\chapter{Introduction} This thesis is a case-study in the application of cubical Agda in the context of category theory. At the center of this is the notion of \nomenindex{equality}. In type-theory there are two pervasive notions of equality: \nomenindex{judgmental equality} and \nomenindex{propositional equality}. Judgmental equality is a property of the type system. Judgmental equality on the other hand is usually defined \emph{within} the system. When introducing definitions this report will use the notation $\defeq$. Judgmental equalities written $=$. For propositional equalities the notation $\equiv$ is used. For judgmental equality there are some properties that it must satisfy. \nomenindex{sound}, enjoy \nomenindex{canonicity} and be a \nomen{congruence relation}. Soundness means that things judged to be equal are equal with respects to the model of the theory (the meta theory). It must be a congruence relation because otherwise the relation certainly does not adhere to our notion of equality. One would be able to conclude things like: $x \equiv y \rightarrow f\ x \nequiv f\ y$. 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. To work as a programming languages it is necessary for judgmental equality to be \nomenindex{decidable}. Being decidable simply means that that an algorithm exists to decide whether two terms are equal. For any practical implementation the decidability must also be effectively computable. For propositional equality the decidability requirement is relaxed. It is not in general possible to decide the correctness of logical propositions (cf.\ Hilbert's \emph{entscheidigungsproblem}). There are two flavors of type-theory. \emph{Intensional-} and \emph{extensional-} type theory. Identity types in extensional type theory are required to be \nomen{propositions}{proposition}. That is, a type with at most one inhabitant. In extensional type thoery the principle of reflection % $$a ≡ b → a = b$$ % is enough to make type checking undecidable. This report focuses on Agda which at a glance can be thought of a version of intensional type theory. Pattern-matching in regular Agda let's one prove \nomenindex{axiom K}. Axiom K states that any two identity proofs are propositionally identical. The usual notion of propositional equality in \nomenindex{Intensional Type Theory} (ITT) is quite restrictive. In the next section a few motivating examples will highlight this. There exist techniques to circumvent these problems, as we shall see. This thesis will explore an extension to Agda that redefines the notion of propositional equality and as such is an alternative to these other techniques. The extension is called cubical Agda. Cubical Agda drops Axiom K as this does not permit \nomenindex{functional extensionality} and \nomenindex{univalence}. What makes this extension particularly interesting is that it gives a \emph{constructive} interpretation of univalence. What all this means will be elaborated in the following sections. % \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 $f \equiv g$. There is no way to construct a proof asserting the obvious equivalence of $f$ and $g$. 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. We cannot show that they are equal, even though we can prove them equal for all points. For functions this is exactly the notion of equality that we are interested in: Functions are considered equal when they are equal for all inputs. This is called \nomenindex{point wise equality}, where the \emph{points} of a function refer 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 is defined for a fixed category $\bC$ and an %% object $X \in \bC$. It's map on objects is defined thus: %% % %% \begin{align*} %% \lambda\ A \to \Arrow\ X\ A %% \end{align*} %% % %% That is, it maps objects to arrows. So, it's map on arrows must map an arrow $\Arrow\ A\ B$ to an %% The map on objects is defined thus: %% % %% \begin{align*} %% \lambda f \to %% \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 A formalization in Agda using the setoid approach: \url{https://github.com/copumpkin/categories} \item A formalization in Agda with univalence and functional extensionality as postulates: \url{https://github.com/pcapriotti/agda-categories} \item A formalization in Coq in the homotopic setting: \url{https://github.com/HoTT/HoTT/tree/master/theories/Categories} \item A formalization in \emph{CubicalTT} -- a language designed for cubical type theory. Formalizes many different things, but only a few concepts from category theory: \url{https://github.com/mortberg/cubicaltt} \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. 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} (\cite[p. 3]{huber-2016}). 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 a priori. 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 a priori. That is, the developer must manually show that e.g.\ the relation is a congruence. Equational proofs $a \sim_{X} b$ are in some sense `local' to the extensional set $(X , \sim)$. To e.g.\ prove that $x ∼ y → f\ x ∼ f\ y$ for some function $f \tp A → B$ between two extensional sets $A$ and $B$ it must be shown that $f$ is a groupoid homomorphism. This makes it very cumbersome to work with in practice (\cite[p. 4]{huber-2016}). \section{Conventions} 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}