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Quotient toposes of discrete dynamical systems

Ryuya Hora, Yuhi Kamio

Abstract

This paper gives a classification of classes of discrete dynamical systems (a set equipped with an endofunction) closed under finite limits and small colimits. The conclusion is simple: they bijectively correspond to the ideals of the product poset $\mathbb{N} \times \mathbb{N}$, where the first $\mathbb{N}$ is ordered by the usual order and the second is by the divisibility. Our method is based on a detailed analysis of the behaviors of states, especially non-periodic behaviors, in discrete dynamical systems. Specifically, extending the fundamental quantity, time until entering a loop, to even those states that do not enter a loop plays a crucial role. Our classification is closely related to epimorphisms from $\mathbb{N}$ in the category of monoids. There are countably many injective epimorphisms from $\mathbb{N}$, including $\mathbb{N} \to \mathbb{Z}$. Those injective epimorphisms correspond to the non-periodic behaviors of states of discrete dynamical systems. We discuss this point at the end of the paper. This fun puzzle is motivated by an open problem in topos theory. Lawvere left open problems in topos theory on his webpage, and the first problem is called quotient toposes. The main theorem of this paper provides a non-trivial example of this problem, which is not implied by any known results. This paper also provides a theoretical framework to address the open problem. We define a preorder among the objects of a Grothendieck topos (which we have named generative order), which enables us to reduce calculations of quotient toposes to calculations of objects. Our method is its application to the topos of discrete dynamical systems.

Quotient toposes of discrete dynamical systems

Abstract

This paper gives a classification of classes of discrete dynamical systems (a set equipped with an endofunction) closed under finite limits and small colimits. The conclusion is simple: they bijectively correspond to the ideals of the product poset , where the first is ordered by the usual order and the second is by the divisibility. Our method is based on a detailed analysis of the behaviors of states, especially non-periodic behaviors, in discrete dynamical systems. Specifically, extending the fundamental quantity, time until entering a loop, to even those states that do not enter a loop plays a crucial role. Our classification is closely related to epimorphisms from in the category of monoids. There are countably many injective epimorphisms from , including . Those injective epimorphisms correspond to the non-periodic behaviors of states of discrete dynamical systems. We discuss this point at the end of the paper. This fun puzzle is motivated by an open problem in topos theory. Lawvere left open problems in topos theory on his webpage, and the first problem is called quotient toposes. The main theorem of this paper provides a non-trivial example of this problem, which is not implied by any known results. This paper also provides a theoretical framework to address the open problem. We define a preorder among the objects of a Grothendieck topos (which we have named generative order), which enables us to reduce calculations of quotient toposes to calculations of objects. Our method is its application to the topos of discrete dynamical systems.
Paper Structure (38 sections, 47 theorems, 61 equations, 7 figures)

This paper contains 38 sections, 47 theorems, 61 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Main problem and its answer
  3. Relationship with monoid theory
  4. Lawvere’s open problem on quotient toposes
  5. The open problem and current situations
  6. Why discrete dynamical systems?
  7. Future works that our results suggest
  8. Technical Novelty, the generative order
  9. Preliminaries and notations
  10. Discrete dynamical systems
  11. Definition and Examples
  12. Connectedness
  13. Period and Height
  14. List of all quotients
  15. Definition of quotients
  16. ...and 23 more sections

Key Result

Proposition 3.10

For a function $\alpha\colon \overline{\mathbb{P}} \to \overline{\mathbb{N}}$, the full subcategory of $\mathrm{Set}^{{\mathbb{N}}^{\mathrm{op}}}$ that consists of a discrete dynamical system $\mathbb{X}=(X,f)$ that satisfies the following conditions is a quotient. Condition: For every $x\in X$,

Figures (7)

  • Figure 1: Associated graph of $(\mathbb{Z}/4\mathbb{Z}, - \times 2)$
  • Figure 2: Associated graph of the subobject classifier $\Omega$
  • Figure 3: Associated graph of the Collatz map
  • Figure 4: Visualization of $\mathbb{T}_{a,b}$ for some $(a,b)$
  • Figure 5: Visualized examples of period, height, and core
  • ...and 2 more figures

Theorems & Definitions (142)

  • Definition 2.1
  • Example 2.2: Canonical endomorphism
  • Example 2.3: Free dynamical system
  • Example 2.4: Monoid
  • Example 2.5: As a directed graph
  • Remark 2.6
  • Example 2.7: Subobject classifier
  • Definition 2.8: Connectedness
  • Example 2.9: Collatz map
  • Definition 2.10: Period and Height of an element
  • ...and 132 more