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\author{Partha Pratim Ghosh}
\date{\today}
\title{Notes on Sup-lattices}
\hypersetup{
 pdfauthor={Partha Pratim Ghosh},
 pdftitle={Notes on Sup-lattices},
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 pdfcreator={Emacs 27.2 (Org mode 9.4.5)}, 
 pdflang={English}}
\begin{document}

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\tableofcontents


\section{Monoidal categories}
\label{sec:org942f09d}

These notes are based on the book \cite[2-dimensional categories (OUP, 2021)]{JohnsonYau2021}.

\begin{enumerate}
\item \label{}
A \emph{monoidal category} is \(\moncat{\Bb{A}}{\otimes}{\alpha}{\lambda}{\rho}\), where:

\begin{itemize}
\item \(\Arr{\otimes}{\Bb{A} \times \Bb{A}}{\Bb{A}}\) is a bifunctor, called the
\emph{monoidal product},

\item an object \(\unit\) of \(\Bb{A}\), called the \emph{monoidal unit},

\item three natural isomorphisms --- the \emph{left unit isomorphism} \(\langle\Arr{\lambda_{X}}{\unit \otimes X}{X}: X \in \Bb{A}_0 \rangle\), the \emph{right unit isomorphism} \(\langle\Arr{\rho_{X}}{X \otimes \unit}{X}: X \in \Bb{A}_0 \rangle\) and the \emph{associativity isomorphism} \(\langle\Arr{\alpha_{X, Y, Z}}{(X \otimes Y) \otimes Z}{X \otimes (Y \otimes Z)}: X, Y, Z \in \Bb{A}_0 \rangle\),

\item these are subject to:

\[
       \label{eq:pentagon-axiom}
       \xymatrixcolsep{5.4em}
       \xymatrixrowsep{3em}
       \xymatrix{
       {((X \otimes Y) \otimes Z) \otimes W} \ar[d]_-{\alpha_{X \otimes Y, Z, W}}
       \ar[rr]^-{\alpha_{X, Y, Z}\otimes\id{W}} & &
       {(X \otimes (Y \otimes Z)) \otimes W} \ar[d]^-{\alpha_{X, Y \otimes Z, W}} \\
       {(X \otimes Y) \otimes (Z \otimes W)} \ar[r]_-{\alpha_{X, Y, Z \otimes W}} &
       {X \otimes (Y \otimes (Z \otimes W))} &
       {X \otimes ((Y \otimes Z) \otimes W)} \ar[l]^-{\id{X} \otimes \alpha_{Y, Z, W}}
       }
       \]

and

\[
       \label{eq:unity-axiom}
       \xymatrixcolsep{5.4em}
       \xymatrixrowsep{3em}
       \xymatrix{
       {(X \otimes \unit) \otimes Y} \ar[rr]^-{\alpha_{X, I, Y}}
       \ar[dr]_-{\rho_X \otimes \id{Y}} & & 
       {X \otimes (\unit \otimes Y)} \ar[dl]^-{\id{X} \otimes \lambda_Y} \\
       & {X \otimes Y}
       }.
       \]
\end{itemize}
\end{enumerate}



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