654 lines
15 KiB
TeX
654 lines
15 KiB
TeX
\documentclass[a4paper,handout]{beamer}
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\usetheme{metropolis}
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\beamertemplatenavigationsymbolsempty
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%% \usecolortheme[named=seagull]{structure}
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\input{packages.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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\newcommand{\myname}{Frederik Hangh{\o}j Iversen}
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\author[\myname]{
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\myname\\
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\footnotesize Supervisors: Thierry Coquand, Andrea Vezzosi\\
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Examiner: Andreas Abel
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}
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\institute{Chalmers University of Technology}
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\begin{document}
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\frame{\titlepage}
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\begin{frame}
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\frametitle{Motivating example}
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\framesubtitle{Functional extensionality}
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Consider the functions
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\begin{align*}
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\var{zeroLeft} & ≜ \lambda (n \tp \bN) \mto (0 + n \tp \bN) \\
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\var{zeroRight} & ≜ \lambda (n \tp \bN) \mto (n + 0 \tp \bN)
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\end{align*}
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\pause
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We have
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%
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$$
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∏_{n \tp \bN} \var{zeroLeft}\ n ≡ \var{zeroRight}\ n
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$$
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%
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\pause
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But not
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%
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$$
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\var{zeroLeft} ≡ \var{zeroRight}
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$$
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%
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\pause
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We need
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%
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$$
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\funExt \tp ∏_{a \tp A} f\ a ≡ g\ a → f ≡ g
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$$
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\end{frame}
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\begin{frame}
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\frametitle{Motivating example}
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\framesubtitle{Univalence}
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Consider the set
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$\{x \mid \phi\ x \land \psi\ x\}$
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\pause
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If we show $∀ x . \psi\ x ≡ \top$
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then we want to conclude
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$\{x \mid \phi\ x \land \psi\ x\} ≡ \{x \mid \phi\ x\}$
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\pause
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We need univalence:
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$$(A ≃ B) ≃ (A ≡ B)$$
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\pause
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%
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We will return to $≃$, but for now think of it as an
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isomorphism, so it induces maps:
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\begin{align*}
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\var{toPath} & \tp (A ≃ B) → (A ≡ B) \\
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\var{toEquiv} & \tp (A ≡ B) → (A ≃ B)
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\end{align*}
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\end{frame}
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\begin{frame}
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\frametitle{Paths}
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\framesubtitle{Definition}
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Heterogeneous paths
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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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\end{equation*}
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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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inhabitants of $\Path\ P\ a_0\ a_1$ are like functions
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%
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$$
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p \tp ∏_{i \tp I} P\ i
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$$
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%
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Which satisfy $p\ 0 & = a_0$ and $p\ 1 & = a_1$
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\pause
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Homogenous paths
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$$
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a_0 ≡ a_1 ≜ \Path\ (\var{const}\ A)\ a_0\ a_1
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$$
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\end{frame}
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\begin{frame}
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\frametitle{Paths}
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\framesubtitle{Functional extenstionality}
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$$
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\funExt & \tp ∏_{a \tp A} f\ a ≡ g\ a → f ≡ g
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$$
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\pause
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$$
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\funExt\ p ≜ λ i\ a → p\ a\ i
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$$
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\pause
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$$
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\funExt\ (\var{const}\ \refl)
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\tp
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\var{zeroLeft} ≡ \var{zeroRight}
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$$
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\end{frame}
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\begin{frame}
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\frametitle{Paths}
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\framesubtitle{Homotopy levels}
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\begin{align*}
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& \isContr && \tp \MCU → \MCU \\
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& \isContr\ A && ≜ ∑_{c \tp A} ∏_{a \tp A} a ≡ c
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\end{align*}
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\pause
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\begin{align*}
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& \isProp && \tp \MCU → \MCU \\
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& \isProp\ A && ≜ ∏_{a_0, a_1 \tp A} a_0 ≡ a_1
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\end{align*}
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\pause
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\begin{align*}
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& \isSet && \tp \MCU → \MCU \\
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& \isSet\ A && ≜ ∏_{a_0, a_1 \tp A} \isProp\ (a_0 ≡ a_1)
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\end{align*}
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\begin{align*}
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& \isGroupoid && \tp \MCU → \MCU \\
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& \isGroupoid\ A && ≜ ∏_{a_0, a_1 \tp A} \isSet\ (a_0 ≡ a_1)
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\end{align*}
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\end{frame}
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\begin{frame}
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\frametitle{Paths}
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\framesubtitle{A few lemmas}
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Let $D$ be a type-family:
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$$
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D \tp ∏_{b \tp A} ∏_{p \tp a ≡ b} \MCU
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$$
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%
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\pause
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And $d$ and in inhabitant of $D$ at $\refl$:
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%
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$$
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d \tp D\ a\ \refl
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$$
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%
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\pause
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We then have the function:
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%
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$$
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\pathJ\ D\ d \tp ∏_{b \tp A} ∏_{p \tp a ≡ b} D\ b\ p
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$$
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\end{frame}
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\begin{frame}
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\frametitle{Paths}
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\framesubtitle{A few lemmas}
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Given
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\begin{align*}
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A & \tp \MCU \\
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P & \tp A → \MCU \\
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\var{propP} & \tp ∏_{x \tp A} \isProp\ (P\ x) \\
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p & \tp a_0 ≡ a_1 \\
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p_0 & \tp P\ a_0 \\
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p_1 & \tp P\ a_1
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\end{align*}
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%
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We have
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$$
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\lemPropF\ \var{propP}\ p
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\tp
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\Path\ (\lambda\; i \mto P\ (p\ i))\ p_0\ p_1
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$$
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%
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\end{frame}
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\begin{frame}
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\frametitle{Paths}
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\framesubtitle{A few lemmas}
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$∏$ preserves $\isProp$:
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$$
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\mathit{propPi}
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\tp
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\left(∏_{a \tp A} \isProp\ (P\ a)\right)
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→ \isProp\ \left(∏_{a \tp A} P\ a\right)
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$$
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\pause
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$∑$ preserves $\isProp$:
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$$
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\mathit{propSig} \tp \isProp\ A → \left(∏_{a \tp A} \isProp\ (P\ a)\right) → \isProp\ \left(∑_{a \tp A} P\ a\right)
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$$
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\end{frame}
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\begin{frame}
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\frametitle{Pre categories}
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\framesubtitle{Definition}
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Data:
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\begin{align*}
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\Object & \tp \Type \\
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\Arrow & \tp \Object → \Object → \Type \\
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\identity & \tp \Arrow\ A\ A \\
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\lll & \tp \Arrow\ B\ C → \Arrow\ A\ B → \Arrow\ A\ C
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\end{align*}
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%
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\pause
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Laws:
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%
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$$
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h \lll (g \lll f) ≡ (h \lll g) \lll f
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$$
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$$
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(\identity \lll f ≡ f)
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×
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(f \lll \identity ≡ f)
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$$
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\pause
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1-categories:
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$$
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\isSet\ (\Arrow\ A\ B)
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$$
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\end{frame}
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\begin{frame}
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\frametitle{Pre categories}
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\framesubtitle{Propositionality}
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$$
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\isProp\ \left( (\identity \comp f ≡ f) × (f \comp \identity ≡ f) \right)
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$$
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\pause
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\begin{align*}
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\isProp\ \IsPreCategory
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\end{align*}
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\pause
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\begin{align*}
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\var{isAssociative} & \tp \var{IsAssociative}\\
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\isIdentity & \tp \var{IsIdentity}\\
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\var{arrowsAreSets} & \tp \var{ArrowsAreSets}
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\end{align*}
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\pause
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\begin{align*}
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& \var{propIsAssociative} && a.\var{isAssociative}\
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&& b.\var{isAssociative} && i \\
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& \propIsIdentity && a.\isIdentity\
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&& b.\isIdentity && i \\
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& \var{propArrowsAreSets} && a.\var{arrowsAreSets}\
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&& b.\var{arrowsAreSets} && i
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\end{align*}
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\end{frame}
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\begin{frame}
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\frametitle{Categories}
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\framesubtitle{Univalence}
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\begin{align*}
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\var{IsIdentity} & ≜
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∏_{A\ B \tp \Object} ∏_{f \tp \Arrow\ A\ B} \phi\ f
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%% \\
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%% & \mathrel{\ } \identity \lll f ≡ f × f \lll \identity ≡ f
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\end{align*}
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where
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$$
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\phi\ f ≜ \identity
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( \lll f ≡ f )
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×
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( f \lll \identity ≡ f)
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$$
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\pause
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Let $\approxeq$ denote ismorphism of objects. We can then construct
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the identity isomorphism in any category:
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$$
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\identity , \identity , \var{isIdentity} \tp A \approxeq A
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$$
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\pause
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Likewise since paths are substitutive we can promote a path to an isomorphism:
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$$
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\idToIso \tp A ≡ B → A ≊ B
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$$
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\pause
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For a category to be univalent we require this to be an equivalence:
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%
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$$
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\isEquiv\ (A ≡ B)\ (A \approxeq B)\ \idToIso
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$$
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%
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\end{frame}
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\begin{frame}
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\frametitle{Categories}
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\framesubtitle{Univalence, cont'd}
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$$\isEquiv\ (A ≡ B)\ (A \approxeq B)\ \idToIso$$
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\pause%
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$$(A ≡ B) ≃ (A \approxeq B)$$
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\pause%
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$$(A ≡ B) ≅ (A \approxeq B)$$
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\pause%
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Name the above maps:
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$$\idToIso \tp A ≡ B → A ≊ B$$
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%
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$$\isoToId \tp (A \approxeq B) → (A ≡ B)$$
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\end{frame}
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\begin{frame}
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\frametitle{Categories}
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\framesubtitle{Propositionality}
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$$
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\isProp\ \IsCategory = ∏_{a, b \tp \IsCategory} a ≡ b
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$$
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\pause
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So, for
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$$
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a\ b \tp \IsCategory
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$$
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the proof obligation is the pair:
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%
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\begin{align*}
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p & \tp a.\isPreCategory ≡ b.\isPreCategory \\
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& \mathrel{\ } \Path\ (\lambda\; i → (p\ i).Univalent)\ a.\isPreCategory\ b.\isPreCategory
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\end{align*}
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\end{frame}
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\begin{frame}
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\frametitle{Categories}
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\framesubtitle{Propositionality, cont'd}
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First path given by:
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$$
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p
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≜
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\var{propIsPreCategory}\ a\ b
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\tp
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a.\isPreCategory ≡ b.\isPreCategory
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$$
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\pause
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Use $\lemPropF$ for the latter.
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\pause
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%
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Univalence is indexed by an identity proof. So $A ≜
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IsIdentity\ identity$ and $B ≜ \var{Univalent}$.
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\pause
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%
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$$
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\lemPropF\ \var{propUnivalent}\ p
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$$
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\end{frame}
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\begin{frame}
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\frametitle{Categories}
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\framesubtitle{A theorem}
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%
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Let the isomorphism $(ι, \inv{ι}) \tp A \approxeq B$.
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%
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\pause
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%
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The isomorphism induces the path
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%
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$$
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p ≜ \idToIso\ (\iota, \inv{\iota}) \tp A ≡ B
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$$
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%
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\pause
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and consequently an arrow:
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%
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$$
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p_{\var{dom}} ≜ \congruence\ (λ x → \Arrow\ x\ X)\ p
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\tp
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\Arrow\ A\ X ≡ \Arrow\ B\ X
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$$
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%
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\pause
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The proposition is:
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%
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\begin{align}
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\label{eq:coeDom}
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\tag{$\var{coeDom}$}
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∏_{f \tp A → X}
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\var{coe}\ p_{\var{dom}}\ f ≡ f \lll \inv{\iota}
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\end{align}
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\end{frame}
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\begin{frame}
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\frametitle{Categories}
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\framesubtitle{A theorem, proof}
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\begin{align*}
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\var{coe}\ p_{\var{dom}}\ f
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& ≡ f \lll \inv{(\idToIso\ p)} && \text{By path-induction} \\
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& ≡ f \lll \inv{\iota}
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&& \text{$\idToIso$ and $\isoToId$ are inverses}\\
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\end{align*}
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\pause
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%
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Induction will be based at $A$. Let $\widetilde{B}$ and $\widetilde{p}
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\tp A ≡ \widetilde{B}$ be given.
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%
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\pause
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%
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Define the family:
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%
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$$
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D\ \widetilde{B}\ \widetilde{p} ≜
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\var{coe}\ \widetilde{p}_{\var{dom}}\ f
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≡
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f \lll \inv{(\idToIso\ \widetilde{p})}
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$$
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\pause
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%
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The base-case becomes:
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$$
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d \tp D\ A\ \refl =
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\var{coe}\ \refl_{\var{dom}}\ f ≡ f \lll \inv{(\idToIso\ \refl)}
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$$
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\end{frame}
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\begin{frame}
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\frametitle{Categories}
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\framesubtitle{A theorem, proof, cont'd}
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$$
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d \tp
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\var{coe}\ \refl_{\var{dom}}\ f ≡ f \lll \inv{(\idToIso\ \refl)}
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$$
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\pause
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\begin{align*}
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\var{coe}\ \refl^*\ f
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& ≡ f
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&& \text{$\refl$ is a neutral element for $\var{coe}$}\\
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& ≡ f \lll \identity \\
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& ≡ f \lll \var{subst}\ \refl\ \identity
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&& \text{$\refl$ is a neutral element for $\var{subst}$}\\
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& ≡ f \lll \inv{(\idToIso\ \refl)}
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&& \text{By definition of $\idToIso$}\\
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\end{align*}
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\pause
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In conclusion, the theorem is inhabited by:
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$$
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\label{eq:pathJ-example}
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\pathJ\ D\ d\ B\ p
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$$
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\end{frame}
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\begin{frame}
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\frametitle{Span category} \framesubtitle{Definition} Given a base
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category $\bC$ and two objects in this category $\pairA$ and $\pairB$
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we can construct the \nomenindex{span category}:
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%
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\pause
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Objects:
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$$
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∑_{X \tp Object} \Arrow\ X\ \pairA × \Arrow\ X\ \pairB
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$$
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\pause
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%
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Arrows between objects $A ,\ a_{\pairA} ,\ a_{\pairB}$ and
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$B ,\ b_{\pairA} ,\ b_{\pairB}$:
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%
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$$
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∑_{f \tp \Arrow\ A\ B}
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b_{\pairA} \lll f ≡ a_{\pairA} ×
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b_{\pairB} \lll f ≡ a_{\pairB}
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$$
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\end{frame}
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\begin{frame}
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\frametitle{Span category}
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\framesubtitle{Univalence}
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\begin{align*}
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\label{eq:univ-0}
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(X , x_{𝒜} , x_{ℬ}) ≡ (Y , y_{𝒜} , y_{ℬ})
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\end{align*}
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\begin{align*}
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\label{eq:univ-1}
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\begin{split}
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p \tp & X ≡ Y \\
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& \Path\ (λ i → \Arrow\ (p\ i)\ 𝒜)\ x_{𝒜}\ y_{𝒜} \\
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& \Path\ (λ i → \Arrow\ (p\ i)\ ℬ)\ x_{ℬ}\ y_{ℬ}
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\end{split}
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\end{align*}
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\begin{align*}
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\begin{split}
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\var{iso} \tp & X \approxeq Y \\
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& \Path\ (λ i → \Arrow\ (\widetilde{p}\ i)\ 𝒜)\ x_{𝒜}\ y_{𝒜} \\
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& \Path\ (λ i → \Arrow\ (\widetilde{p}\ i)\ ℬ)\ x_{ℬ}\ y_{ℬ}
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\end{split}
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\end{align*}
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\begin{align*}
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(X , x_{𝒜} , x_{ℬ}) ≊ (Y , y_{𝒜} , y_{ℬ})
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\end{align*}
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\end{frame}
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\begin{frame}
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\frametitle{Span category}
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\framesubtitle{Univalence, proof}
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%
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\begin{align*}
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%% (f, \inv{f}, \var{inv}_f, \var{inv}_{\inv{f}})
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%% \tp
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(X, x_{𝒜}, x_{ℬ}) \approxeq (Y, y_{𝒜}, y_{ℬ})
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\to
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\begin{split}
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\var{iso} \tp & X \approxeq Y \\
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& \Path\ (λ i → \Arrow\ (\widetilde{p}\ i)\ 𝒜)\ x_{𝒜}\ y_{𝒜} \\
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& \Path\ (λ i → \Arrow\ (\widetilde{p}\ i)\ ℬ)\ x_{ℬ}\ y_{ℬ}
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\end{split}
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\end{align*}
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\pause
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%
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Let $(f, \inv{f}, \var{inv}_f, \var{inv}_{\inv{f}})$ be an inhabitant
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of the antecedent.\pause
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Projecting out the first component gives us the isomorphism
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%
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$$
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(\fst\ f, \fst\ \inv{f}
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, \congruence\ \fst\ \var{inv}_f
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||
, \congruence\ \fst\ \var{inv}_{\inv{f}}
|
||
)
|
||
\tp X \approxeq Y
|
||
$$
|
||
\pause
|
||
%
|
||
This gives rise to the following paths:
|
||
%
|
||
\begin{align*}
|
||
\begin{split}
|
||
\widetilde{p} & \tp X ≡ Y \\
|
||
\widetilde{p}_{𝒜} & \tp \Arrow\ X\ 𝒜 ≡ \Arrow\ Y\ 𝒜 \\
|
||
\end{split}
|
||
\end{align*}
|
||
%
|
||
\end{frame}
|
||
\begin{frame}
|
||
\frametitle{Span category}
|
||
\framesubtitle{Univalence, proof, cont'd}
|
||
It remains to construct:
|
||
%
|
||
\begin{align*}
|
||
\begin{split}
|
||
\label{eq:product-paths}
|
||
& \Path\ (λ i → \widetilde{p}_{𝒜}\ i)\ x_{𝒜}\ y_{𝒜}
|
||
\end{split}
|
||
\end{align*}
|
||
\pause
|
||
%
|
||
This is achieved with the following lemma:
|
||
%
|
||
\begin{align*}
|
||
∏_{q \tp A ≡ B} \var{coe}\ q\ x_{𝒜} ≡ y_{𝒜}
|
||
→
|
||
\Path\ (λ i → q\ i)\ x_{𝒜}\ y_{𝒜}
|
||
\end{align*}
|
||
%
|
||
Which is used without proof.\pause
|
||
|
||
So the construction reduces to:
|
||
%
|
||
\begin{align*}
|
||
\var{coe}\ \widetilde{p}_{𝒜}\ x_{𝒜} ≡ y_{𝒜}
|
||
\end{align*}%
|
||
\pause%
|
||
This is proven with:
|
||
%
|
||
\begin{align*}
|
||
\var{coe}\ \widetilde{p}_{𝒜}\ x_{𝒜}
|
||
& ≡ x_{𝒜} \lll \fst\ \inv{f} && \text{\ref{eq:coeDom}} \\
|
||
& ≡ y_{𝒜} && \text{Property of span category}
|
||
\end{align*}
|
||
\end{frame}
|
||
\begin{frame}
|
||
\frametitle{Propositionality of products}
|
||
We have
|
||
%
|
||
$$
|
||
\isProp\ \var{Terminal}
|
||
$$\pause
|
||
%
|
||
We can show:
|
||
\begin{align*}
|
||
\var{Terminal} ≃ \var{Product}\ ℂ\ 𝒜\ ℬ
|
||
\end{align*}
|
||
\pause
|
||
And since equivalences preserve homotopy levels we get:
|
||
%
|
||
$$
|
||
\isProp\ \left(\var{Product}\ \bC\ 𝒜\ ℬ\right)
|
||
$$
|
||
\end{frame}
|
||
\begin{frame}
|
||
\frametitle{Monads}
|
||
\framesubtitle{Monoidal form}
|
||
%
|
||
\begin{align*}
|
||
\EndoR & \tp \Endo ℂ \\
|
||
\pureNT
|
||
& \tp \NT{\EndoR^0}{\EndoR} \\
|
||
\joinNT
|
||
& \tp \NT{\EndoR^2}{\EndoR}
|
||
\end{align*}
|
||
\pause
|
||
%
|
||
Let $\fmap$ be the map on arrows of $\EndoR$. Likewise
|
||
$\pure$ and $\join$ are the maps of the natural transformations
|
||
$\pureNT$ and $\joinNT$ respectively.
|
||
%
|
||
\begin{align*}
|
||
\join \lll \fmap\ \join
|
||
& ≡ \join \lll \join \\
|
||
\join \lll \pure\ & ≡ \identity \\
|
||
\join \lll \fmap\ \pure & ≡ \identity
|
||
\end{align*}
|
||
\end{frame}
|
||
\begin{frame}
|
||
\frametitle{Monads}
|
||
\framesubtitle{Kleisli form}
|
||
%
|
||
\begin{align*}
|
||
\omapR & \tp \Object → \Object \\
|
||
\pure & \tp % ∏_{X \tp Object}
|
||
\Arrow\ X\ (\omapR\ X) \\
|
||
\bind & \tp
|
||
\Arrow\ X\ (\omapR\ Y)
|
||
\to
|
||
\Arrow\ (\omapR\ X)\ (\omapR\ Y)
|
||
\end{align*}\pause
|
||
%
|
||
\begin{align*}
|
||
\fish & \tp
|
||
\Arrow\ A\ (\omapR\ B)
|
||
→
|
||
\Arrow\ B\ (\omapR\ C)
|
||
→
|
||
\Arrow\ A\ (\omapR\ C) \\
|
||
f \fish g & ≜ f \rrr (\bind\ g)
|
||
\end{align*}
|
||
\pause
|
||
%
|
||
\begin{align*}
|
||
\label{eq:monad-kleisli-laws-0}
|
||
\bind\ \pure & ≡ \identity_{\omapR\ X} \\
|
||
\label{eq:monad-kleisli-laws-1}
|
||
\pure \fish f & ≡ f \\
|
||
\label{eq:monad-kleisli-laws-2}
|
||
(\bind\ f) \rrr (\bind\ g) & ≡ \bind\ (f \fish g)
|
||
\end{align*}
|
||
\end{frame}
|
||
\begin{frame}
|
||
\frametitle{Monads}
|
||
\framesubtitle{Equivalence}
|
||
In the monoidal formulation we can define $\bind$:
|
||
%
|
||
$$
|
||
\bind\ f ≜ \join \lll \fmap\ f
|
||
$$
|
||
\pause
|
||
%
|
||
And likewise in the Kleisli formulation we can define $\join$:
|
||
%
|
||
$$
|
||
\join ≜ \bind\ \identity
|
||
$$
|
||
\pause
|
||
The laws are logically equivalent. So we get:
|
||
%
|
||
$$
|
||
\var{Monoidal} ≃ \var{Kleisli}
|
||
$$
|
||
%
|
||
\end{frame}
|
||
\end{document}
|