\chapter{Introduction}
\section{Motivating examples}
%
In the following two sections I present two examples that illustrate some
limitations inherent in ITT and -- by extension -- Agda.
%
\subsection{Functional extensionality}
\label{sec:functional-extensionality}%
Consider the functions:
%
\begin{multicols}{2}
  \noindent%
  \begin{equation*}%
    f \defeq \lambda\ (n \tp \bN) \to (0 + n \tp \bN)
  \end{equation*}%
  \begin{equation*}%
    g \defeq \lambda\ (n \tp \bN) \to (n + 0 \tp \bN)
  \end{equation*}%
\end{multicols}%
%
The term $n + 0$ is 
\nomenindex{definitionally} equal to $n$, which we
write as $n + 0 = n$. This is also called 
\nomenindex{judgmental equality}.
We call it definitional equality because the \emph{equality} arises
from the \emph{definition} of $+$ which is:
%
\begin{align*}
  +           & \tp \bN \to \bN \to \bN      \\
  n + 0       & \defeq n                   \\
  n + (\suc{m}) & \defeq \suc{(n + m)}
\end{align*}
%
Note that $0 + n$ is \emph{not} definitionally equal to $n$. $0 + n$
is in normal form. I.e.; there is no rule for $+$ whose left hand side
matches this expression. We \emph{do}, however, have that they are
\nomen{propositionally}{propositional equality} equal, which we write
as $n + 0 \equiv n$. Propositional equality means that there is a
proof that exhibits this relation. Since equality is a transitive
relation we have that $n + 0 \equiv 0 + n$.

Unfortunately we don't have $f \equiv g$.\footnote{Actually showing this is
outside the scope of this text. Essentially it would involve giving a model
for our type theory that validates all our axioms but where $f \equiv g$ is
not true.} There is no way to construct a proof asserting the obvious
equivalence of $f$ and $g$ -- even though we can prove them equal for all
points. This is exactly the notion of equality of functions that we are
interested in; that they are equal for all inputs. We call this

\nomenindex{point-wise equality}, where the \emph{points} of a function refers
to its arguments.

In the context of category theory functional extensionality is e.g. needed to
show that representable functors are indeed functors. The representable functor
for a category $\bC$ and a fixed object in $A \in \bC$ is defined to be:
%
\begin{align*}
\fmap \defeq \lambda\ X \to \Hom_{\bC}(A, X)
\end{align*}
%
The proof obligation that this satisfies the identity law of functors
($\fmap\ \idFun \equiv \idFun$) thus becomes:
%
\begin{align*}
\Hom(A, \idFun_{\bX}) = (\lambda\ g \to \idFun \comp g) \equiv \idFun_{\Sets}
\end{align*}
%
One needs functional extensionality to ``go under'' the function arrow and apply
the (left) identity law of the underlying category to prove $\idFun \comp g
\equiv g$ and thus close the goal.
%
\subsection{Equality of isomorphic types}
%
Let $\top$ denote the unit type -- a type with a single constructor.
In the propositions as types interpretation of type theory $\top$ is
the proposition that is always true. The type $A \x \top$ and $A$ has
an element for each $a \tp A$. So in a sense they have the same shape
(Greek; 
\nomenindex{isomorphic}). The second element of the pair does not
add any ``interesting information''. It can be useful to identify such
types. In fact, it is quite commonplace in mathematics. Say we look at
a set $\{x \mid \phi\ x \land \psi\ x\}$ and somehow conclude that
$\psi\ x \equiv \top$ for all $x$. A mathematician would immediately
conclude $\{x \mid \phi\ x \land \psi\ x\} \equiv \{x \mid \phi\ x\}$
without thinking twice. Unfortunately such an identification can not
be performed in ITT.

More specifically what we are interested in is a way of identifying

\nomenindex{equivalent} types. I will return to the definition of equivalence later
in section \S\ref{sec:equiv}, but for now it is sufficient to think of an
equivalence as a one-to-one correspondence. We write $A \simeq B$ to assert that
$A$ and $B$ are equivalent types. The principle of univalence says that:
%
$$\mathit{univalence} \tp (A \simeq B) \simeq (A \equiv B)$$
%
In particular this allows us to construct an equality from an equivalence
($\mathit{ua} \tp (A \simeq B) \to (A \equiv B)$) and vice versa.

\section{Formalizing Category Theory}
%
The above examples serve to illustrate a limitation of ITT. One case where these
limitations are particularly prohibitive is in the study of Category Theory. At
a glance category theory can be described as ``the mathematical study of
(abstract) algebras of functions'' (\cite{awodey-2006}). By that token
functional extensionality is particularly useful for formulating Category
Theory. In Category theory it is also common to identify isomorphic structures
and univalence gives us a way to make this notion precise. In fact we can
formulate this requirement within our formulation of categories by requiring the
\emph{categories} themselves to be univalent as we shall see.

\section{Context}
\label{sec:context}
%
The idea of formalizing Category Theory in proof assistants is not new. There
are a multitude of these available online. Just as a first reference see this
question on Math Overflow: \cite{mo-formalizations}. Notably these
implementations of category theory in Agda:
%
\begin{itemize}
\item
  \url{https://github.com/copumpkin/categories}

  A formalization in Agda using the setoid approach
\item
  \url{https://github.com/pcapriotti/agda-categories}

  A formalization in Agda with univalence and functional extensionality as
  postulates.
\item
  \url{https://github.com/HoTT/HoTT/tree/master/theories/Categories}

  A formalization in Coq in the homotopic setting
\item
  \url{https://github.com/mortberg/cubicaltt}

  A formalization in CubicalTT - a language designed for cubical type theory.
  Formalizes many different things, but only a few concepts from category
  theory.

\end{itemize}
%
The contribution of this thesis is to explore how working in a cubical setting
will make it possible to prove more things and to reuse proofs and to try and
compare some aspects of this formalization with the existing ones.\TODO{How can
  I live up to this?}

There are alternative approaches to working in a cubical setting where one can
still have univalence and functional extensionality. One option is to postulate
these as axioms. This approach, however, has other shortcomings, e.g.; you lose

\nomenindex{canonicity} (\TODO{Pageno!} \cite{huber-2016}). Canonicity means that any
well typed term evaluates to a \emph{canonical} form. For example for a closed
term $e \tp \bN$ it will be the case that $e$ reduces to $n$ applications of
$\mathit{suc}$ to $0$ for some $n$; $e = \mathit{suc}^n\ 0$. Without canonicity
terms in the language can get ``stuck'' -- meaning that they do not reduce to a
canonical form.

Another approach is to use the \emph{setoid interpretation} of type
theory (\cite{hofmann-1995,huber-2016}). With this approach one works
with 
\nomenindex{extensional sets} $(X, \sim)$, that is a type $X \tp \MCU$
and an equivalence relation $\sim\ \tp X \to X \to \MCU$ on that type.
Under the setoid interpretation the equivalence relation serve as a
sort of ``local'' propositional equality. Since the developer gets to
pick this relation it is not guaranteed to be a congruence relation
apriori. So this must be verified manually by the developer.
Furthermore, functions between different setoids must be shown to be
setoid homomorphism, that is; they preserve the relation.

This approach has other drawbacks; it does not satisfy
all propositional equalities of type theory (\TODO{Citation needed}), is
cumbersome to work with in practice (\cite[p. 4]{huber-2016}) and makes
equational proofs less reusable since equational proofs $a \sim_{X} b$ are
inherently `local' to the extensional set $(X , \sim)$.

\section{Conventions}
\TODO{Talk a bit about terminology. Find a good place to stuff this little
  section.}

In the remainder of this paper I will use the term 
\nomenindex{Type} to describe --
well, types. Thereby diverging from the notation in Agda where the keyword
\texttt{Set} refers to types. 
\nomenindex{Set} on the other hand shall refer to the
homotopical notion of a set. I will also leave all universe levels implicit.

And I use the term 
\nomenindex{arrow} to refer to morphisms in a category,
whereas the terms 
\nomenindex{morphism}, 
\nomenindex{map} or 
\nomenindex{function}
shall be reserved for talking about type theoretic functions; i.e.
functions in Agda.

$\defeq$ will be used for introducing definitions. $=$ will be used to for
judgmental equality and $\equiv$ will be used for propositional equality.

All this is summarized in the following table:

\begin{center}
\begin{tabular}{ c c c }
Name & Agda & Notation \\
\hline

\varindex{Type}            & \texttt{Set}         & $\Type$            \\

\varindex{Set}             & \texttt{Σ Set IsSet} & $\Set$             \\
Function, morphism, map & \texttt{A → B}       & $A → B$            \\
Dependent- ditto        & \texttt{(a : A) → B} & $∏_{a \tp A} B$  \\

\varindex{Arrow}           & \texttt{Arrow A B}   & $\Arrow\ A\ B$     \\

\varindex{Object}          & \texttt{C.Object}    & $̱ℂ.Object$     \\
Definition              & \texttt{=}           & $̱\defeq$       \\
Judgmental equality     & \null                & $̱=$            \\
Propositional equality  & \null                & $̱\equiv$
\end{tabular}
\end{center}
