← A topos theoretic view of Representation theory
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\title{A topos theoretic view of Representation theory}
% \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}}
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\subjclass[2020]{MSC}
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\begin{document}
\maketitle
\begin{abstract}
This note aims to broadly summarize and share the author's ideas about the intriguing potential relationship between topos-internal logic and representation theory.
\end{abstract}
\tableofcontents
\section{Informal Ideas}
This article is based on the idea that the following two theories correspond:
\begin{itemize}
\item Inverse mathematics of linear algebra based on constructive 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 category \(\C\), the category of internal vector spaces of the topos \(\Set^{\C^{\mathrm{op}}}\) coincides with the catgeory of $\C$-representations \(\mathrm{Vect}_{\mathbb{C}}^{\C^{\mathrm{op}}}\).
\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. With this, we want to correspond the logical properties of the corresponding topos with the properties of the category of representations.
Some goals are listed as follows:
\begin{enumerate}
\item \textbf{Complete reducibility}: In the representations of 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 is well-known to be classified by Jordan 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{Tasks}
Our first task may be to prove (or disprove) the following conjecture
% \begin{conjecture}[False]
% For a Grothendieck topos $\E$, the following conditions are equivalent:
% \begin{itemize}
% \item The excluded middle is valid in $\E$, i.e., $\E$ is boolean.
% \item The category of internal $\C$-vector spaces $\E$ has complete reducibility.
% \end{itemize}
% \end{conjecture}
\begin{conjecture}
For a Grothendieck topos $\E$, the following conditions are equivalent:
\begin{itemize}
\item The excluded middle is valid in $\E$, i.e., $\E$ is boolean.
\item Internal $\mathbb{C}$-vector spaces are internally completely reducible.
\end{itemize}
\end{conjecture}
\begin{conjecture}
For a finite category $\C$ and its presheaf topos $\E$, the following conditions are equivalent:
\begin{itemize}
\item The excluded middle is valid in $\E$, i.e., $\E$ is boolean.
\item The category of internal $\C$-vector spaces $\E$ has complete reducibility.
\end{itemize}
\end{conjecture}
\section{internal linear algebra}
\printbibliography
\end{document}