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\begin{document}

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\section*{poly-morphism}

\begin{quote}%
This page is about the notion of poly-morphism considered by [[Shinichi Mochizuki]]. For the concept in [[computer science]] see [[polymorphism]]. Or see [[universe polymorphism]] for the concept in [[type theory]].


\end{quote}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{abstract_approach}{Abstract approach}\dotfill \pageref*{abstract_approach} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For a [[category]] $C$, a \textbf{poly-morphism} is a collection of [[morphisms]] with common [[source]] and [[target]], considered as a morphism in a new category $C^{poly}$ with the same [[objects]], but with [[hom-sets]] $C^{poly}(a,b) \coloneqq P(C(a,b))$, the [[power set]] of the original hom-sets.

These play a substantive rôle in [[Shinichi Mochizuki]]`s [[inter-universal Teichmüller theory]] (\hyperlink{Mochizuki12}{Mochizuki 12, section 0}). Here we put these into the broader context of [[change of enriching categories]] along [[lax monoidal functors]] $F$ as the special case that $F$ is a [[power set]]-functor, or an [[G-set|equivariant]] version thereof.

\hypertarget{abstract_approach}{}\subsection*{{Abstract approach}}\label{abstract_approach}

In the following, fix a [[monoidal category]] $V$ and a $V$-[[enriched category]]

\begin{displaymath}
C \in V Cat
  \,.
\end{displaymath}
Examples to keep in mind for $V$ are the category [[Set]] of sets considered as a [[cartesian monoidal category]], or the category [[G-set]] for a given [[group]] $G$, again with the cartesian monoidal structure.

Recall:

\begin{defn}
\label{ChangeOfEnrichingCategory}\hypertarget{ChangeOfEnrichingCategory}{}
\textbf{([[change of enriching category]] from $V$ to itself)}

For $F\colon V\to V$ a [[lax monoidal functor]] from $V$ to itself, the image $F_\ast(C)$ of $C$ under the corresponding [[change of enriching category]] is the $V$-category with the same [[objects]] as $C$, and with $V$-[[hom objects]] given by

\begin{displaymath}
F_\ast(C)(a,b) \coloneqq F(C(a,b))
  \,.
\end{displaymath}
The [[composition]]-, [[unit]]-, [[associator]]- and [[unitor]]-[[morphisms]] of $F_\ast(C)$ are the [[images]] of those of $C$ under $P$ suitably composed with the structure morphisms of the lax monoidal functor $P$.

\end{defn}
For the case of poly-morphisms the relevant [[lax monoidal functors]] are [[power set]]-functors

\begin{example}
\label{PowerSetFunctors}\hypertarget{PowerSetFunctors}{}
\textbf{([[lax monoidal functor|lax monoidal]] [[power set]]-[[functors]])}

\begin{itemize}%
\item For $V=$[[Set]] equipped with its [[cartesian monoidal category]] [[structure]], $C$ an arbitrary category, take $F \coloneqq P \;\colon\; Set \to Set$ the covariant [[power set]] functor. This is lax monoidal since every [[pair]] of [[subsets]] $U \subseteq S$ and $V\subseteq T$ gives a subset $U\times V \subseteq S\times T$.


\item For $G$ a group, $V=$[[GSet]] with the [[cartesian monoidal category]] [[structure]], $C$ a [[enriched category|category enriched in]] [[G-sets]], let $F \coloneqq P\;\colon\; GSet \to GSet$ also be the covariant [[power set]] functor, but for $S$ a $G$-set, equip $P(S)$ with the $G$-[[action]]

\begin{displaymath}
U\mapsto g\cdot U 
= 
\{g\cdot x \in S \mid x\in U\}
\,.
\end{displaymath}


\end{itemize}
\end{example}
\begin{defn}
\label{PolyMorphisms}\hypertarget{PolyMorphisms}{}
\textbf{(poly-morphisms)}

For $C$ a plain category and $F \coloneqq P \colon Set \to Set$ the [[power set]]-functor (Def. \ref{PowerSetFunctors}), we write

\begin{displaymath}
C^{poly}
  \;\coloneqq\;
   P_\ast(C)
\end{displaymath}
for the image of $C$ under the [[change of enriching category]] (Def. \ref{ChangeOfEnrichingCategory}) along $P$.

We say that the [[morphisms]] of $C^{poly}$ are the \textbf{poly-morphisms} of $C$.

Explicitly, this means [[composition]] is defined to be

\begin{displaymath}
C^{poly}(a,b) \otimes C^{poly}(b,c) = P(C(a,b)) \otimes P(C(b,c)) \to P(C(a,b)\otimes B(b,c)) \to P(C(a,c)) = C^{poly}(a,c)
\end{displaymath}
and the [[unit]] map is

\begin{displaymath}
I \to P(I) \to P(C(a,a)) = C^{poly}(a,a)
  \,.
\end{displaymath}
A \textbf{poly-isomorphism} of $C$ is defined to be a poly-morphism of the [[core]] of $C$, hence a morphism of $(Core(C))^{poly}$. Hence a poly-isomorphism is a collection of \emph{[[invertible morphisms]]} of $C$. (Note that these are \textbf{not} in general the [[isomorphisms]] in $C^{poly}$: all the isomorphisms in $C^{poly}$ are actual isomorphisms in $C$.)

\end{defn}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The construction for enrichment in plain sets is considered (without the category-theoretic formulation) in:

\begin{itemize}%
\item [[Shinichi Mochizuki]], section 0 of \emph{Inter-universal Teichm\"u{}ller theory I, Construction of Hodge theaters} (2012) (\href{http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20I.pdf}{pdf})

\end{itemize}
The construction for enrichment in [[posets]] is considered in

\begin{itemize}%
\item Alveen Chand, [[Ittay Weiss]], \emph{An ordered framework for partial multivalued functors}, Computer Science and Engineering (APWC on CSE), 2015 2nd Asia-Pacific World Congress on Computer Science (\href{https://arxiv.org/abs/1511.00746}{arXiv:1511.00746})

\end{itemize}
[[!redirects poly-morphisms]]



\end{document}