temporarily-removed-sections.tex

\section{Miscellany}\label{section-miscellany}
\subsection{Concrete Categories}\label{subsection-concrete-categories}
\begin{definition}{Concrete Categories}{concrete-categories}%
    A category $\CatFont{C}$ is \index[categories]{category!concrete}\textbf{concrete} if there exists a faithful functor $F\colon\CatFont{C}\to\Sets$.
\end{definition}
\subsection{Balanced Categories}\label{subsection-balanced-categories}
\begin{definition}{Balanced Categories}{balanced-categories}%
    A category is \index[categories]{category!balanced}\textbf{balanced} if every morphism which is both a monomorphism and an epimorphism is an isomorphism.
\end{definition}
\subsection{Monoid Actions on Objects of Categories}\label{subsection-monoid-actions-on-objects-of-categories}
Let $A$ be a monoid, let $\CatFont{C}$ be a category, and let $X\in\Obj(\CatFont{C})$.
\begin{definition}{Monoid Actions on Objects of Categories}{monoid-actions-on-objects-of-categories}%
    An \index[categories]{monoid action!on an object of a category}\textbf{$A$-action on $X$} is a functor $\lambda\colon\B{A}\to\CatFont{C}$ with $\lambda(\point)=X$.
\end{definition}
\begin{remark}{Unwinding \cref{monoid-actions-on-objects-of-categories}}{unwinding-monoid-actions-on-objects-of-categories}%
    In detail, an \textbf{$A$-action on $X$} is an $A$-action on $\sfEnd_{\CatFont{C}}(X)$, consisting of a morphism
    \[
        \lambda
        \colon
        A
        \to
        \underbrace{\sfEnd_{\CatFont{C}}(X)}_{\defeq\Hom_{\CatFont{C}}(X,X)}
    \]%
    satisfying the following conditions:
    \begin{enumerate}
        \item\SloganFont{Preservation of Identities. }We have
            \[
                \lambda_{1_{A}}
                =
                \id_{X}.
            \]%
        \item\SloganFont{Preservation of Composition. }For each $a,b\in A$, we have
            \begin{webcompile}
                \lambda_{b}\circ\lambda_{a}
                =
                \lambda_{ab},
                \quad
                \begin{tikzcd}[row sep={5.0*\the\DL,between origins}, column sep={5.0*\the\DL,between origins}, background color=backgroundColor, ampersand replacement=\&]
                    X
                    \arrow[r,"\lambda_{a}"]
                    \arrow[rd,"\lambda_{ab}"']
                    \&
                    X
                    \arrow[d,"\lambda_{b}"]
                    \\
                    \&
                    X\mrp{.}
                \end{tikzcd}
            \end{webcompile}
    \end{enumerate}
\end{remark}
\subsection{Group Actions on Objects of Categories}\label{subsection-group-actions-on-objects-of-categories}
Let $G$ be a group, let $\CatFont{C}$ be a category, and let $X\in\Obj(\CatFont{C})$.
\begin{definition}{Group Actions on Objects of Categories}{group-actions-on-objects-of-categories}%
    A \index[categories]{group action!on an object of a category}\textbf{$G$-action on $X$} is a functor $\lambda\colon\B{G}\to\CatFont{C}$ with $\lambda(\point)=X$.
\end{definition}
\begin{remark}{Unwinding \cref{group-actions-on-objects-of-categories}}{unwinding-group-actions-on-objects-of-categories}%
    In detail, a \textbf{$G$-action on $X$} is a $G$-action on $\sfAut_{\CatFont{C}}(X)$, consisting of a morphism
    \[
        \lambda
        \colon
        G
        \to
        \underbrace{\sfEnd_{\CatFont{C}}(X)}_{\defeq\Hom_{\CatFont{C}}(X,X)}
    \]%
    satisfying the following conditions:
    \begin{enumerate}
        \item\SloganFont{Preservation of Identities. }We have
            \[
                \lambda_{1_{A}}
                =
                \id_{X}.
            \]%
        \item\SloganFont{Preservation of Composition. }For each $a,b\in A$, we have
            \begin{webcompile}
                \lambda_{b}\circ\lambda_{a}
                =
                \lambda_{ab},
                \quad
                \begin{tikzcd}[row sep={5.0*\the\DL,between origins}, column sep={5.0*\the\DL,between origins}, background color=backgroundColor, ampersand replacement=\&]
                    X
                    \arrow[r,"\lambda_{a}"]
                    \arrow[rd,"\lambda_{ab}"']
                    \&
                    X
                    \arrow[d,"\lambda_{b}"]
                    \\
                    \&
                    X\mrp{.}
                \end{tikzcd}
            \end{webcompile}
    \end{enumerate}
\end{remark}