Remove tex warnings
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@ -38,12 +38,11 @@
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\newcommand*{\department}[1]{\gdef\@department{#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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\newcommand*{\researchgroup}[1]{\gdef\@researchgroup{#1}}
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\newcommand*{\subtitle}[1]{\gdef\@subtitle{#1}}
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\newcommand*{\subtitle}[1]{\gdef\@subtitle{#1}}
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%% FRONTMATTER
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\newcommand*{\myfrontmatter}{%
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\newgeometry{top=3cm, bottom=3cm,left=2.25 cm, right=2.25cm}
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\newgeometry{top=3cm, bottom=3cm,left=2.25 cm, right=2.25cm}
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\begingroup
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\begingroup
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\thispagestyle{empty}
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\thispagestyle{empty}
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\usepackage{noto}
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\fontseries{sb}
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%% \fontfamily{noto}\selectfont
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{\Huge\@title}\\[.5cm]
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{\Huge\@title}\\[.5cm]
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{\Large A formalization of category theory in Cubical Agda}\\[2.5cm]
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{\Large A formalization of category theory in Cubical Agda}\\[2.5cm]
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\begin{center}
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\begin{center}
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@ -55,17 +54,16 @@
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{\Large\@author}\\[.5cm]
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{\Large\@author}\\[.5cm]
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Master's thesis in Computer Science
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Master's thesis in Computer Science
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\endgroup
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\endgroup
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%% \renewcommand{\familydefault}{\rmdefault} \normalfont % Reset standard font
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%% \end{titlepage}
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%% \end{titlepage}
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% BACK OF COVER PAGE (BLANK PAGE)
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% BACK OF COVER PAGE (BLANK PAGE)
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\newpage
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\newpage
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\newgeometry{a4paper} % Temporarily change margins
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%% \newgeometry{a4paper} % Temporarily change margins
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\restoregeometry
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%% \restoregeometry
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\thispagestyle{empty}
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\thispagestyle{empty}
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\null
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\null
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}
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\renewcommand*{\maketitle}{%
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\renewcommand*{\maketitle}{%
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\begin{titlepage}
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\begin{titlepage}
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@ -89,10 +89,12 @@ interest.
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With this definition we can also recover reflexivity. That is, for any $A \tp
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With this definition we can also recover reflexivity. That is, for any $A \tp
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\MCU$ and $a \tp A$:
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\MCU$ and $a \tp A$:
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%
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%
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\begin{align}
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\begin{equation}
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\begin{aligned}
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\refl & \tp \Path (\lambda i \to A)\ a\ a \\
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\refl & \tp \Path (\lambda i \to A)\ a\ a \\
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\refl & \defeq \lambda i \to a
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\refl & \defeq \lambda i \to a
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\end{align}
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\end{aligned}
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\end{equation}
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%
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%
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Or, in other terms; reflexivity is the path in $A$ that is $a$ at the left
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Or, in other terms; reflexivity is the path in $A$ that is $a$ at the left
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endpoint as well as at the right endpoint. It is inhabited by the path which
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endpoint as well as at the right endpoint. It is inhabited by the path which
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@ -113,10 +115,12 @@ structure''. At the bottom of this hierarchy we have the set of contractible
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types:
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types:
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%
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%
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\begin{equation}
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\begin{equation}
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\begin{alignat}{2}
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\begin{aligned}
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%% \begin{split}
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& \isContr && \tp \MCU \to \MCU \\
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& \isContr && \tp \MCU \to \MCU \\
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& \isContr\ A && \defeq \sum_{c \tp A} \prod_{a \tp A} a \equiv c
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& \isContr\ A && \defeq \sum_{c \tp A} \prod_{a \tp A} a \equiv c
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\end{alignedat}
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%% \end{split}
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\end{aligned}
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\end{equation}
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\end{equation}
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%
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%
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The first component of $\isContr\ A$ is called ``the center of contraction''.
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The first component of $\isContr\ A$ is called ``the center of contraction''.
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@ -127,10 +131,10 @@ contractible, then it is isomorphic to the unit-type $\top$.
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The next step in the hierarchy is the set of mere propositions:
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The next step in the hierarchy is the set of mere propositions:
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%
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%
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\begin{equation}
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\begin{equation}
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\begin{alignat}{2}
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\begin{aligned}
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& \isProp && \tp \MCU \to \MCU \\
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& \isProp && \tp \MCU \to \MCU \\
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& \isProp\ A && \defeq \prod_{a_0, a_1 \tp A} a_0 \equiv a_1
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& \isProp\ A && \defeq \prod_{a_0, a_1 \tp A} a_0 \equiv a_1
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\end{alignedat}
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\end{aligned}
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\end{equation}
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\end{equation}
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%
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%
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$\isProp\ A$ can be thought of as the set of true and false propositions. It is
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$\isProp\ A$ can be thought of as the set of true and false propositions. It is
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@ -144,10 +148,10 @@ stress that we have $\isProp\ A$.
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Then comes the set of homotopical sets:
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Then comes the set of homotopical sets:
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%
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%
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\begin{equation}
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\begin{equation}
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\begin{alignat}{2}
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\begin{aligned}
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& \isSet && \tp \MCU \to \MCU \\
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& \isSet && \tp \MCU \to \MCU \\
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& \isSet\ A && \defeq \prod_{a_0, a_1 \tp A} \isProp\ (a_0 \equiv a_1)
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& \isSet\ A && \defeq \prod_{a_0, a_1 \tp A} \isProp\ (a_0 \equiv a_1)
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\end{alignedat}
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\end{aligned}
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\end{equation}
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\end{equation}
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%
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%
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At this point it should be noted that the term ``set'' is somewhat conflated;
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At this point it should be noted that the term ``set'' is somewhat conflated;
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@ -158,10 +162,10 @@ if $\isSet\ A$ is inhabited.
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The next step in the hierarchy is, as the reader might've guessed, the type:
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The next step in the hierarchy is, as the reader might've guessed, the type:
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%
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%
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\begin{equation}
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\begin{equation}
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\begin{alignat}{2}
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\begin{aligned}
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& \isGroupoid && \tp \MCU \to \MCU \\
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& \isGroupoid && \tp \MCU \to \MCU \\
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& \isGroupoid\ A && \defeq \prod_{a_0, a_1 \tp A} \isSet\ (a_0 \equiv a_1)
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& \isGroupoid\ A && \defeq \prod_{a_0, a_1 \tp A} \isSet\ (a_0 \equiv a_1)
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\end{alignedat}
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\end{aligned}
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\end{equation}
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\end{equation}
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%
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%
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And so it continues. In fact we can generalize this family of types by indexing
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And so it continues. In fact we can generalize this family of types by indexing
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@ -195,7 +195,7 @@ Name & Agda & Notation \\
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\nomen{Type} & \texttt{Set} & $\Type$ \\
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\nomen{Type} & \texttt{Set} & $\Type$ \\
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\nomen{Set} & \texttt{Σ Set IsSet} & $\Set$ \\
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\nomen{Set} & \texttt{Σ Set IsSet} & $\Set$ \\
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Function, morphism, map & \texttt{A → B} & $A → B$ \\
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Function, morphism, map & \texttt{A → B} & $A → B$ \\
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Dependent- ditto & \texttt{(a : A) → B} & $∏_{a \tp A} → B$ \\
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Dependent- ditto & \texttt{(a : A) → B} & $∏_{a \tp A} B$ \\
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\nomen{Arrow} & \texttt{Arrow A B} & $\Arrow\ A\ B$ \\
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\nomen{Arrow} & \texttt{Arrow A B} & $\Arrow\ A\ B$ \\
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\nomen{Object} & \texttt{C.Object} & $̱ℂ.Object$ \\
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\nomen{Object} & \texttt{C.Object} & $̱ℂ.Object$ \\
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Definition & \texttt{=} & $̱\defeq$ \\
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Definition & \texttt{=} & $̱\defeq$ \\
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@ -25,6 +25,7 @@
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%% department=Department of Computer Science and Engineering,
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%% department=Department of Computer Science and Engineering,
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%% researchgroup=Programming Logic Group
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%% researchgroup=Programming Logic Group
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%% ]{chalmerstitle}
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%% ]{chalmerstitle}
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\usepackage{chalmerstitle}
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\usepackage{chalmerstitle}
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\subtitle{A formalization of category theory in Cubical Agda}
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\subtitle{A formalization of category theory in Cubical Agda}
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\authoremail{hanghj@student.chalmers.se}
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\authoremail{hanghj@student.chalmers.se}
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@ -48,7 +49,7 @@
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\newcommand*{\rom}[1]{\expandafter\@slowroman\romannumeral #1@}
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\newcommand*{\rom}[1]{\expandafter\@slowroman\romannumeral #1@}
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\makeatother
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\makeatother
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\begin{document}
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\begin{document}
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\myfrontmatter
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\pagenumbering{roman}
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\pagenumbering{roman}
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\maketitle
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\maketitle
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\addtocontents{toc}{\protect\thispagestyle{empty}}
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\addtocontents{toc}{\protect\thispagestyle{empty}}
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@ -45,7 +45,7 @@
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%% \setmonofont{FreeMono.otf}
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%% \setmonofont{FreeMono.otf}
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\pagestyle{fancyplain}
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%% \pagestyle{fancyplain}
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\setlength{\headheight}{15pt}
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\setlength{\headheight}{15pt}
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\renewcommand{\chaptermark}[1]{\markboth{\textsc{Chapter \thechapter. #1}}{}}
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\renewcommand{\chaptermark}[1]{\markboth{\textsc{Chapter \thechapter. #1}}{}}
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\renewcommand{\sectionmark}[1]{\markright{\textsc{\thesection\ #1}}}
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\renewcommand{\sectionmark}[1]{\markright{\textsc{\thesection\ #1}}}
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