491 lines
11 KiB
TeX
491 lines
11 KiB
TeX
\documentclass[a4paper]{beamer}
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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}
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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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\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{Introduction}
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Category Theory: The study of abstract functions. Slogan: ``It's the
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arrows that matter''\pause
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Objects are equal ``up to isomorphism''. Univalence makes this notion
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precise.\pause
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Agda does not permit proofs of univalence. Cubical Agda admits
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this.\pause
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Goal: Construct a category whose terminal objects are (equivalent to)
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products. Use this to conclude that products are propositions, not a
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structure on a category.
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\end{frame}
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\begin{frame}
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\frametitle{Outline}
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The path type
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Definition of a (pre-) category
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1-categories
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Univalent (proper) categories
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The category of spans
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\end{frame}
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\section{Paths}
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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 $P \tp \I → \MCU$ and $a_0 \tp P\ 0$, $a_1 \tp P\ 1$
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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{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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\end{frame}
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\begin{frame}
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\frametitle{Pre categories}
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\framesubtitle{1-categories}
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Cubical Agda does not admit \emph{Uniqueness of Identity Proofs}
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(UIP). Rather there is a hierarchy of \emph{Homotopy Types}:
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Contractible types, mere propositions, sets, groupoids, \dots
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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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\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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\end{frame}
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\begin{frame}
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\frametitle{Outline}
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The path type \ensuremath{\checkmark}
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Definition of a (pre-) category \ensuremath{\checkmark}
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1-categories \ensuremath{\checkmark}
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Univalent (proper) categories
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The category of spans
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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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Let $\approxeq$ denote isomorphism 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 inverse of $\idToIso$:
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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{Propositionality of products}
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Construct a category for which it is the case that the terminal
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objects are equivalent to products:
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\begin{align*}
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\var{Terminal} ≃ \var{Product}\ ℂ\ 𝒜\ ℬ
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\end{align*}
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\pause
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And since equivalences preserve homotopy levels we get:
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%
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$$
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\isProp\ \left(\var{Product}\ \bC\ 𝒜\ ℬ\right)
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$$
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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 $(ι, \inv{ι}, \var{inv}) \tp A \approxeq B$.
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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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$$
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p ≜ \isoToId\ (\iota, \inv{\iota}, \var{inv}) \tp A ≡ B
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$$
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%
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\pause
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and consequently a path on arrows:
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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 (\idToIso\ p)_1 && \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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$$
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d \tp D\ A\ \refl =
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\left(\var{coe}\ \refl_{\var{dom}}\ f ≡ f \lll \inv{(\idToIso\ \refl)}\right)
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$$
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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_{\var{dom}}\ f
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& =
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\var{coe}\ \refl\ f \\
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& ≡ f
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&& \text{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{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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$$
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\var{pathInd}\ D\ d\ B\ p
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$$
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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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(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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\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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$$
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(\fst\ f, \fst\ \inv{f}
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, \congruence\ \fst\ \var{inv}_f
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, \congruence\ \fst\ \var{inv}_{\inv{f}}
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)
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\tp X \approxeq Y
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$$
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\pause
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%
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This gives rise to the following paths:
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%
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\begin{align*}
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\begin{split}
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\widetilde{p} & \tp X ≡ Y \\
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\widetilde{p}_{𝒜} & \tp \Arrow\ X\ 𝒜 ≡ \Arrow\ Y\ 𝒜 \\
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\end{split}
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\end{align*}
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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, proof, cont'd}
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It remains to construct:
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%
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\begin{align*}
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\begin{split}
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& \Path\ (λ i → \widetilde{p}_{𝒜}\ i)\ x_{𝒜}\ y_{𝒜}
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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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This is achieved with the following lemma:
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%
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\begin{align*}
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∏_{q \tp A ≡ B} \var{coe}\ q\ x_{𝒜} ≡ y_{𝒜}
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→
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\Path\ (λ i → q\ i)\ x_{𝒜}\ y_{𝒜}
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\end{align*}
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%
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Which is used without proof.\pause
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So the construction reduces to:
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%
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\begin{align*}
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\var{coe}\ \widetilde{p}_{𝒜}\ x_{𝒜} ≡ y_{𝒜}
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\end{align*}%
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\pause%
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This is proven with:
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%
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\begin{align*}
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\var{coe}\ \widetilde{p}_{𝒜}\ x_{𝒜}
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& ≡ x_{𝒜} \lll \fst\ \inv{f} && \text{\ref{eq:coeDom}} \\
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& ≡ y_{𝒜} && \text{Property of span category}
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\end{align*}
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\end{frame}
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\begin{frame}
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\frametitle{Propositionality of products}
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We have
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%
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$$
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\isProp\ \var{Terminal}
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$$\pause
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%
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We can show:
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\begin{align*}
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\var{Terminal} ≃ \var{Product}\ ℂ\ 𝒜\ ℬ
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\end{align*}
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\pause
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And since equivalences preserve homotopy levels we get:
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%
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$$
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\isProp\ \left(\var{Product}\ \bC\ 𝒜\ ℬ\right)
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$$
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\end{frame}
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\begin{frame}
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\frametitle{Monads}
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\framesubtitle{Monoidal form}
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%
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\begin{align*}
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\EndoR & \tp \Functor\ ℂ\ ℂ \\
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\pureNT
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& \tp \NT{\widehat{\identity}}{\EndoR} \\
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\joinNT
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& \tp \NT{(\EndoR \oplus \EndoR)}{\EndoR}
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\end{align*}
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\pause
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%
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Let $\fmap$ be the map on arrows of $\EndoR$.
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%
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\begin{align*}
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\join \lll \fmap\ \join
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& ≡ \join \lll \join \\
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\join \lll \pure\ & ≡ \identity \\
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\join \lll \fmap\ \pure & ≡ \identity
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\end{align*}
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\end{frame}
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\begin{frame}
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\frametitle{Monads}
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\framesubtitle{Kleisli form}
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%
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\begin{align*}
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\omapR & \tp \Object → \Object \\
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\pure & \tp % ∏_{X \tp Object}
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\Arrow\ X\ (\omapR\ X) \\
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\bind & \tp
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\Arrow\ X\ (\omapR\ Y)
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\to
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\Arrow\ (\omapR\ X)\ (\omapR\ Y)
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\end{align*}\pause
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%
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\begin{align*}
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\fish & \tp
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\Arrow\ A\ (\omapR\ B)
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→
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\Arrow\ B\ (\omapR\ C)
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→
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\Arrow\ A\ (\omapR\ C) \\
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f \fish g & ≜ f \rrr (\bind\ g)
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\end{align*}
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\pause
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%
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\begin{align*}
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\bind\ \pure & ≡ \identity_{\omapR\ X} \\
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\pure \fish f & ≡ f \\
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(\bind\ f) \rrr (\bind\ g) & ≡ \bind\ (f \fish g)
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\end{align*}
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\end{frame}
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\begin{frame}
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\frametitle{Monads}
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\framesubtitle{Equivalence}
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In the monoidal formulation we can define $\bind$:
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%
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$$
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\bind\ f ≜ \join \lll \fmap\ f
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$$
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\pause
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%
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And likewise in the Kleisli formulation we can define $\join$:
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%
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$$
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\join ≜ \bind\ \identity
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$$
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\pause
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The laws are logically equivalent. Since logical equivalence is
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enough for as an equivalence of types for propositions we get:
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%
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$$
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\var{Monoidal} ≃ \var{Kleisli}
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$$
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%
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\end{frame}
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\end{document}
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