possible systems that can inhabit particular interfaces, with global sections again encoding admissible compositions. Feedback and parallel/serial composition correspond to the categorical monoidal and operadic structures (Schultz et al., 2016).

3.3 States, Transitions, and Behaviors as Sheaves

Sheaves can encode not only raw trajectories but the entire data of states, allowed transitions, and time-indexed behaviors for concurrent or modular systems. For instance, sheaves assign to each open collection of interacting subsystems the set of local states and transitions, and the gluing of these sheaves corresponds to the synchronization and consistency of behaviors across the system (0810.2877).

4. Cohomology, Consistency, and Inference

Sheaf cohomology provides algebraic invariants for dynamical families and detects obstructions to gluing local data. For parametric families of dynamical systems, an abelianization (e.g., via functors to boolean or monoid rings) enables the construction of cohomology groups whose nontriviality signals bifurcations—parameter values for which invariant structures (such as attractors) undergo a qualitative change (Dowling et al., 2021). Higher cohomology classes measure the global failure to consistently patch together local solutions or measurements in the presence of noise, missing data, or group action (Robinson et al., 6 Nov 2025, Garofali, 2022).

The following summarizes this paradigm:

Context	Sheaf Structure	Significance
Composite system via netlist (Robinson et al., 6 Nov 2025)	Assigns state/data spaces to nodes; restrictions via system wiring	Global sections = complete system solutions; cohomology = uncertainty/inconsistency
Parametric family bifurcations (Dowling et al., 2021)	Sheaves of lattices/abelianizations of attractors over parameter space	Nontrivial $H^1$ signals bifurcation locus
Group/semigroup action, quotient site (Garofali, 2022)	Sheaves equivariant for action; sites $[X/\Sigma]$	Extension, deformation, and rigidity theory; symmetry reduction

5. Applications in Arithmetic, Geometry, and Analysis

5.1 Arithmetic Schemes and Canonical Dynamical Systems

For a scheme $X$, the sheafified rational Witt vector construction $\mathcal W_{\mathrm{rat}}$ yields a ringed space $(X, \mathcal W_{\mathrm{rat}})$ whose $\mathbb C$-valued points form an infinite-dimensional real dynamical system, with periodic orbits corresponding to the closed points of $X$ and orbit lengths $\log N(x)$ (for $N(x)$ the norm) (Deninger, 2018). Flow structures are sheaf-theoretically suspended from the Frobenius operator, and gluing over affines recovers global dynamics.

5.2 Holomorphic Dynamics, Deformation, and Rigidity

The formalism of dynamical sheaves on quotient sites $[X/\Sigma]$ encodes semigroup actions such as iteration under rational maps. Sheaf cohomology in this setting yields deformation (Ext) groups, enables computation of the étale fundamental group, and underlies results such as Infinitesimal Thurston Rigidity and the Fatou–Shishikura inequalities for dynamics on $\mathbb P^1$ (Garofali, 2022).

5.3 Finite-Gap Integration and Differential Geometry

Sheaves of (rank-one, torsion-free) modules over reducible spectral curves allow the explicit construction of orthogonal curvilinear coordinate systems in $\mathbb R^n$, with local data at singularities encoding the moduli of the corresponding coordinate systems via finite-gap integration (Mironov et al., 2023).

5.4 Systems Verification and Geometric Logic

Sheaf semantics allows the transfer of local verification results (safety, liveness, invariance) for component systems into global guarantees for their interconnections. Geometric logical formulas which are true locally in each stalk (e.g., collision-freedom in coupled train controllers) lift to true global properties of the full system (0810.2877).

6. Twisted Dynamical Systems, Gerbes, and C*-Algebraic Extensions

In the context of operator algebras and quantum field theory, sheaf-theoretic and gerbe-theoretic methods encode families of C*-dynamical systems, with twisting arising from nontrivial 2-group cocycles over base posets. The global holonomy produces "twisted" actions on C*-algebras (e.g., Cuntz algebras), integrating local symmetries and obstructions into the global dynamical framework (Vasselli, 2017).

7. Methodological Considerations and Computational Aspects

The general workflow in the construction and use of sheaves of dynamical systems involves the following steps:

1. Selection of Base Site/Topology: Based on the system decomposition (components, subsystems, time intervals, parameter space, group action).
2. Sheaf Assignment: Specification of state/data/dynamics categories to opens; definition of restriction maps following subsystem inclusion or data flow.
3. Sheaf Verification: Checking locality and gluing via standard axioms—often directly computable for sets, modules, or categories.
4. Solution/Section Extraction: Global sections are computed, often as fixed points or solutions to compatibilized subsystem equations.
5. Cohomology and Gluing Obstruction: Computation via Godement or Čech resolutions; in parametric cases this provides bifurcation/degeneracy loci.
6. Inference and Consistency: Optimization of consistency radii to reconstruct global behaviors from incomplete or local observations (Robinson et al., 6 Nov 2025).

The sheaf-theoretic paradigm unifies algebraic geometry, differential equations, operator algebras, systems theory, and topological data analysis, providing a rigorous language to paper global behaviors arising from local data and their interdependencies.