← A topos theoretic view of Representation theory
Old Versions__Ivan_Ver__Ivan_Ver.tex
\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{array}
\usepackage[style=alphabetic,sorting=nyt]{biblatex}
\renewbibmacro{in:}{}
\addbibresource{biblio.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{conjecture}[theorem]{Conjecture}
\newtheorem{todo}[theorem]{Todo}
\theoremstyle{definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\dq}[1]{``#1"}
\newcommand{\memo}[1]{\textcolor{red}{memo: #1}}
\newcommand{\invmemo}[1]{\textcolor{blue}{memo: #1}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\M}{\mathbb{M}}
\newcommand{\D}{\mathcal{D}}
\newcommand{\E}{\mathcal{E}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\id}[1]{\mathrm{id}_{#1}}
\newcommand{\Set}{\mathrm{Set}}
\newcommand{\Fun}[2]{[#1,#2]}
\newcommand{\Sh}[1]{\mathrm{Sh}_{#1}}
\newcommand{\Vect}[1]{\mathrm{Vect}_{#1}}
\newcommand{\CMod}{\C\mathrm{Mod}}
\newcommand{\Conti}{\mathrm{Top}}
\title{A topos theoretic view of Representation theory\memo{Mada kanari metyakutya!!}}
% \title{A possibility of a topos-theoretic representation theory}
\author{Ryuya Hora}
\thanks{Graduate School of Mathematical Sciences, University of Tokyo. \url{hora@ms.u-tokyo}}
% \date{\today}
\subjclass[2020]{MSC}
\keywords{Keywords}
\begin{document}
\maketitle
\begin{abstract}
This note aims to roughly summarize and share the author's ideas about the intriguing potential relationship between topos-internal logic and representation theory.
\end{abstract}
\tableofcontents
\section{Informal Idea}
This article is based on the idea that the following two theories may correspond:
\begin{itemize}
\item Inverse mathematics of linear algebra based on constructive (or topos-internal) mathematics
\item Various properties of the category of representations of groups, monoids, quivers, categories, etc.
\end{itemize}
The starting point is the following lemma:
\begin{lemma}
For a small category $\D$, the category of internal $\C$-vector spaces of the topos $\Fun{\D}{\Set}$ is equivalent to the catgeory of $\D$-representations $\Fun{\D}{\Vect{\C}}$.
\end{lemma}
In other words, the category of representations of groups (, monoids, quivers, categories)
% as (C-)representations
is to be considered as the category of internal vector spaces of the group (, monoid, quiver, category) action topos. Through this equivalence of the categories, the logical properties of the corresponding topos and the properties of representations might correspond.
For example, if we could prove a property of $\C$-vector spaces $P$ is equivalent to the excluded middle, (with the base theory of topos-internal mathematics,) we would obtain the equivalence between
\begin{enumerate}
\item The excluded middle is valid in the topos $\Fun{\D}{\Set}$.
\begin{itemize}
\item $\iff$ The topos $\Fun{\D}{\Set}$ is boolean.
\item $\iff$ The category $\D$ is a groupoid.
\end{itemize}
\item $\D$-representations have the property $P$.
\end{enumerate}
% \section{Optimistic Goals}
% Some goals are listed as follows:
% \begin{enumerate}
% \item \textbf{Complete reducibility}: In the representations of finite groups, all indecomposable representations are irreducible. This is, of course, true in ordinary ($\Set$-)linear algebra as well. (In the usual linear algebra, a vector space is indecomposable iff irreducible iff one-dimensional.)
% % The usual proof is to pick an element "$x$" not included in a subrepresentation \(V\) and consider the subspace spanned by \(x\), repeating this operation. \memo{Maybe no? Quite suspicious.}
% However, in the representation theory of the monoid \(\mathbb{N}\), complete reducibility does not hold. (Instead, it's well-known that they are classified by Jordan's normal form). Could this be related to the fact that the law of excluded middle does not hold in the topos \(\Set^{\mathbb{N}}\)? In \(\Set^{\mathbb{N}}\), instead of a "Yes" or "No" response to the predicate whether an element \(x\) is included in a subrepresentation \(V\), one might answer "Yes in 3 seconds." That is, could the existence of the law of excluded middle (being a Boolean topos) be essential for complete reducibility?
% \begin{conjecture}
% Within the framework of topos internal, using the law of excluded middle, one can prove that directly irreducible vector spaces are irreducible.
% \end{conjecture}
% \begin{conjecture}
% From this, complete reducibility holds immediately in the category of internal vector spaces of a Boolean topos.
% \end{conjecture}
% \begin{conjecture}
% From this, the complete reducibility of group representations can be proven immediately.
% \end{conjecture}
% \item \textbf{Krull-Schmidt property}: In the topos internal context, when can one prove "a vector space can be directly irreducibly decomposed"?
% \begin{conjecture}
% Under the finiteness of \(C\), Krull-Schmidt can be constructively stated almost unconditionally.
% \end{conjecture}
% \begin{conjecture}
% From this, Krull-Schmidt for the category of representations follows immediately.
% \end{conjecture}
% \item \textbf{Gabriel's Theorem}: What are the logical conditions necessary to show "there are only finitely many irreducible representations"?
% \begin{conjecture}
% The logical axiom \(Ax\) that is exactly necessary to show "there are only finitely many irreducible representations" exists, and the validity of \(Ax\) in the action topos \(PSh(Q)\) of a quiver \(Q\) is equivalent to \(Q\) being a Dynkin diagram.
% \end{conjecture}
% \begin{conjecture}
% This provides (1) a topos-theoretic proof of Gabriel's Theorem, (2) a categorical logical characterization of Dynkin diagrams, and (3) a topos-theoretic extension concept of Dynkin diagrams.
% \end{conjecture}
% \end{enumerate}
% \begin{remark}
% Some kind of finiteness may be necessary.
% \end{remark}
% \begin{remark}
% Gabriel's Theorem is quite a fanciful tale.
% \end{remark}
\section{Definitions}
\begin{definition}[Internal ring of complex numbers, and its internal modules]\label{DefinitionInternalComplexNumber}
For a Grothendieck topos $\E$,
\begin{itemize}
\item $\R_{\E}$ denotes the internal ring of Dedekind reals. See \cite[][Section VI.8]{maclane1994sheaves}.
\item $\C_{\E}$ denotes the internal ring of complex numbers, defined by $\C_{\E} \coloneqq \R_{\E}\times \R_{\E}$ with usual operations.
% \item Internal $\C$-module is an module object in $\E$. That is an abelian group object $\M=(M, 0, +, -)$ equipped with an $\C_{\E}$-action morphism $\C_{\E}\times M \to M$ that satisfies the usual axioms.
\item Internal $\C$-module is an module object in $\E$. That is an object $M$ equipped with an abelian group structure $0,+,-$ and $\C_{\E}$-action morphism $\C_{\E}\times M \to M$ that satisfy the usual axioms. An internal $\C_{\E}$-module homomorphism is a morphism in $\E$ that satisfies the usual properties. \memo{The object of internal homomorphisms has much more information.}
\item $\CMod(\E)$ denotes the \memo{$\Set$-enriched} category of internal $\C_{\E}$-modules and their homomorphisms.
\end{itemize}
% $\C_{\E}$ denotes the product of
\end{definition}
\begin{example}[Complex numbers in a presheaf topos]\label{ExampleCinPresheaf}
In a presheaf topos $\E = \Fun{\D}{\Set}$, the complex number object $\C_{\E}$ is (isomorphisc to) the constant presheaf $\Delta_{\C}\colon \D \to \Set$.
% In particular, if $\D$ is a group, $\C_{\E}$ is the trivial representation
\end{example}
\begin{example}[Complex numbers in a sheaf topos over a topological space]\label{ExampleCinTopsp}
In a sheaf topos $\E = \Sh{X}$ over a topological space $X$, the complex number object $\C_{\E}$ is (isomorphic to) the sheaf of rings of $\C$-valued continuous functions $U \mapsto \Conti (U,\C)$.
% In particular, if $\D$ is a group, $\C_{\E}$ is the trivial representation
\end{example}
\begin{todo}
Calculate the internal statement \dq{A module $M$ is finitely generated}. And state Gabriel's theorem in topos-theoretic terminology.
\end{todo}
\section{Obstacles}
\begin{proposition}
An indecomposable representation of an infinite group is not necessarily irreducible, even if it is finite-dimensional.
% There is a finite-dimensional indecomposable representation of $\Z$ that is not irreducible.
\end{proposition}
\begin{proof}
The group homomorphism
\[\Z \ni n \mapsto
\begin{pmatrix}
1 & n \\
0 & 1 \\
\end{pmatrix}
\in \mathrm{GL}_{2}(\C)
\]
gives a $2$-dimensional representation of $\Z$. The subspace $\C \times \{0\}$ is a $1$-dimensional subrepresentation.
\end{proof}
\begin{proposition}
There is a Grothendieck topos $\E$ in which the complex number object $\C_{\E}$ is not a (discrete) field:
\[\C_{\E}\not \vdash \forall x \in \C_{\E}, (x=0 \lor \exists y \in \C_{\E}, xy = 1).\]
\end{proposition}
\printbibliography
\end{document}