Add abstract

doc/abstract.tex

@@ -0,0 +1,21 @@
+\chapter*{Abstract}
+The usual notion of propositional equality in intensional type-theory is
+restrictive. For instance it does not admit functional extensionality or
+univalence. This poses a severe limitation on both what is \emph{provable} and
+the \emph{re-usability} of proofs. Recent developments have, however, resulted
+in cubical type theory which permits a constructive proof of these two important
+notions. The programming language Agda has been extended with capabilities for
+working in such a cubical setting. This thesis will explore the usefulness of
+this extension in the context of category theory.
+
+The thesis will motivate and explain why propositional equality in cubical Agda
+is more expressive than in standard Agda. Alternative approaches to Cubical Agda
+will be presented and their pros and cons will be explained. It will emphasize
+why it is useful to have a constructive interpretation of univalence. As an
+example of this two formulations of monads will be presented: Namely monaeds in
+the monoidal form an monads in the Kleisli form.
+
+Finally the thesis will explain the challenges that a developer will face when
+working with cubical Agda and give some techniques to overcome these
+difficulties. It will also try to suggest how furhter work can help allievate
+some of these challenges.

doc/introduction.tex

@@ -1,22 +1,11 @@
 \chapter{Introduction}
-Functional extensionality and univalence is not expressible in
-\nomen{Intensional Martin Löf Type Theory} (ITT). This poses a severe limitation
-on both what is \emph{provable} and the \emph{re-usability} of proofs. Recent
-developments have, however, resulted in \nomen{Cubical Type Theory} (CTT) which
-permits a constructive proof of these two important notions.
-
-Furthermore an extension has been implemented for the proof assistant Agda
-(\cite{agda}, \cite{cubical-agda}) that allows us to work in such a ``cubical
-setting''. This thesis will explore the usefulness of this extension in the
-context of category theory.
-%
 \section{Motivating examples}
 %
 In the following two sections I present two examples that illustrate some
 limitations inherent in ITT and -- by extension -- Agda.
 %
 \subsection{Functional extensionality}
-\label{sec:functional-extensionality}
+\label{sec:functional-extensionality}%
 Consider the functions:
 %
 \begin{multicols}{2}
@@ -30,11 +19,10 @@ Consider the functions:
 \end{multicols}
 %
 $n + 0$ is \nomen{definitionally} equal to $n$, which we write as $n + 0 = n$.
-This is also \called \nomen{judgmental} equality. We call it definitional
+This is also called \nomen{judgmental} equality. We call it definitional
 equality because the \emph{equality} arises from the \emph{definition} of $+$
 which is:
 %
-\newcommand{\suc}[1]{\mathit{suc}\ #1}
 \begin{align*}
   +           & \tp \bN \to \bN \to \bN      \\
   n + 0       & \defeq n                   \\

doc/macros.tex

@@ -91,3 +91,4 @@
 \newcommand\Endo[1]{\varindex{Endo}\ #1}
 \newcommand\EndoR{\mathcal{R}}
 \newcommand\funExt{\varindex{funExt}}
+\newcommand{\suc}[1]{\mathit{suc}\ #1}

doc/main.tex

@@ -48,7 +48,7 @@
 \frontmatter
 \myfrontmatter
 \maketitle
-\addtocontents{toc}{\protect\thispagestyle{empty}}
+\input{abstract.tex}
 \tableofcontents
 \mainmatter
 %