\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}
\begin{conjecture}
    For a small category $\D$, its 
\end{conjecture}