\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{applied category theory}

\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{references}{References}\dotfill \pageref*{references} \linebreak


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

The whole point of [[category theory]] is to study fundamental [[general abstract]] patterns and phenomena that (re-)appear throughout [[mathematics]]. Hence applications of [[theorems]] of category theory are ubiquituous in [[mathematics]] and in subjects with a mathematical basis, such as [[physics]] and [[computer science]]. Often this goes without saying.

In the introduction to \hyperlink{Bradley18}{Bradley 18} it says:

\begin{quote}%
\ldots{} ideas and results from category theory have found applications in computer science and quantum physics (not to mention pure mathematics itself), but these are not the only applications to which the word \emph{applied} in \emph{applied category theory} is being applied\ldots{}

category theory has found applications in a wide range of disciplines outside of pure mathematics—even beyond the closely related fields of computer science and quantum physics. These disciplines include chemistry, neuroscience, systems biology, natural language processing, causality, network theory, dynamical systems, and database theory to name a few. And what do they all have in common? \ldots{} In other words, the techniques, tools, and ideas of category theory are being used to identify recurring themes across these various disciplines with the purpose of making them a little more formal.


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

\begin{itemize}%
\item [[Tai-Danae Bradley]], \emph{What is applied category theory?}, \href{https://arxiv.org/abs/1809.05923}{arXiv:1809.05923}


\item [[Brendan Fong]] and [[David I Spivak]], \emph{Seven Sketches in Compositionality: An Invitation to Applied Category Theory}, \href{https://arxiv.org/abs/1803.05316}{arXiv:1803.05316}.



\end{itemize}
[[John Baez]] ran a series of lectures based on this book:

\begin{itemize}%
\item lectures on chapter 1

\begin{itemize}%
\item \href{https://forum.azimuthproject.org/discussion/1807/lecture-1-introduction/p1}{Lecture 1 - Introduction}
\item \href{https://forum.azimuthproject.org/discussion/1808/lecture-2-what-is-applied-category-theory/p1}{Lecture 2 - What is Applied Category Theory?}
\item \href{https://forum.azimuthproject.org/discussion/1812/lecture-3-chapter-1-posets/p1}{Lecture 3 - Chapter 1: Preorders}
\item \href{https://forum.azimuthproject.org/discussion/1828/lecture-4-chapter-1-galois-connections/p1}{Lecture 4 - Chapter 1: Galois Connections}
\item \href{https://forum.azimuthproject.org/discussion/1845/lecture-5-chapter-1-galois-connections/p1}{Lecture 5 - Chapter 1: Galois Connections}
\item \href{https://forum.azimuthproject.org/discussion/1901/lecture-6-chapter-1-computing-adjoints/p1}{Lecture 6 - Chapter 1: Computing Adjoints}
\item \href{https://forum.azimuthproject.org/discussion/1909/lecture-7-chapter-1-logic/p1}{Lecture 7 - Chapter 1: Logic}
\item \href{https://forum.azimuthproject.org/discussion/1921/lecture-8-chapter-1-the-logic-of-subsets/p1}{Lecture 8 - Chapter 1: The Logic of Subsets}
\item \href{https://forum.azimuthproject.org/discussion/1931/lecture-9-chapter-1-adjoints-and-the-logic-of-subsets/p1}{Lecture 9 - Chapter 1: Adjoints and the Logic of Subsets}
\item \href{https://forum.azimuthproject.org/discussion/1963/lecture-10-the-logic-of-partitions/p1}{Lecture 10 - Chapter 1: The Logic of Partitions}
\item \href{https://forum.azimuthproject.org/discussion/1991/lecture-11-chapter-1-the-poset-of-partitions/p1}{Lecture 11 - Chapter 1: The Poset of Partitions}
\item \href{https://forum.azimuthproject.org/discussion/1999/lecture-12-chapter-1-generative-effects/p1}{Lecture 12 - Chapter 1: Generative Effects}
\item \href{https://forum.azimuthproject.org/discussion/2008/lecture-13-chapter-1-pulling-back-partitions/p1}{Lecture 13 - Chapter 1: Pulling Back Partitions}
\item \href{https://forum.azimuthproject.org/discussion/2013/lecture-14-adjoints-joins-and-meets/p1}{Lecture 14 - Chapter 1: Adjoints, Joins and Meets}
\item \href{https://forum.azimuthproject.org/discussion/2027/lecture-15-chapter-1-preserving-joins-and-meets/p1}{Lecture 15 - Chapter 1: Preserving Joins and Meets}
\item \href{https://forum.azimuthproject.org/discussion/2031/lecture-16-chapter-1-the-adjoint-functor-theorem-for-posets/p1}{Lecture 16 - Chapter 1: The Adjoint Functor Theorem for Posets}
\item \href{https://forum.azimuthproject.org/discussion/2037/lecture-17-chapter-1-the-grand-synthesis/p1}{Lecture 17 - Chapter 1: The Grand Synthesis}

\end{itemize}

\item lectures on chapter 2

\begin{itemize}%
\item \href{https://forum.azimuthproject.org/discussion/2075/lecture-18-chapter-2-resource-theories/p1}{Lecture 18 - Chapter 2: Resource Theories}
\item \href{https://forum.azimuthproject.org/discussion/2079/lecture-19-chapter-2-chemistry-and-scheduling/p1}{Lecture 19 - Chapter 2: Chemistry and Scheduling}
\item \href{https://forum.azimuthproject.org/discussion/2081/lecture-20-chapter-2-manufacturing/p1}{Lecture 20 - Chapter 2: Manufacturing}
\item \href{https://forum.azimuthproject.org/discussion/2082/lecture-21-chapter-2-monoidal-preorders/p1}{Lecture 21 - Chapter 2: Monoidal Preorders}
\item \href{https://forum.azimuthproject.org/discussion/2084/lecture-22-chapter-2-symmetric-monoidal-preorders/p1}{Lecture 22 - Chapter 2: Symmetric Monoidal Preorders}
\item \href{https://forum.azimuthproject.org/discussion/2086/lecture-23-chapter-2-commutative-monoidal-posets/p1}{Lecture 23 - Chapter 2: Commutative Monoidal Posets}
\item \href{https://forum.azimuthproject.org/discussion/2089/lecture-24-chapter-2-pricing-resources/p1}{Lecture 24 - Chapter 2: Pricing Resources}
\item \href{https://forum.azimuthproject.org/discussion/2090/lecture-25-chapter-2-reaction-networks/p1}{Lecture 25 - Chapter 2: Reaction Networks}
\item \href{https://forum.azimuthproject.org/discussion/2095/lecture-26-chapter-2-monoidal-monotones/p1}{Lecture 26 - Chapter 2: Monoidal Monotones}
\item \href{https://forum.azimuthproject.org/discussion/2098/lecture-27-chapter-2-adjoints-of-monoidal-monotones/p1}{Lecture 27 - Chapter 2: Adjoints of Monoidal Monotones}
\item \href{https://forum.azimuthproject.org/discussion/2105/lecture-28-chapter-2-ignoring-externalities/p1}{Lecture 28 - Chapter 2: Ignoring Externalities}
\item \href{https://forum.azimuthproject.org/discussion/2121/lecture-29-chapter-2-enriched-categories/p1}{Lecture 29 - Chapter 2: Enriched Categories}
\item \href{https://forum.azimuthproject.org/discussion/2124/lecture-30-chapter-1-preorders-as-enriched-categories/p1}{Lecture 30 - Chapter 2: Preorders as Enriched Categories}
\item \href{https://forum.azimuthproject.org/discussion/2128/lecture-31-chapter-2-lawvere-metric-spaces/p1}{Lecture 31 - Chapter 2: Lawvere Metric Spaces}
\item \href{https://forum.azimuthproject.org/discussion/2169/lecture-32-chapter-2-enriched-functors/p1}{Lecture 32 - Chapter 2: Enriched Functors}
\item \href{https://forum.azimuthproject.org/discussion/2192/lecture-33-chapter-2-tying-up-loose-ends/p1}{Lecture 33 - Chapter 2: Tying Up Loose Ends}

\end{itemize}

\item [[David I Spivak]], \emph{Category theory for the sciences.} MIT Press, 2014.


\item [[Brandon Coya]], \emph{Circuits, Bond Graphs, and Signal-Flow Diagrams: A Categorical Perspective}, \href{https://arxiv.org/abs/1805.08290}{arXiv:1805.08290}


\item [[John Baez]] and [[Brendan Fong]], \emph{A Compositional Framework for Passive Linear Networks}, \href{https://arxiv.org/abs/1504.05625}{arXiv:1504.05625}


\item [[Blake Pollard]], \emph{Open Markov processes: A compositional perspective on non-equilibrium steady states in biology}, \href{https://arxiv.org/abs/1601.00711}{arXiv:1601.00711}


\item [[Brendan Fong]], \emph{The Algebra of Open and Interconnected Systems}, \href{https://arxiv.org/abs/1609.05382}{arXiv:arXiv:1609.05382}


\item [[John Baez]] and [[Blake Pollard]], \emph{A Compositional Framework for Reaction Networks}, \href{https://arxiv.org/abs/1704.02051}{arXiv:1704.02051}


\item [[John C. Baez]], [[Brendan Fong]] and [[Blake Pollard]], \emph{A Compositional Framework for Markov Processes}, \href{https://arxiv.org/abs/1508.06448}{arXiv:1508.06448}


\item [[Jules Hedges]] and [[Martha Lewis]], \emph{Towards Functorial Language-Games}, \href{https://arxiv.org/abs/1807.07828}{arXiv:1807.07828}



\end{itemize}


\end{document}