Fix spacing after 'e.g.' and 'i.e.'

doc/cubical.tex

@@ -61,7 +61,7 @@ $I$. I will sometimes refer to $P$ as the \nomenindex{path space} of some path $
 \Path\ P\ a\ b$. By this token $P\ 0$ then corresponds to the type at the
 left-endpoint and $P\ 1$ as the type at the right-endpoint. The type is called
 $\Path$ because it is connected with paths in homotopy theory. The intuition
-behind this is that $\Path$ describes paths in $\MCU$ -- i.e. between types. For
+behind this is that $\Path$ describes paths in $\MCU$ -- i.e.\ between types. For
 a path $p$ for the point $p\ i$ the index $i$ describes how far along the path
 one has moved. An inhabitant of $\Path\ P\ a_0\ a_1$ is a (dependent-) function,
 $p$, from the index-space to the path space:
@@ -104,7 +104,7 @@ With this definition we can also recover reflexivity. That is, for any $A \tp
 Here the path space is $P \defeq \lambda i \to A$ and it satsifies $P\ i = A$
 definitionally. So to inhabit it, is to give a path $I \to A$ which is
 judgmentally $a$ at either endpoint. This is satisfied by the constant path;
-i.e. the path that stays at $a$ at any index $i$.
+i.e.\ the path that stays at $a$ at any index $i$.
 
 It is also surpisingly easy to show functional extensionality with which we can
 construct a path between $f$ and $g$ -- the functions defined in the
@@ -207,7 +207,7 @@ Then comes the set of homotopical sets:
 \end{equation}
 %
 I will not give an example of a set at this point. It turns out that proving
-e.g. $\isProp\ \bN$ is not so straight-forward (see \cite[\S3.1.4]{hott-2013}).
+e.g.\ $\isProp\ \bN$ is not so straight-forward (see \cite[\S3.1.4]{hott-2013}).
 There will be examples of sets later in this report. At this point it should be
 noted that the term ``set'' is somewhat conflated; there is the notion of sets
 from set-theory, in Agda types are denoted \texttt{Set}. I will use it

doc/discussion.tex

@@ -15,7 +15,7 @@ formalized this common result about monads:
 
 By transporting this to the Kleisli formulation we get a result that we can use
 to compute with. This is particularly useful because the Kleisli formulation
-will be more familiar to programmers e.g. those coming from a background in
+will be more familiar to programmers e.g.\ those coming from a background in
 Haskell. Whereas the theory usually talks about monoidal monads.
 
 \TODO{Mention that with postulates we cannot do this}

doc/halftime.tex

@@ -3,7 +3,7 @@ I've written this as an appendix because 1) the aim of the thesis changed
 drastically from the planning report/proposal 2) partly I'm not sure how to
 structure my thesis.
 
-My work so far has very much focused on the formalization, i.e. coding. It's
+My work so far has very much focused on the formalization, i.e.\ coding. It's
 unclear to me at this point what I should have in the final report. Here I will
 describe what I have managed to formalize so far and what outstanding challenges
 I'm facing.

doc/implementation.tex

@@ -202,7 +202,7 @@ set.
 This example illustrates nicely how we can use these combinators to reason about
 `canonical' types like $\sum$ and $\prod$. Similar combinators can be defined at
 the other homotopic levels. These combinators are however not applicable in
-situations where we want to reason about other types - e.g. types we have
+situations where we want to reason about other types - e.g.\ types we have
 defined ourselves. For instance, after we have proven that all the projections
 of pre categories are propositions, then we would like to bundle this up to show
 that the type of pre categories is also a proposition. Formally:
@@ -247,7 +247,7 @@ $$
 So to give the continuous function $I \to \IsPreCategory$, which is our goal, we
 introduce $i \tp I$ and proceed by constructing an element of $\IsPreCategory$
 by using the fact that all the projections are propositions to generate paths
-between all projections. Once we have such a path e.g. $p \tp a.\isIdentity
+between all projections. Once we have such a path e.g.\ $p \tp a.\isIdentity
 \equiv b.\isIdentity$ we can eliminate it with $i$ and thus obtain $p\ i \tp
 (p\ i).\isIdentity$. This element satisfies exactly that it corresponds to the
 corresponding projections at either endpoint. Thus the element we construct at
@@ -436,7 +436,7 @@ That is, we must demonstrate that there is an isomorphism (on types) between
 equalities and isomorphisms (on arrows). It is worthwhile to dwell on this for a
 few seconds. This type looks very similar to univalence for types and is
 therefore perhaps a bit more intuitive to grasp the implications of. Of course
-univalence for types (which is a proposition -- i.e. provable) does not imply
+univalence for types (which is a proposition -- i.e.\ provable) does not imply
 univalence of all pre category since morphisms in a category are not regular
 functions -- in stead they can be thought of as a generalization hereof. The univalence criterion therefore is simply a way of restricting arrows
 to behave similarly to maps.
@@ -551,7 +551,7 @@ same as the one in the underlying category (they have the same type). Function
 composition will be reverse function composition from the underlying category.
 
 I will refer to things in terms of the underlying category, unless they have an
-over-bar. So e.g. $\idToIso$ is a function in the underlying category and the
+over-bar. So e.g.\ $\idToIso$ is a function in the underlying category and the
 corresponding thing is denoted $\wideoverbar{\idToIso}$ in the opposite
 category.
 
@@ -1333,7 +1333,7 @@ f \rrr (\bind\ g)$ . (\TODO{Better way to typeset $\fish$?})
 \subsection{Equivalence of formulations}
 %
 The notation I have chosen here in the report
-overloads e.g. $\pure$ to both refer to a natural transformation and an arrow.
+overloads e.g.\ $\pure$ to both refer to a natural transformation and an arrow.
 This is of course not a coincidence as the arrow in the Kleisli formulation
 shall correspond exactly to the map on arrows from the natural transformation
 called $\pure$.

doc/introduction.tex

@@ -185,16 +185,16 @@ will make it possible to prove more things and to reuse proofs and to try and
 compare some aspects of this formalization with the existing ones.\TODO{How can
   I live up to this?}
 
-There are alternative approaches to working in a cubical setting where one can
+There are alternative approaches to working in a cubical setting where
-still have univalence and functional extensionality. One option is to postulate
+one can still have univalence and functional extensionality. One
-these as axioms. This approach, however, has other shortcomings, e.g.; you lose
+option is to postulate these as axioms. This approach, however, has
-
+other shortcomings, e.g. you lose \nomenindex{canonicity}
-\nomenindex{canonicity} (\TODO{Pageno!} \cite{huber-2016}). Canonicity means that any
+(\TODO{Pageno!} \cite{huber-2016}). Canonicity means that any well
-well typed term evaluates to a \emph{canonical} form. For example for a closed
+typed term evaluates to a \emph{canonical} form. For example for a
-term $e \tp \bN$ it will be the case that $e$ reduces to $n$ applications of
+closed term $e \tp \bN$ it will be the case that $e$ reduces to $n$
-$\mathit{suc}$ to $0$ for some $n$; $e = \mathit{suc}^n\ 0$. Without canonicity
+applications of $\mathit{suc}$ to $0$ for some $n$; $e =
-terms in the language can get ``stuck'' -- meaning that they do not reduce to a
+\mathit{suc}^n\ 0$. Without canonicity terms in the language can get
-canonical form.
+``stuck'' -- meaning that they do not reduce to a canonical form.
 
 Another approach is to use the \emph{setoid interpretation} of type
 theory (\cite{hofmann-1995,huber-2016}). With this approach one works

doc/presentation.tex

@@ -0,0 +1,236 @@
+\documentclass[a4paper,handout]{beamer}
+\input{packages.tex}
+\input{macros.tex}
+\title{Univalent Categories}
+\author{Frederik Hangh{\o}j Iversen}
+\institute{Chalmers University of Technology}
+\begin{document}
+\frame{\titlepage}
+
+\begin{frame}
+  \frametitle{Motivating example}
+  \framesubtitle{Functional extensionality}
+Consider the functions
+\begin{align*}
+  \var{zeroLeft} & \defeq (n \tp \bN) \mto (0 + n \tp \bN) \\
+  \var{zeroRight} & \defeq (n \tp \bN) \mto (n + 0 \tp \bN)
+\end{align*}
+\pause
+We have
+%
+$$
+\prod_{n \tp \bN} n + 0 \equiv 0 + n
+$$
+%
+\pause
+But not
+%
+$$
+\var{zeroLeft} \equiv \var{zeroRight}
+$$
+%
+\pause
+We need
+%
+$$
+\funExt \tp \prod_{a \tp A} f\ a \equiv g\ a \to f \equiv g
+$$
+
+\end{frame}
+\begin{frame}
+  \frametitle{Motivating example}
+  \framesubtitle{Univalence}
+  Consider the set
+  $\{x \mid \phi\ x \land \psi\ x\}$
+  \pause
+
+  If we show $\forall x . \psi\ x \equiv \top$
+  then we want to conclude
+  $\{x \mid \phi\ x \land \psi\ x\} \equiv \{x \mid \phi\ x\}$
+  \pause
+
+  We need univalence:
+  $$(A \simeq B) \simeq (A \equiv B)$$
+  \pause
+%
+  We will return to $\simeq$, but for not, think of it as an
+  isomorphism, so it induces maps:
+  \begin{align*}
+    \var{toPath}  & \tp (A \simeq B) \to (A \equiv B) \\
+    \var{toEquiv} & \tp (A \equiv B) \to (A \simeq B)
+  \end{align*}
+\end{frame}
+\begin{frame}
+  \frametitle{Paths}
+  \framesubtitle{Definition}
+Heterogeneous paths
+\begin{equation*}
+  \Path \tp (P \tp I → \MCU) → P\ 0 → P\ 1 → \MCU
+\end{equation*}
+\pause
+  For $P \tp I \to \MCU$, $A \tp \MCU$ and $a_0, a_1 \tp A$
+  inhabitants of $\Path\ P\ a_0\ a_1$ are like functions
+%
+$$
+p \tp \prod_{i \tp I} P\ i
+$$
+%
+Which satisfy $p\ 0 & = a_0$ and $p\ 1 & = a_1$
+\pause
+
+Homogenous paths
+$$
+a_0 \equiv a_1 \defeq \Path\ (\var{const}\ A)\ a_0\ a_1
+$$
+\end{frame}
+\begin{frame}
+\frametitle{Paths}
+\framesubtitle{Functional extenstionality}
+$$
+\funExt & \tp \prod_{a \tp A} f\ a \equiv g\ a \to f \equiv g
+$$
+\pause
+$$
+\funExt\ p \defeq λ i\ a → p\ a\ i
+$$
+\pause
+$$
+\funExt\ (\var{const}\ \refl)
+\tp
+\var{zeroLeft} \equiv \var{zeroRight}
+$$
+\end{frame}
+\begin{frame}
+  \frametitle{Paths}
+  \framesubtitle{Homotopy levels}
+\begin{align*}
+& \isContr    && \tp    \MCU \to \MCU \\
+& \isContr\ A && \defeq \sum_{c \tp A} \prod_{a \tp A} a \equiv c
+\end{align*}
+\pause
+\begin{align*}
+& \isProp    && \tp \MCU \to \MCU \\
+& \isProp\ A && \defeq \prod_{a_0, a_1 \tp A} a_0 \equiv a_1
+\end{align*}
+\pause
+\begin{align*}
+& \isSet    && \tp \MCU \to \MCU \\
+& \isSet\ A && \defeq \prod_{a_0, a_1 \tp A} \isProp\ (a_0 \equiv a_1)
+\end{align*}
+\pause
+\end{frame}
+\begin{frame}
+\frametitle{Paths}
+\framesubtitle{A few lemmas}
+Let $D$ be a type-family:
+$$
+D \tp \prod_{b \tp A} \prod_{p \tp a ≡ b} \MCU
+$$
+%
+\pause
+And $d$ and in inhabitant of $D$ at $\refl$:
+%
+$$
+d \tp D\ a\ \refl
+$$
+%
+\pause
+We then have the function:
+%
+\begin{equation}
+\pathJ\ D\ d \tp \prod_{b \tp A} \prod_{p \tp a ≡ b} D\ a\ p
+\end{equation}
+\end{frame}
+\begin{frame}
+\frametitle{Paths}
+\framesubtitle{A few lemmas}
+Given
+\begin{align*}
+  A           & \tp \MCU \\
+  P           & \tp A \to \MCU \\
+  \var{propP} & \tp \prod_{x \tp A} \isProp\ (P\ x) \\
+  p           & \tp a_0 \equiv a_1 \\
+  p_0         & \tp P\ a_0 \\
+  p_1         & \tp P\ a_1
+\end{align*}
+%
+We have
+$$
+\lemPropF\ \var{propP}\ p
+\tp
+\Path\ (\lambda\; i \mto P\ (p\ i))\ p_0\ p_1
+$$
+%
+\end{frame}
+\begin{frame}
+\frametitle{Paths}
+\framesubtitle{A few lemmas}
+$\prod$ preserves $\isProp$:
+$$
+\mathit{propPi}
+\tp
+\left(\prod_{a \tp A} \isProp\ (P\ a)\right)
+\to \isProp\ \left(\prod_{a \tp A} P\ a\right)
+$$
+\pause
+$\sum$ preserves $\isProp$:
+$$
+\mathit{propSig} \tp \isProp\ A \to \left(\prod_{a \tp A} \isProp\ (P\ a)\right) \to \isProp\ \left(\sum_{a \tp A} P\ a\right)
+$$
+\end{frame}
+\begin{frame}
+\frametitle{Categories}
+\framesubtitle{Definition}
+Data:
+\begin{align*}
+  \Object   & \tp \Type \\
+  \Arrow    & \tp \Object \to \Object \to \Type \\
+  \identity & \tp \Arrow\ A\ A \\
+  \lll      & \tp \Arrow\ B\ C \to \Arrow\ A\ B \to \Arrow\ A\ C
+\end{align*}
+%
+Laws:
+%
+$$
+h \lll (g \lll f) ≡ (h \lll g) \lll f
+$$
+$$
+\identity \lll f ≡ f \x
+f \lll \identity ≡ f
+$$
+\pause
+1-categories:
+$$
+\isSet\ (\Arrow\ A\ B)
+$$
+\pause
+Univalent categories:
+$$
+\isEquiv\ (A \equiv B)\ (A \approxeq B)\ \idToIso
+$$
+\end{frame}
+\begin{frame}
+\frametitle{Categories}
+\framesubtitle{Univalence}
+\begin{align*}
+\var{IsIdentity} & \defeq
+\prod_{A\ B \tp \Object} \prod_{f \tp \Arrow\ A\ B} \phi\ f
+%% \\
+%%   & \mathrel{\ } \identity \lll f \equiv f \x f \lll \identity \equiv f
+\end{align*}
+where
+$$
+\phi\ f \defeq \identity \lll f \equiv f \x f \lll \identity \equiv f
+$$
+Let $\approxeq$ denote ismorphism of objects. We can then construct
+the identity isomorphism in any category:
+$$
+\identity , \identity , \var{isIdentity} \tp A \approxeq A
+$$
+Likewise since paths are substitutive we can promote a path to an isomorphism:
+$$
+\idToIso \tp A ≡ B → A ≊ B
+$$
+For a category to be univalent we require this to be an equivalence: 
+\end{frame}
+\end{document}

doc/title.tex

@@ -0,0 +1,95 @@
+%% FRONTMATTER
+\frontmatter
+%% \newgeometry{top=3cm, bottom=3cm,left=2.25 cm, right=2.25cm}
+\begingroup
+\thispagestyle{empty}
+{\Huge\thetitle}\\[.5cm]
+{\Large A formalization of category theory in Cubical Agda}\\[6cm]
+\begin{center}
+\includegraphics[width=\linewidth,keepaspectratio]{isomorphism.png}
+\end{center}
+% Cover text
+\vfill
+%% \renewcommand{\familydefault}{\sfdefault} \normalfont % Set cover page font
+{\Large\theauthor}\\[.5cm]
+Master's thesis in Computer Science
+\endgroup
+%% \end{titlepage}
+
+
+% BACK OF COVER PAGE (BLANK PAGE)
+\newpage
+%% \newgeometry{a4paper}	% Temporarily change margins
+%% \restoregeometry
+\thispagestyle{empty}
+\null
+
+%% \begin{titlepage}
+
+% TITLE PAGE
+\newpage
+\thispagestyle{empty}
+\begin{center}
+	\textsc{\LARGE Master's thesis \the\year}\\[4cm]		% Report number is currently not in use
+	\textbf{\LARGE \thetitle} \\[1cm]
+	{\large \subtitle}\\[1cm]
+	{\large \theauthor}
+	
+	\vfill	
+	\centering
+	\includegraphics[width=0.2\pdfpagewidth]{logo_eng.pdf}
+  \vspace{5mm}
+	
+	\textsc{Department of Computer Science and Engineering}\\
+	\textsc{{\researchgroup}}\\
+	%Name of research group (if applicable)\\
+	\textsc{\institution} \\
+	\textsc{Gothenburg, Sweden \the\year}\\
+\end{center}
+
+
+% IMPRINT PAGE (BACK OF TITLE PAGE)
+\newpage
+\thispagestyle{plain}
+\textit{\thetitle}\\
+\subtitle\\
+\copyright\ \the\year ~ \textsc{\theauthor}
+\vspace{4.5cm}
+
+\setlength{\parskip}{0.5cm}
+\textbf{Author:}\\
+\theauthor\\
+\href{mailto:\authoremail>}{\texttt{<\authoremail>}}
+
+\textbf{Supervisor:}\\
+\supervisor\\
+\href{mailto:\supervisoremail>}{\texttt{<\supervisoremail>}}\\
+\supervisordepartment
+
+\textbf{Co-supervisor:}\\
+\cosupervisor\\
+\href{mailto:\cosupervisoremail>}{\texttt{<\cosupervisoremail>}}\\
+\cosupervisordepartment
+
+\textbf{Examiner:}\\
+\examiner\\
+\href{mailto:\examineremail>}{\texttt{<\examineremail>}}\\
+\examinerdepartment
+
+\vfill
+Master's Thesis \the\year\\	% Report number currently not in use
+\department\\
+%Division of Division name\\
+%Name of research group (if applicable)\\
+\institution\\
+SE-412 96 Gothenburg\\
+Telephone +46 31 772 1000 \setlength{\parskip}{0.5cm}\\
+% Caption for cover page figure if used, possibly with reference to further information in the report
+%% Cover: Wind visualization constructed in Matlab showing a surface of constant wind speed along with streamlines of the flow. \setlength{\parskip}{0.5cm}
+%Printed by [Name of printing company]\\
+Gothenburg, Sweden \the\year
+
+%% \restoregeometry
+%% \end{titlepage}
+
+\tableofcontents