Change title-page, write stuff about products
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@ -4,45 +4,85 @@
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\newcommand*{\authoremail}[1]{\gdef\@authoremail{#1}}
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\newcommand*{\supervisor}[1]{\gdef\@supervisor{#1}}
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\newcommand*{\supervisoremail}[1]{\gdef\@supervisoremail{#1}}
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\newcommand*{\supervisordepartment}[1]{\gdef\@supervisordepartment{#1}}
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\newcommand*{\cosupervisor}[1]{\gdef\@cosupervisor{#1}}
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\newcommand*{\cosupervisoremail}[1]{\gdef\@cosupervisoremail{#1}}
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\newcommand*{\cosupervisordepartment}[1]{\gdef\@cosupervisordepartment{#1}}
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\newcommand*{\examiner}[1]{\gdef\@examiner{#1}}
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\newcommand*{\examineremail}[1]{\gdef\@examineremail{#1}}
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\newcommand*{\examinerdepartment}[1]{\gdef\@examinerdepartment{#1}}
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\newcommand*{\institution}[1]{\gdef\@institution{#1}}
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\newcommand*{\department}[1]{\gdef\@department{#1}}
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\newcommand*{\researchgroup}[1]{\gdef\@researchgroup{#1}}
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\renewcommand*{\maketitle}{%
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\begin{titlepage}
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% TITLE PAGE
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\newpage
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\thispagestyle{empty}
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\begin{center}
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{\scshape\LARGE Master thesis\\}
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\vspace{0.5cm}
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{\LARGE\bfseries \@title\\}
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\vspace{2cm}
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{\Large \@author\\ \href{mailto:\@authoremail>}{\texttt{<\@authoremail>}} \\}
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% \vspace{0.2cm}
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%
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% {\Large name and email adress of student 2\\}
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\vspace{1.0cm}
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{\large Supervisor: \@supervisor\\ \href{mailto:\@supervisoremail>}{\texttt{<\@supervisoremail>}}\\}
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\vspace{0.2cm}
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{\large Co-supervisor: \@cosupervisor\\ \href{mailto:\@cosupervisoremail>}{\texttt{<\@cosupervisoremail>}}\\}
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\vspace{1.5cm}
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\textsc{\LARGE Master's thesis \the\year}\\[4cm] % Report number is currently not in use
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\textbf{\LARGE \@title} \\[1cm]
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{\large \@subtitle}\\[1cm]
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{\large \@author}
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\vfill
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\centering
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\includegraphics[width=0.2\pdfpagewidth]{logo_eng.pdf}
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\vspace{5mm}
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{\large \@institution\\\today\\}
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Department of Computer Science and Engineering\\
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\emph{\@researchgroup}\\
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%Name of research group (if applicable)\\
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\textsc{\@institution} \\
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Gothenburg, Sweden \the\year \\
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\end{center}
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\end{titlepage}
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}
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% IMPRINT PAGE (BACK OF TITLE PAGE)
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\newpage
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\thispagestyle{plain}
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\vspace*{4.5cm}
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\@title\\
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\@subtitle\\
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\textbf{Author:}\\
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\@author\\
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\href{mailto:\@authoremail>}{\texttt{<\@authoremail>}}
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\setlength{\parskip}{1cm}
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\copyright ~ \MakeUppercase{\@author}, \the\year.
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\setlength{\parskip}{0.5cm}
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\textbf{Supervisor:}\\
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\@supervisor\\
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\href{mailto:\@supervisoremail>}{\texttt{<\@supervisoremail>}}\\
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\@supervisordepartment
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\textbf{Co-supervisor:}\\
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\@cosupervisor\\
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\href{mailto:\@cosupervisoremail>}{\texttt{<\@cosupervisoremail>}}\\
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\@cosupervisordepartment
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\textbf{Examiner:}\\
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\@examiner\\
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\href{mailto:\@examineremail>}{\texttt{<\@examineremail>}}\\
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\@examinerdepartment\setlength{\parskip}{1cm}
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Master's Thesis \the\year\\ % Report number currently not in use
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\@department\\
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%Division of Division name\\
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%Name of research group (if applicable)\\
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\@institution\\
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SE-412 96 Gothenburg\\
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Telephone +46 31 772 1000 \setlength{\parskip}{0.5cm}
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\vfill
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% Caption for cover page figure if used, possibly with reference to further information in the report
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%% Cover: Wind visualization constructed in Matlab showing a surface of constant wind speed along with streamlines of the flow. \setlength{\parskip}{0.5cm}
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%Printed by [Name of printing company]\\
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Gothenburg, Sweden \the\year
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\end{titlepage}}
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@ -691,6 +691,7 @@ are propositional in a proper category and equivalences preservehomotopy level,
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then we know that products also are propositions. But before we get to that,
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let's recall the definition of products.
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\subsection{Products}
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Given a category $\bC$ and two objects $A$ and $B$ in $bC$ we define the product
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of $A$ and $B$ to be an object $A \x B$ in $\bC$ and two arrows $\pi_1 \tp A \x
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B \to A$ and $\pi_2 \tp A \x B \to B$ called the projections of the product. The projections must satisfy the following property:
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@ -709,6 +710,54 @@ that there exists a unique arrow $\pi \tp \Arrow\ X\ (A \x B)$ satisfying
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$
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$\pi$ is called the product (arrow) of $f$ and $g$.
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\subsection{Pair category}
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\newcommand\pairA{\mathcal{A}}
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\newcommand\pairB{\mathcal{B}}
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Given a base category $\bC$ and two objects in this category $\pairA$ and
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$\pairrB$ we can construct the ``pair category'': \TODO{This is a working title,
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it's nice to have a name for this thing to refer back to}
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The type objects in this category will be an object in the underlying category,
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$X$, and two arrows (also from the underlying category)
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$\Arrow\ X\ \pairA$ and $\Arrow\ X\ \pairB$.
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\newcommand\pairf{\ensuremath{f}}
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\newcommand\pairFst{\mathcal{\pi_1}}
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\newcommand\pairSnd{\mathcal{\pi_2}}
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An arrow between objects $A ,\ a_0 ,\ a_1$ and $B ,\ b_0 ,\ b_1$ in this
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category will consist of an arrow from the underlying category $\pairf \tp
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\Arrow\ A\ B$ satisfying:
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%
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\begin{align}
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\begin{split}
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\label{eq:pairArrowLaw}
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b_0 \lll f & \equiv a_0 \\
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b_1 \lll f & \equiv a_1
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\end{split}
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\end{align}
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The identity morphism is the identity morphism from the underlying category.
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This choice satisfies \ref{eq:pairArrowLaw} because of the right-identity law
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from the underlying category.
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For composition of arrows $f \tp \Arrow\ A\ B$ and $g \tp \Arrow\ B\ C$ we
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choose $g \lll f$ and we must now verify that it satisfies
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\ref{eq:pairArrowLaw}:
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%
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\begin{align*}
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c_0 \lll (f \lll g)
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& \equiv
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(c_0 \lll f) \lll g
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&& \text{Associativity} \\
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& \equiv
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b_0 \lll g
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&& \text{$f$ satisfies \ref{eq:pairArrowLaw}} \\
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& \equiv
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a_0
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&& \text{$g$ satisfies \ref{eq:pairArrowLaw}} \\
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\end{align*}
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\section{Monads}
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@ -22,46 +22,46 @@
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\newcommand{\tp}{\;\mathord{:}\;}
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\newcommand{\Type}{\mathcal{U}}
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\newcommand{\var}[1]{\mathit{#1}}
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\newcommand{\Hom}{\mathit{Hom}}
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\newcommand{\fmap}{\mathit{fmap}}
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\newcommand{\idFun}{\mathit{id}}
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\newcommand{\Sets}{\mathit{Sets}}
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\newcommand{\Set}{\mathit{Set}}
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\newcommand{\hSet}{\mathit{hSet}}
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\newcommand{\id}{\mathit{id}}
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\newcommand{\isEquiv}{\mathit{isEquiv}}
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\newcommand{\idToIso}{\mathit{idToIso}}
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\newcommand{\isSet}{\mathit{isSet}}
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\newcommand{\isContr}{\mathit{isContr}}
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\newcommand\Object{\mathit{Object}}
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\newcommand\Functor{\mathit{Functor}}
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\newcommand\isProp{\mathit{isProp}}
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\newcommand\propPi{\mathit{propPi}}
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\newcommand\propSig{\mathit{propSig}}
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\newcommand\PreCategory{\mathit{PreCategory}}
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\newcommand\IsPreCategory{\mathit{IsPreCategory}}
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\newcommand\isIdentity{\mathit{isIdentity}}
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\newcommand\propIsIdentity{\mathit{propIsIdentity}}
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\newcommand\IsCategory{\mathit{IsCategory}}
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\newcommand\Gl{\mathit{\lambda}}
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\newcommand\lemPropF{\mathit{lemPropF}}
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\newcommand\isPreCategory{\mathit{isPreCategory}}
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\newcommand\congruence{\mathit{cong}}
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\newcommand\identity{\mathit{identity}}
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\newcommand\isequiv{\mathit{isequiv}}
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\newcommand\qinv{\mathit{qinv}}
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\newcommand\fiber{\mathit{fiber}}
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\newcommand\shuffle{\mathit{shuffle}}
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\newcommand\Univalent{\mathit{Univalent}}
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\newcommand\refl{\mathit{refl}}
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\newcommand\isoToId{\mathit{isoToId}}
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\newcommand{\var}[1]{\ensuremath{\mathit{#1}}}
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\newcommand{\Hom}{\var{Hom}}
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\newcommand{\fmap}{\var{fmap}}
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\newcommand{\idFun}{\var{id}}
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\newcommand{\Sets}{\var{Sets}}
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\newcommand{\Set}{\var{Set}}
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\newcommand{\hSet}{\var{hSet}}
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\newcommand{\id}{\var{id}}
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\newcommand{\isEquiv}{\var{isEquiv}}
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\newcommand{\idToIso}{\var{idToIso}}
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\newcommand{\isSet}{\var{isSet}}
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\newcommand{\isContr}{\var{isContr}}
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\newcommand\Object{\var{Object}}
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\newcommand\Functor{\var{Functor}}
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\newcommand\isProp{\var{isProp}}
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\newcommand\propPi{\var{propPi}}
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\newcommand\propSig{\var{propSig}}
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\newcommand\PreCategory{\var{PreCategory}}
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\newcommand\IsPreCategory{\var{IsPreCategory}}
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\newcommand\isIdentity{\var{isIdentity}}
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\newcommand\propIsIdentity{\var{propIsIdentity}}
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\newcommand\IsCategory{\var{IsCategory}}
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\newcommand\Gl{\var{\lambda}}
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\newcommand\lemPropF{\var{lemPropF}}
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\newcommand\isPreCategory{\var{isPreCategory}}
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\newcommand\congruence{\var{cong}}
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\newcommand\identity{\var{identity}}
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\newcommand\isequiv{\var{isequiv}}
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\newcommand\qinv{\var{qinv}}
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\newcommand\fiber{\var{fiber}}
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\newcommand\shuffle{\var{shuffle}}
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\newcommand\Univalent{\var{Univalent}}
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\newcommand\refl{\var{refl}}
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\newcommand\isoToId{\var{isoToId}}
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\newcommand\rrr{\ggg}
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\newcommand\fst{\mathit{fst}}
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\newcommand\snd{\mathit{snd}}
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\newcommand\Path{\mathit{Path}}
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\newcommand\Category{\mathit{Category}}
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\newcommand\fst{\var{fst}}
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\newcommand\snd{\var{snd}}
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\newcommand\Path{\var{Path}}
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\newcommand\Category{\var{Category}}
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\newcommand\TODO[1]{TODO: \emph{#1}}
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\newcommand*{\QED}{\hfill\ensuremath{\square}}%
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\newcommand\uexists{\exists!}
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\newcommand\Arrow{\mathit{Arrow}}
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\newcommand\Arrow{\var{Arrow}}
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23
doc/main.tex
23
doc/main.tex
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@ -1,22 +1,34 @@
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\documentclass{report}
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\documentclass[a4paper]{report}
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%% \documentclass[compact,a4paper]{article}
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\input{packages.tex}
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\input{macros.tex}
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\title{Univalent categories}
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\title{Univalent categories in cubical Agda}
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\subtitle{}
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\author{Frederik Hanghøj Iversen}
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\authoremail{hanghj@student.chalmers.se}
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\newcommand{\chalmers}{Chalmers University of Technology}
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\supervisor{Thierry Coquand}
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\supervisoremail{coquand@chalmers.se}
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\supervisordepartment{\chalmers}
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\cosupervisor{Andrea Vezzosi}
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\cosupervisoremail{vezzosi@chalmers.se}
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\institution{Chalmers University of Technology}
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\cosupervisordepartment{\chalmers}
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\examiner{Andreas Abel}
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\examineremail{abela@chalmers.se}
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\examinerdepartment{\chalmers}
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\institution{\chalmers}
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\department{Department of Computer Science and Engineering}
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\researchgroup{Programming Logic Group}
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\begin{document}
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\pagenumbering{roman}
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\maketitle
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\tableofcontents
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%
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\pagenumbering{arabic}
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\chapter{Introduction}
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\input{introduction.tex}
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@ -27,7 +39,10 @@
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\nocite{cubical-demo}
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\nocite{coquand-2013}
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\bibliography{refs}
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\begin{appendices}
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\setcounter{page}{1}
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\pagenumbering{roman}
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%% \input{planning.tex}
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%% \input{halftime.tex}
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\end{appendices}
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@ -1,7 +1,12 @@
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\usepackage[utf8]{inputenc}
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\usepackage{natbib}
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\usepackage[hidelinks]{hyperref}
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\usepackage[
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hidelinks,
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pdfusetitle,
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pdfsubject={category theory},
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pdfkeywords={type theory, homotopy theory, category theory, agda}]
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{hyperref}
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\usepackage{graphicx}
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@ -10,6 +15,7 @@
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\usepackage{amssymb,amsmath,amsthm,stmaryrd,mathrsfs,wasysym}
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\usepackage[toc,page]{appendix}
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\usepackage{xspace}
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%% \usepackage{geometry}
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% \setlength{\parskip}{10pt}
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