Dynamical Systems and Sheaves

TL;DR

This paper introduces a categorical framework using sheaf theory and lax monoidal functors to model, analyze, and compose diverse dynamical systems with inputs and outputs over time.

Contribution

It develops a unified categorical approach to model various dynamical systems and their interconnections, ensuring compositionality and determinism.

Findings

Framework accommodates continuous and discrete systems.

Provides conditions for deterministic composition.

Uses sheaf theory to model temporal aspects.

Abstract

A categorical framework for modeling and analyzing systems in a broad sense is proposed. These systems should be thought of as `machines' with inputs and outputs, carrying some sort of signal that occurs through some notion of time. Special cases include continuous and discrete dynamical systems (e.g. Moore machines). Additionally, morphisms between the different types of systems allow their translation in a common framework. A central goal is to understand the systems that result from arbitrary interconnection of component subsystems, possibly of different types, as well as establish conditions that ensure totality and determinism compositionally. The fundamental categorical tools used here include lax monoidal functors, which provide a language of compositionality, as well as sheaf theory, which flexibly captures the crucial notion of time.