\documentclass[a4paper,handout]{beamer}
\input{packages.tex}
\input{macros.tex}
\title{Univalent Categories}
\author{Frederik Hangh{\o}j Iversen}
\institute{Chalmers University of Technology}
\begin{document}
\frame{\titlepage}

\begin{frame}
  \frametitle{Motivating example}
  \framesubtitle{Functional extensionality}
Consider the functions
\begin{align*}
  \var{zeroLeft} & \defeq (n \tp \bN) \mto (0 + n \tp \bN) \\
  \var{zeroRight} & \defeq (n \tp \bN) \mto (n + 0 \tp \bN)
\end{align*}
\pause
We have
%
$$
\prod_{n \tp \bN} n + 0 \equiv 0 + n
$$
%
\pause
But not
%
$$
\var{zeroLeft} \equiv \var{zeroRight}
$$
%
\pause
We need
%
$$
\funExt \tp \prod_{a \tp A} f\ a \equiv g\ a \to f \equiv g
$$

\end{frame}
\begin{frame}
  \frametitle{Motivating example}
  \framesubtitle{Univalence}
  Consider the set
  $\{x \mid \phi\ x \land \psi\ x\}$
  \pause

  If we show $\forall x . \psi\ x \equiv \top$
  then we want to conclude
  $\{x \mid \phi\ x \land \psi\ x\} \equiv \{x \mid \phi\ x\}$
  \pause

  We need univalence:
  $$(A \simeq B) \simeq (A \equiv B)$$
  \pause
%
  We will return to $\simeq$, but for not, think of it as an
  isomorphism, so it induces maps:
  \begin{align*}
    \var{toPath}  & \tp (A \simeq B) \to (A \equiv B) \\
    \var{toEquiv} & \tp (A \equiv B) \to (A \simeq B)
  \end{align*}
\end{frame}
\begin{frame}
  \frametitle{Paths}
  \framesubtitle{Definition}
Heterogeneous paths
\begin{equation*}
  \Path \tp (P \tp I → \MCU) → P\ 0 → P\ 1 → \MCU
\end{equation*}
\pause
  For $P \tp I \to \MCU$, $A \tp \MCU$ and $a_0, a_1 \tp A$
  inhabitants of $\Path\ P\ a_0\ a_1$ are like functions
%
$$
p \tp \prod_{i \tp I} P\ i
$$
%
Which satisfy $p\ 0 & = a_0$ and $p\ 1 & = a_1$
\pause

Homogenous paths
$$
a_0 \equiv a_1 \defeq \Path\ (\var{const}\ A)\ a_0\ a_1
$$
\end{frame}
\begin{frame}
\frametitle{Paths}
\framesubtitle{Functional extenstionality}
$$
\funExt & \tp \prod_{a \tp A} f\ a \equiv g\ a \to f \equiv g
$$
\pause
$$
\funExt\ p \defeq λ i\ a → p\ a\ i
$$
\pause
$$
\funExt\ (\var{const}\ \refl)
\tp
\var{zeroLeft} \equiv \var{zeroRight}
$$
\end{frame}
\begin{frame}
  \frametitle{Paths}
  \framesubtitle{Homotopy levels}
\begin{align*}
& \isContr    && \tp    \MCU \to \MCU \\
& \isContr\ A && \defeq \sum_{c \tp A} \prod_{a \tp A} a \equiv c
\end{align*}
\pause
\begin{align*}
& \isProp    && \tp \MCU \to \MCU \\
& \isProp\ A && \defeq \prod_{a_0, a_1 \tp A} a_0 \equiv a_1
\end{align*}
\pause
\begin{align*}
& \isSet    && \tp \MCU \to \MCU \\
& \isSet\ A && \defeq \prod_{a_0, a_1 \tp A} \isProp\ (a_0 \equiv a_1)
\end{align*}
\pause
\end{frame}
\begin{frame}
\frametitle{Paths}
\framesubtitle{A few lemmas}
Let $D$ be a type-family:
$$
D \tp \prod_{b \tp A} \prod_{p \tp a ≡ b} \MCU
$$
%
\pause
And $d$ and in inhabitant of $D$ at $\refl$:
%
$$
d \tp D\ a\ \refl
$$
%
\pause
We then have the function:
%
\begin{equation}
\pathJ\ D\ d \tp \prod_{b \tp A} \prod_{p \tp a ≡ b} D\ a\ p
\end{equation}
\end{frame}
\begin{frame}
\frametitle{Paths}
\framesubtitle{A few lemmas}
Given
\begin{align*}
  A           & \tp \MCU \\
  P           & \tp A \to \MCU \\
  \var{propP} & \tp \prod_{x \tp A} \isProp\ (P\ x) \\
  p           & \tp a_0 \equiv a_1 \\
  p_0         & \tp P\ a_0 \\
  p_1         & \tp P\ a_1
\end{align*}
%
We have
$$
\lemPropF\ \var{propP}\ p
\tp
\Path\ (\lambda\; i \mto P\ (p\ i))\ p_0\ p_1
$$
%
\end{frame}
\begin{frame}
\frametitle{Paths}
\framesubtitle{A few lemmas}
$\prod$ preserves $\isProp$:
$$
\mathit{propPi}
\tp
\left(\prod_{a \tp A} \isProp\ (P\ a)\right)
\to \isProp\ \left(\prod_{a \tp A} P\ a\right)
$$
\pause
$\sum$ preserves $\isProp$:
$$
\mathit{propSig} \tp \isProp\ A \to \left(\prod_{a \tp A} \isProp\ (P\ a)\right) \to \isProp\ \left(\sum_{a \tp A} P\ a\right)
$$
\end{frame}
\begin{frame}
\frametitle{Categories}
\framesubtitle{Definition}
Data:
\begin{align*}
  \Object   & \tp \Type \\
  \Arrow    & \tp \Object \to \Object \to \Type \\
  \identity & \tp \Arrow\ A\ A \\
  \lll      & \tp \Arrow\ B\ C \to \Arrow\ A\ B \to \Arrow\ A\ C
\end{align*}
%
Laws:
%
$$
h \lll (g \lll f) ≡ (h \lll g) \lll f
$$
$$
\identity \lll f ≡ f \x
f \lll \identity ≡ f
$$
\pause
1-categories:
$$
\isSet\ (\Arrow\ A\ B)
$$
\pause
Univalent categories:
$$
\isEquiv\ (A \equiv B)\ (A \approxeq B)\ \idToIso
$$
\end{frame}
\begin{frame}
\frametitle{Categories}
\framesubtitle{Univalence}
\begin{align*}
\var{IsIdentity} & \defeq
\prod_{A\ B \tp \Object} \prod_{f \tp \Arrow\ A\ B} \phi\ f
%% \\
%%   & \mathrel{\ } \identity \lll f \equiv f \x f \lll \identity \equiv f
\end{align*}
where
$$
\phi\ f \defeq \identity \lll f \equiv f \x f \lll \identity \equiv f
$$
Let $\approxeq$ denote ismorphism of objects. We can then construct
the identity isomorphism in any category:
$$
\identity , \identity , \var{isIdentity} \tp A \approxeq A
$$
Likewise since paths are substitutive we can promote a path to an isomorphism:
$$
\idToIso \tp A ≡ B → A ≊ B
$$
For a category to be univalent we require this to be an equivalence: 
\end{frame}
\end{document}