\documentclass{amsart} \usepackage[left=2cm, right=2cm]{geometry} \usepackage[utf8]{inputenc} \usepackage{amsfonts, amsthm, amssymb, mathtools,etoolbox} \usepackage{blindtext} \usepackage[colorlinks=true, urlcolor=blue, linkcolor=blue, citecolor=blue]{hyperref} \usepackage{tikz,tikz-cd} \usepackage{cleveref} \usepackage{array} \usepackage[style=alphabetic,sorting=nyt]{biblatex} \renewbibmacro{in:}{} % \addbibresource{biblio.bib} \addbibresource{CommonBiblio20240922.bib} \tikzset{pullback/.style={minimum size=1.2ex,path picture={ \draw[opacity=1,black,-,#1] (-0.5ex,-0.5ex) -- (0.5ex,-0.5ex) -- (0.5ex,0.5ex);% }}} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{todo}[theorem]{Todo} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{fact}[theorem]{Fact} \theoremstyle{definition} \newtheorem{example}[theorem]{Example} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{notation}[theorem]{Notation} \newtheorem{question}[theorem]{Question} \newtheorem{idea}[theorem]{Idea} \newcommand{\dq}[1]{``#1"} \newcommand{\memo}[1]{\textcolor{red}{memo: #1}} \newcommand{\invmemo}[1]{\textcolor{blue}{memo: #1}} \newcommand{\para}[1]{\paragraph{\textbf{#1}}} \newcommand{\N}{\mathbb{N}} \newcommand{\2}{\mathbf{2}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathcal{C}} \newcommand{\D}{\mathcal{D}} \newcommand{\E}{\mathcal{E}} \newcommand{\F}{\mathcal{F}} \newcommand{\id}{\mathrm{id}} \newcommand{\Pow}{\mathcal{P}} \newcommand{\op}{\mathrm{op}} \newcommand{\ob}{\mathrm{ob}} \newcommand{\Set}{\mathbf{Set}} \newcommand{\FinSet}{\mathbf{FinSet}} \newcommand{\dSet}{\sigma{\text{-}}\mathbf{Set}} \newcommand{\PSh}{\mathbf{PSh}} \newcommand{\dPSh}{\sigma{\text{-}}\mathbf{PSh}} \newcommand{\Sh}{\mathbf{Sh}} \newcommand{\Top}{\mathbf{Top}} \newcommand{\preTop}{\mathbf{preTop}} \newcommand{\Int}{\mathrm{Int}} \newcommand{\res}{\mathrm{res}} \newcommand{\Cont}{\mathbf{Cont}} \newcommand{\Func}[2]{[#1,#2]} \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\demph}[1]{\textbf{#1}} \font\maljapanese=dmjhira at 2.5ex \newcommand{\yo}{\textrm{\!\maljapanese\char"48}} \title{Dynamical system on a pretopological space} \author{Ryuya Hora} \thanks{ZEN University. \url{ryuya_hora@zen.ac.jp}} % \date{\today} \subjclass[2020]{MSC} \keywords{Keywords} \begin{document} \begin{abstract} \end{abstract} \maketitle \tableofcontents \begin{quote} \cite[][WHY ARE PEOPLE?]{dawkins2016selfish}: Intelligent life on a planet comes of age when it first works out the reason for its own existence. If superior creatures from space ever visit earth, the first question they will ask, in order to assess the level of our civilization, is: `Have they discovered evolution yet?' \end{quote} \section{Introduction} % The ultimate goal of this project is a mathematical formulation of the phenomenon described in the Selfish gene \cite{dawkins2016selfish}. There are already numerous mathematical models of evolution theory. However, (in terms of \cite{spivak2017categories},) \demph{structual models} are much less than \demph{numerical models}. \subsection{Speculative Motivation} One of the ultimate goals of mathematics is to elucidate (the mathematical aspects of) the phenomena that exist in the real world. Indeed, mathematics has provided fundamental tools to diverse fields such as physics, economics, and public health. However, the author cannot help but feel that the most important phenomenon, \demph{ourselves}, is being overlooked, though % That phenomenon is \textbf{ourselves}. we as human beings would be explained by a (mathematical) structure, the principle of evolutionary theory\footnote{Although this is widely recognized as a fact, it seems to be almost entirely misunderstood. Reducing our problems to evolutionary principles is not a simplistic analytical reduction, but rather something that becomes inevitable through continuous self-criticism.}. This note is the first in a series of projects aimed at providing a mathematical formalization of evolutionary theory, particularly Dawkins' concept of the `selfish gene' \cite{dawkins2016selfish}. Of course, countless mathematical models of evolutionary theory have been proposed. However, to the author's knowledge, most of them are ``numerical models" rather than ``conceptual models." \begin{center}\textbf{ Conceptual model (= structural model) vs Numerical model }\cite{spivak2017categories} \end{center} Our goal is to construct a conceptual model (= structural model) for the principle of evolutionary theory, natural selection. More colloquially, our aim is to refer to the remarkable consequence of evolution, the existence of complex creatures (or ``our existence"), from a structural perspective. More concretely, we seek to mathematically describe how complex individuals emerge solely from simple dynamical systems under the principle of natural selection. The reason why this question (i.e., describing the principles by which complex individuals emerge from dynamical systems) has not been extensively researched involves philosophical difficulties. In standard numerical models, partial differential equations are set up based on various numerical values (such as population size or the proportion of genes in the gene pool), which are then analyzed numerically. In this numerical approach, concepts like ``individual" or ``species" are presupposed (transcendently) before the partial differential equations themselves, and thus the reason for the occurrence of the phenomenon of natural selection itself is not addressed. The truly remarkable aspect of evolutionary theory lies in its structural (or qualitative) consequences—that is, the emergence of complex life from dynamical systems alone—and numerical investigations say nothing about its structural origins. By discussing evolutionary theory structurally (or more precisely, within the paradigm of modern mathematics), we aim to mathematically formalize the dynamical structure of evolutionary theory that leads to our existence in a materialistic world. The author cannot help but feel that this project is one of the most important themes in mathematics, a field that investigates the abstract structures of phenomena. \subsection{The implicit geometry (and logic)} In tackling this highly popular yet simultaneously vague problem, it is important to clarify how our approach differs from previous ones. First and foremost, we confront the fundamental issue of the emergence of individuality, and for this reason, our approach to dynamical systems is inherently geometrical. We don't ignore the geometry implicit in natural selection. As we will discuss later, the first step in the principles of evolutionary theory lies in the emergence of individuality. The emergence of individuals with temporal/spatial self-identity within a system that is merely given time evolution is far from trivial. (In fact, history has shown the danger of presupposing the concept of the individual. The errors in assuming individuality, such as organisms or species, have been pointed out in The Selfish Gene!) % We believe there is still room for geometric investigation here. The key concept is temporal/spatial self-identity. Time evolution, spatial extension, and the “logic imbued with ambiguous identity” that arises from them—I only know of one mathematical language that inherently possesses these qualities: \demph{topos}\footnote{Note to my friend: You might think that I'm trying to apply topos theory to every random theory, here in particular, evolutionary theory. % and topos theory because I like both of them. However, it’s exactly the opposite: \demph{Evolutionary theory is the reason why I chose topos theory. } % It is because I view the world through the ideas of evolutionary theory that I chose the field of topos. To explain this fully, I would need to speak seriously about my materialist and skeptical beliefs as well as my geometric convictions, something that is beyond my current writing abilities.}. Alain Connes has said that the remarkable property of topos as a space is that it is closed under monoid actions. We will challenge the geometric and dynamical systems formulation of evolutionary theory by assigning a monoid operation, in this case, time action, to space. \memo{ Why, then, is space necessary in the problem of individuality? As will be explained later, because individuals must possess the following properties, space and time are essential to the problem of "emergence of individuals": \begin{itemize} \item Continuity (Two balls moving in parallel are not swapping places every three seconds.) \item Similarity (Similarity between past and future, i.e., self-identity) \item Separability (Filling the space with matter) \end{itemize} Here, we cannot ignore the epistemological aspect, which is another principle of topos-geometry. } \subsection{First Step: Objective of this document} As the first step towards this grand speculative goal, we will begin by discussing the ``individual." This is a decisive departure from conventional numerical models. Since the structure we wish to discuss is the astonishing emergence of ourselves from a truly materialistic (flat and homogeneous) world, the existence of the ``individual" should not be an axiom but the first theorem of our theory. In other words, we aim to derive the ``individual" only from the assumption of a ``time-evolving system." The problem of the ``emergence of individuals" is, of course, a topic with a long history of research and is by no means a novel idea. A well-known example is Conway's Life Game, which serves as an important toy example. In the Life Game, space is modeled by $\Z^2$ with local rules, and even though each cell merely blinks, we can refer to `gliders moving to the upper right,' thus referring to ``individuals." First, we will consider methods to describe such discrete-time/discrete-space systems. \section{An implicit geometry behind Conway's life game} \subsection{Naive observations % : what is the geometry of Conway's life game? }\label{ssec:NaiveObservations} How can we structuralize Conway's life game? One naive idea is just consider it as an endofunction \[\Pow(\Z^2)\to \Pow(\Z^2).\] However, this does not capture the geometric aspect of the Conway's life game, so that it seems impossible to formulate that ``a glider is an indivisual." We need to capture two aspects: \begin{itemize} \item The rules are defined \demph{locally}, and \item The whole state changes \demph{globally}. \end{itemize} So another naive idea to capture the geometric aspect is to consider \demph{sheaves} on a space $\Z^2$ valued in $\2\coloneqq \{0,1\}$ \[ \Z^2 \supset U \to \2. \] But there is an obvious problem: what is the topology on $\Z^2$? The author does not think there is a suitable topology on $\Z^2$ so that we can capture the geometry of life game. (For example, if $\{(x+i,y+j)\mid -1\leq i,j, \leq 1\}$ is open for every $(x,y) \in \Z^2$, the space becomes discrete.) Next, and lastly, we consider a generalized notion of interior operator. For a subset $S\subset \Z^2$ and a configulation $S \to\2$, the next state \[ \Int(S)\to \2 \] is defined on a smaller space \[ \Int(S) \coloneqq \{(x,y)\in \Z^2\mid \forall i,j \in \{-1,0,1\}^2, (x+i,y+j)\in S \} \subset S. \] So we want to define $\Int$ to be the interior operator on the space $\Z^2$. But this is NOT idempotent $\Int\Int \neq \Int$. So this does not define a topological space, but defines a \demph{pretopological space}. \subsection{Pretopological spaces} \begin{lemma} Topologies on a set $X$ are in one-to-one correspondence with \demph{interior operators}, i.e., a function $\Int \colon \Pow(X) \to \Pow(X)$ such that \begin{description} \item[order-preserving] $S\subset T \implies \Int(S) \subset \Int(T)$ \item[lex] $\Int$ preserves finite inf. $\Int(S\cap T)=\Int(S)\cap \Int(T)$ and $\Int(X) =X$ % ($=$ binary meet and top). \item[counit] $\Int(S) \subset S$ \item[idempotent] $\Int(S) = \Int(\Int(S))$ \end{description} \end{lemma} (This is a lex comonad on $\Pow(X)$.) % \begin{remark} % A \demph{topology} (or interior operator) on a set $X$ is a function $\Int \colon \Pow(X) \to \Pow(X)$ such that % \begin{description} % \item[order-preserving] $S\subset T \implies \Int(S) \subset \Int(T)$ % \item[lex] $\Int$ preserves finite inf. $\Int(S\cap T)=\Int(S)\cap \Int(T)$ and $\Int(X) =X$ % % ($=$ binary meet and top). % \item[counit] $\Int(S) \subset S$ % \item[idempotent] $\Int(S) = \Int(\Int(S))$ % \end{description} % \end{remark} \begin{definition} A \demph{pretopology} on a set $X$ is a function $\Int \colon \Pow(X) \to \Pow(X)$ such that \begin{description} \item[order-preserving] $S\subset T \implies \Int(S) \subset \Int(T)$ \item[lex] $\Int$ preserves finite inf. $\Int(S\cap T)=\Int(S)\cap \Int(T)$ and $\Int(X) =X$ % ($=$ binary meet and top). \item[counit] $\Int(S) \subset S$ \end{description} A \demph{pretopological space} is a set $X$ equipped with a pretopology. \end{definition} (This is a lex pointed endofunctor on $\Pow(X)$.) \begin{example} Every topological space is a pretopological space. \end{example} \begin{example} The pair $(\Z^2, \Int)$ in \cref{ssec:NaiveObservations} is a pretopological space. \end{example} \begin{definition} A function between two pretopological spaces $f\colon (X, \Int_X)\to (Y, \Int_Y)$ is \demph{precontinuous} if \[ f^{-1}(\Int_Y (S)) \subset \Int_X (f^{-1}(S)) \] for every $S\subset Y$. The category of pretopological spaces and precontinuous functions is denoted by $\preTop$ \end{definition} The embedding $\Top \to \preTop$ is fully faithful. \begin{remark} The pretopological space $(\Z^2, \Int)$ is homogeneous, in the sense that the group $\Z^2$ (freely and) transitively acts on $(\Z^2, \Int)$. \end{remark} As a slogan, pretopological spaces are dynamical version of topological space, so that ``static" (i.e. $\Int\Int=\Int$) pretopological spaces are topological spaces. \memo{What is the relationship with reflexive Kripke frames? Is the ``pretopological" semantics of modal logic T complete?} \subsection{Dynamical presheaf topos on a pretopological space} \begin{definition} A \demph{dynamical presheaf}\footnote{This name might be confusing when the pretopological space is a topological space, since this does not coincide with the usual notion of presheaves.} on a pretopological space $(X, \Int)$ is a presheaf \[ F\colon \Pow(X)^{\op} \to \Set \] equipped with a family of operators \[ \{\sigma_S \colon F(S) \to F(\Int(S))\}_{S\in \Pow(X)} \] such that \[ \begin{tikzcd} F(S)\ar[d,"\res"]\ar[r,"\sigma_S"]&F(\Int(S))\ar[d,"\res"]\\ F(T)\ar[r,"\sigma_T"]&F(\Int(T)) \end{tikzcd} \] commutes. \end{definition} \begin{conjecture} The notion of dynamical presheaf is equivalent to the notion of internal presheaf over the internal poset $\Pow(X)\to \Pow(X)$ in the topos $\dSet \coloneqq \PSh(\N)$. In particular, the category of dynamical presheaves $\dPSh(X)$ is a relative topos over the base topos $\dSet$. \end{conjecture} \begin{conjecture} \[ \dPSh(X, \Int) \simeq \PSh(\Pow(X)\rtimes_{\Int} \N) \memo{opposite?} \] \end{conjecture} (See \cite{tomasic2020topos, connes2017geometry, johnstone2002sketchesv1}.) This is an example of classifying topos of primary doctrine \cite{} \begin{remark} The category $\Pow(X)\rtimes_{\Int} \N$ is defined by Grothendieck construction of the functor \[ \begin{tikzcd}[column sep = 100 pt] \N \ar[r,"{(\Pow(X), \Int)}"]& \mathbf{Cat} \end{tikzcd} \] More concretely, the objects are the same as $\Pow(X)$. A morphism from $U$ to $V$ is a non-negative integer $n$ such that \[ U \subset \Int^n (V).\] The composition is just the usual addition of integers. For example, endomorphism monoid of an object $U$ is $\N$ if $U$ is a fixed point of $\Int$, and otherwise $\{0\}$. \end{remark} So far, we have not used the properties of $\Int$, except that $\Int$ is order-preserving. \begin{example} The Conway's life game defines a dynamical presheaf over the pretopological space $\Z^2$. \end{example} \subsection{Dynamical sheaf topos on a pretopological space} Conway's life game should be a \demph{dynamical sheaf}, not only dynamical presheaf. In order to define dynamical sheaves and capture the notion of \dq{locally determined system,} We need to talk about the dynamic gluing conditions/ internal Grothendieck topology. \section{Related works} \begin{itemize} \item Sekiyama's dynamical modal logic on minesweeper \item Conway's life game \item \cite{awodey2014topos} \item Selfish gene \end{itemize} \printbibliography \end{document}