Merge branch 'dev'
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commit
33f7e2ebbb
4
Makefile
4
Makefile
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@ -4,8 +4,10 @@ build: src/**.agda
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clean:
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clean:
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find src -name "*.agdai" -type f -delete
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find src -name "*.agdai" -type f -delete
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html:
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html: src/**.agda
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agda --html src/Cat.agda
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agda --html src/Cat.agda
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upload: html
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upload: html
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scp -r html/ remote11.chalmers.se:www/cat/doc/
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scp -r html/ remote11.chalmers.se:www/cat/doc/
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.PHONY: upload clean
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1
doc/.gitignore
vendored
1
doc/.gitignore
vendored
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@ -4,6 +4,7 @@
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*.log
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*.log
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*.out
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*.out
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*.pdf
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*.pdf
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!assets/**
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*.bbl
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*.bbl
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*.blg
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*.blg
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*.toc
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*.toc
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@ -1,3 +1,13 @@
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Presentation
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====
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Find one clear goal.
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Remember crowd-control.
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Leave out:
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lemPropF
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Talk about structure of library:
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Talk about structure of library:
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===
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===
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@ -1 +0,0 @@
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\chapter*{Acknowledgements}
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@ -7,7 +7,7 @@ can conjure up various proofs. I also want to recognize the support
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of Knud Højgaards Fond who graciously sponsored me with a 20.000 DKK
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of Knud Højgaards Fond who graciously sponsored me with a 20.000 DKK
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scholarship which helped toward sponsoring the two years I have spent
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scholarship which helped toward sponsoring the two years I have spent
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studying abroad. I would also like to give a warm thanks to my fellow
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studying abroad. I would also like to give a warm thanks to my fellow
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students Pierre Kraft and Nachiappan Villiappan who have made the time
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students Pierre~Kraft and Nachiappan~Valliappan who have made the time
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spent working on the thesis way more enjoyable. Lastly I would like to
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spent working on the thesis way more enjoyable. Lastly I would like to
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give a special thanks to Valentina Méndez who have been a great moral
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give a special thanks to Valentina~Méndez who have been a great moral
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support throughout the whole process.
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support throughout the whole process.
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BIN
doc/assets/logo_eng.pdf
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BIN
doc/assets/logo_eng.pdf
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@ -80,7 +80,7 @@ Master's thesis in Computer Science
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\vfill
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\vfill
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\centering
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\centering
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\includegraphics[width=0.2\pdfpagewidth]{logo_eng.pdf}
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\includegraphics[width=0.2\pdfpagewidth]{assets/logo_eng.pdf}
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\vspace{5mm}
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\vspace{5mm}
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\textsc{Department of Computer Science and Engineering}\\
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\textsc{Department of Computer Science and Engineering}\\
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@ -75,8 +75,8 @@ some limitations inherent in ITT and -- by extension -- Agda.
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Consider the functions:
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Consider the functions:
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%
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%
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\begin{align*}%
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\begin{align*}%
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\var{zeroLeft} & \defeq \lambda\; (n \tp \bN) \to (0 + n \tp \bN) \\
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\var{zeroLeft} & \defeq λ\; (n \tp \bN) \to (0 + n \tp \bN) \\
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\var{zeroRight} & \defeq \lambda\; (n \tp \bN) \to (n + 0 \tp \bN)
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\var{zeroRight} & \defeq λ\; (n \tp \bN) \to (n + 0 \tp \bN)
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\end{align*}%
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\end{align*}%
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%
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%
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The term $n + 0$ is \nomenindex{definitionally} equal to $n$, which we
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The term $n + 0$ is \nomenindex{definitionally} equal to $n$, which we
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%% \usecolortheme[named=seagull]{structure}
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%% \usecolortheme[named=seagull]{structure}
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\input{packages.tex}
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\input{packages.tex}
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\input{macros.tex}
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\input{macros.tex}
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\title[Univalent Categories]{Univalent Categories\\ \footnotesize A formalization of category theory in Cubical Agda}
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\title{Univalent Categories}
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\subtitle{A formalization of category theory in Cubical Agda}
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\newcommand{\myname}{Frederik Hangh{\o}j Iversen}
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\newcommand{\myname}{Frederik Hangh{\o}j Iversen}
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\author[\myname]{
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\author[\myname]{
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\myname\\
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\myname\\
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\framesubtitle{Definition}
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\framesubtitle{Definition}
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Heterogeneous paths
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Heterogeneous paths
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\begin{equation*}
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\begin{equation*}
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\Path \tp (P \tp I → \MCU) → P\ 0 → P\ 1 → \MCU
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\Path \tp (P \tp \I → \MCU) → P\ 0 → P\ 1 → \MCU
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\end{equation*}
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\end{equation*}
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\pause
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\pause
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For $P \tp I → \MCU$, $A \tp \MCU$ and $a_0, a_1 \tp A$
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For $P \tp \I → \MCU$, $A \tp \MCU$ and $a_0, a_1 \tp A$
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inhabitants of $\Path\ P\ a_0\ a_1$ are like functions
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inhabitants of $\Path\ P\ a_0\ a_1$ are like functions
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%
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%
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$$
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$$
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p \tp ∏_{i \tp I} P\ i
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p \tp ∏_{i \tp \I} P\ i
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$$
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$$
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%
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%
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Which satisfy $p\ 0 & = a_0$ and $p\ 1 & = a_1$
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Which satisfy $p\ 0 & = a_0$ and $p\ 1 & = a_1$
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\end{align*}
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\end{align*}
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where
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where
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$$
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$$
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\phi\ f ≜ \identity
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\phi\ f ≜
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( \lll f ≡ f )
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( \identity \lll f ≡ f )
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×
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×
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( f \lll \identity ≡ f)
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( f \lll \identity ≡ f)
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$$
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$$
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\pause
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\pause
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Let $\approxeq$ denote ismorphism of objects. We can then construct
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Let $\approxeq$ denote isomorphism of objects. We can then construct
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the identity isomorphism in any category:
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the identity isomorphism in any category:
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$$
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$$
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\identity , \identity , \var{isIdentity} \tp A \approxeq A
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\identity , \identity , \var{isIdentity} \tp A \approxeq A
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@ -37,7 +37,7 @@ Master's thesis in Computer Science
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\vfill
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\vfill
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\centering
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\centering
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\includegraphics[width=0.2\pdfpagewidth]{logo_eng.pdf}
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\includegraphics[width=0.2\pdfpagewidth]{assets/logo_eng.pdf}
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\vspace{5mm}
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\vspace{5mm}
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\textsc{Department of Computer Science and Engineering}\\
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\textsc{Department of Computer Science and Engineering}\\
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