#+title: Notes on Sup-lattices
#+author: Partha Pratim Ghosh
#+options:toc:nil
#+latex_header: \usepackage{Documents/tex/essentials/symbols}
#+latex_header: \usepackage[left=2.82cm,top=2.22cm,right=2.82cm,bottom=2.22cm]{geometry}
#+latex_header: \usepackage{parskip}
#+latex_header: \setlength{\parindent}{0em} \setlength{\parskip}{0.6em}
#+latex_header: \usepackage[all,pdf]{xy}
#+latex_header:\usepackage[backend=biber,backref=true, bibencoding=utf8,  sorting=nyt]{biblatex}
#+bibliography:{ ~/Documents/tex/essentials/bib-algebra.bib,~/Documents/tex/essentials/bib-category.bib,~/Documents/tex/essentials/bib-frames.bib,~/Documents/tex/essentials/bib-topology.bib,~/Documents/tex/essentials/bib-self.bib}
#+bibliographystyle:amsplain

* Monoidal categories

  These notes are based on the book [[cite:JohnsonYau2021][2-dimensional categories (OUP, 2021)]].
  
  1. [[label:][monoidal category definition]]
     A /monoidal category/ is $\moncat{\Bb{A}}{\otimes}{\alpha}{\lambda}{\rho}$, where:

     + $\Arr{\otimes}{\Bb{A} \times \Bb{A}}{\Bb{A}}$ is a bifunctor, called the
       /monoidal product/,

     + an object $\unit$ of $\Bb{A}$, called the /monoidal unit/,

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

     + 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

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



bibliography:~/Documents/tex/essentials/bib-algebra.bib,~/Documents/tex/essentials/bib-category.bib,~/Documents/tex/essentials/bib-frames.bib,~/Documents/tex/essentials/bib-topology.bib,~/Documents/tex/essentials/bib-self.bib

\printbibliography