\section{Riegs of Structures}
\subsection{Bicartesian closed category forms a rieg}
We have seen that every Heyting algebra is a rieg. More generally, in this section, we will observe that every \emph{bicartesian closed category} gives an example of riegs.

\begin{definition}[Bicartesian closed category]
    A category $\C$ is \emph{bicartesian closed},
    if $\C$ is 
    \begin{itemize}
        \item bicartesian (i.e., has all finite products and finite coproducts), and
        \item cartesian closed.
    \end{itemize}
\end{definition}

Examples include the category of sets, finite sets, posets, categories, groupoids, directed graphs, and group actions. Every topos is bicartesian closed.

We will see those examples in the following subsections.

\begin{proposition}\label{PropositionbiCCC}
    For a bicartesian closed category $\C$, the set of all isomorphism classes $\K{\C}$ equipped with the five operations induced from bicartesian closed structure, is a rieg.
\end{proposition}

\begin{remark}[Size matter]
    
\end{remark}

\memo{Explain, why not monoidal, but cartesian}

\memo{For bicartesian closed bicategories? finite groupoids, finite groups}

\memo{Rieg of species? Differential operator?}

\subsection{Connectedness and augmentation: enumerative comnibatorics}
\subsection{Riegs of sets}
\subsection{Riegs of functions: Liouville's divisor theorem}

\begin{example}[Multisets of natural numbers]
    Let $\mN$ be a set of all multiset of natural numbers. Each multiset will be denoted just like lists of natural numbers. For example, \[(), (3,1,4), (1,2,3,1,1,0,0) \in \mN.\] Note here that the list may be rearranged as desired, \[(1,2,3,1,1,0,0)  = (0,0,1,1,1,2,3)\in \mN\]

    This set of multisets has a standard rieg structure.
    The addition is concatenations of lists. The product is a component-wise \memo{confusing} product.
    \begin{align*}
        (3,1,4)+ (1,2,7,0)&=(3,1,4,1,2,7,0)\\
        (3,1,4)\ti (1,2,7,0)&= (3,6,21,0,1,2,7,0,4,8,28,0)
        % (3\ti 1,3\ti 2,3\ti 7, 3\ti 0, 1\ti 1, 1\ti 2, 1\ti 7, 1\ti 0, 4\ti 1, 4\ti 2, 4\ti 7, 4\ti 0)
    \end{align*}

    This rig structure is isomorphic to the rig of (formal) Dirichlet polynomials $\N[\frac{1}{n^x}\mid n=0,1,2,\dots]$ \cite{spivak2020dirichlet}. However, exponentials are different:
    \[(n_1, \dots,n_k)^{(m)}=({n_1}^m  , \dots, {n_k}^m ).\]
    There is a unique extension of this exponential to all exponentials. For example,
     \begin{align*}
        (3,1,4)^{ (2,0)} &= (3,1,4)^{ (2)} (3,1,4)^{ (0)} \\
        &=(9,1,16)(1,1,1)\\
        &=(9,9,9,1,1,1,16,16,16).
    \end{align*}
\end{example}
\memo{Write the generalization and $\mult{1}$}

\subsection{Riegs of graphs}
\subsection{Riegs of group actions: Burnside riegs}
\memo{rieg homomorphism to $\N$, which induced by atomic geometric morphism}
\paragraph{Rieg of involutions}
$A^x = n + \frac{n^2 -n}{2} x$, where $n= \#A$.
\subsection{Riegs of loops: Counting repeating dicimals}


\subsection{Riegs of Higher structures}
% \subsection{Bicartesian closed bicategory}
\subsection{Riegs of categories}
\subsection{Riegs of finite groups}
Since the category of finite groupoids $\Groupoidfin$ is bicartesian closed, we have the riegs of finite groupoids, $\K{\Groupoidfin}$.

Furthermore, since all five operations are compatible with equivalences, we have its quotient $\biK{\Groupoidfin}$.

\memo{Using the rieg structure, compute the number of subgroups of order $2$, faster!}

\memo{compute $n$-dimensional representation of a product group}


\subsection{Functorial construction of riegs}
\subsection{Functor from a locally bicartesian closed category}
\subsection{Examples of induced rieg homomorphisms}
\subsection{Categories and discrete fibrations}
\memo{categorification of the multiset construction}
\subsection{Topological spaces and \'etale maps}
\subsection{Quasitoposes}
\memo{Heyting algebra}
\cite{nlab:quasitopos}
\begin{definition}
    
\end{definition}
\begin{example}
    The category of (finite) categories and (finite) discrete fibrations. 
\end{example}
\memo{When is $\FinSet^{\C}$ a topos? \url{https://ncatlab.org/nlab/show/category+of+presheaves\#finite_presheaves}}

\begin{example}
    The category of topological spaces and local homeomorphisms.
\end{example}



% \section{Rieg of multisets, Liouville's divisor theorem}
% \section{Riegs from toposes, repeating decimal}
\memo{CC and CCC functor, \'etendue, logical morphism, directed graph, and group actions}