221 lines
9.2 KiB
TeX
221 lines
9.2 KiB
TeX
\chapter{Introduction}
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\section{Motivating examples}
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%
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In the following two sections I present two examples that illustrate some
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limitations inherent in ITT and -- by extension -- Agda.
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%
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\subsection{Functional extensionality}
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\label{sec:functional-extensionality}%
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Consider the functions:
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%
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\begin{multicols}{2}
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\noindent%
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\begin{equation*}%
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f \defeq \lambda\ (n \tp \bN) \to (0 + n \tp \bN)
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\end{equation*}%
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\begin{equation*}%
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g \defeq \lambda\ (n \tp \bN) \to (n + 0 \tp \bN)
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\end{equation*}%
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\end{multicols}%
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%
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The term $n + 0$ is
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\nomenindex{definitionally} equal to $n$, which we
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write as $n + 0 = n$. This is also called
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\nomenindex{judgmental equality}.
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We call it definitional equality because the \emph{equality} arises
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from the \emph{definition} of $+$ which is:
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%
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\begin{align*}
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+ & \tp \bN \to \bN \to \bN \\
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n + 0 & \defeq n \\
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n + (\suc{m}) & \defeq \suc{(n + m)}
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\end{align*}
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%
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Note that $0 + n$ is \emph{not} definitionally equal to $n$. $0 + n$
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is in normal form. I.e.; there is no rule for $+$ whose left hand side
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matches this expression. We \emph{do}, however, have that they are
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\nomen{propositionally}{propositional equality} equal, which we write
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as $n + 0 \equiv n$. Propositional equality means that there is a
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proof that exhibits this relation. Since equality is a transitive
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relation we have that $n + 0 \equiv 0 + n$.
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Unfortunately we don't have $f \equiv g$.\footnote{Actually showing this is
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outside the scope of this text. Essentially it would involve giving a model
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for our type theory that validates all our axioms but where $f \equiv g$ is
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not true.} There is no way to construct a proof asserting the obvious
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equivalence of $f$ and $g$ -- even though we can prove them equal for all
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points. This is exactly the notion of equality of functions that we are
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interested in; that they are equal for all inputs. We call this
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\nomenindex{point-wise equality}, where the \emph{points} of a function refers
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to its arguments.
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In the context of category theory functional extensionality is e.g. needed to
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show that representable functors are indeed functors. The representable functor
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for a category $\bC$ and a fixed object in $A \in \bC$ is defined to be:
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%
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\begin{align*}
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\fmap \defeq \lambda\ X \to \Hom_{\bC}(A, X)
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\end{align*}
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%
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The proof obligation that this satisfies the identity law of functors
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($\fmap\ \idFun \equiv \idFun$) thus becomes:
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%
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\begin{align*}
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\Hom(A, \idFun_{\bX}) = (\lambda\ g \to \idFun \comp g) \equiv \idFun_{\Sets}
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\end{align*}
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%
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One needs functional extensionality to ``go under'' the function arrow and apply
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the (left) identity law of the underlying category to prove $\idFun \comp g
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\equiv g$ and thus close the goal.
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%
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\subsection{Equality of isomorphic types}
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%
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Let $\top$ denote the unit type -- a type with a single constructor.
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In the propositions as types interpretation of type theory $\top$ is
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the proposition that is always true. The type $A \x \top$ and $A$ has
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an element for each $a \tp A$. So in a sense they have the same shape
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(Greek;
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\nomenindex{isomorphic}). The second element of the pair does not
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add any ``interesting information''. It can be useful to identify such
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types. In fact, it is quite commonplace in mathematics. Say we look at
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a set $\{x \mid \phi\ x \land \psi\ x\}$ and somehow conclude that
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$\psi\ x \equiv \top$ for all $x$. A mathematician would immediately
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conclude $\{x \mid \phi\ x \land \psi\ x\} \equiv \{x \mid \phi\ x\}$
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without thinking twice. Unfortunately such an identification can not
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be performed in ITT.
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More specifically what we are interested in is a way of identifying
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\nomenindex{equivalent} types. I will return to the definition of equivalence later
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in section \S\ref{sec:equiv}, but for now it is sufficient to think of an
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equivalence as a one-to-one correspondence. We write $A \simeq B$ to assert that
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$A$ and $B$ are equivalent types. The principle of univalence says that:
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%
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$$\mathit{univalence} \tp (A \simeq B) \simeq (A \equiv B)$$
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%
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In particular this allows us to construct an equality from an equivalence
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($\mathit{ua} \tp (A \simeq B) \to (A \equiv B)$) and vice versa.
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\section{Formalizing Category Theory}
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%
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The above examples serve to illustrate a limitation of ITT. One case where these
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limitations are particularly prohibitive is in the study of Category Theory. At
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a glance category theory can be described as ``the mathematical study of
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(abstract) algebras of functions'' (\cite{awodey-2006}). By that token
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functional extensionality is particularly useful for formulating Category
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Theory. In Category theory it is also common to identify isomorphic structures
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and univalence gives us a way to make this notion precise. In fact we can
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formulate this requirement within our formulation of categories by requiring the
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\emph{categories} themselves to be univalent as we shall see.
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\section{Context}
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\label{sec:context}
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%
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The idea of formalizing Category Theory in proof assistants is not new. There
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are a multitude of these available online. Just as a first reference see this
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question on Math Overflow: \cite{mo-formalizations}. Notably these
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implementations of category theory in Agda:
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%
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\begin{itemize}
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\item
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\url{https://github.com/copumpkin/categories}
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A formalization in Agda using the setoid approach
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\item
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\url{https://github.com/pcapriotti/agda-categories}
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A formalization in Agda with univalence and functional extensionality as
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postulates.
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\item
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\url{https://github.com/HoTT/HoTT/tree/master/theories/Categories}
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A formalization in Coq in the homotopic setting
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\item
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\url{https://github.com/mortberg/cubicaltt}
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A formalization in CubicalTT - a language designed for cubical type theory.
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Formalizes many different things, but only a few concepts from category
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theory.
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\end{itemize}
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%
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The contribution of this thesis is to explore how working in a cubical setting
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will make it possible to prove more things and to reuse proofs and to try and
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compare some aspects of this formalization with the existing ones.\TODO{How can
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I live up to this?}
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There are alternative approaches to working in a cubical setting where one can
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still have univalence and functional extensionality. One option is to postulate
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these as axioms. This approach, however, has other shortcomings, e.g.; you lose
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\nomenindex{canonicity} (\TODO{Pageno!} \cite{huber-2016}). Canonicity means that any
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well typed term evaluates to a \emph{canonical} form. For example for a closed
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term $e \tp \bN$ it will be the case that $e$ reduces to $n$ applications of
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$\mathit{suc}$ to $0$ for some $n$; $e = \mathit{suc}^n\ 0$. Without canonicity
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terms in the language can get ``stuck'' -- meaning that they do not reduce to a
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canonical form.
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Another approach is to use the \emph{setoid interpretation} of type
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theory (\cite{hofmann-1995,huber-2016}). With this approach one works
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with
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\nomenindex{extensional sets} $(X, \sim)$, that is a type $X \tp \MCU$
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and an equivalence relation $\sim\ \tp X \to X \to \MCU$ on that type.
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Under the setoid interpretation the equivalence relation serve as a
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sort of ``local'' propositional equality. Since the developer gets to
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pick this relation it is not guaranteed to be a congruence relation
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apriori. So this must be verified manually by the developer.
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Furthermore, functions between different setoids must be shown to be
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setoid homomorphism, that is; they preserve the relation.
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This approach has other drawbacks; it does not satisfy
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all propositional equalities of type theory (\TODO{Citation needed}), is
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cumbersome to work with in practice (\cite[p. 4]{huber-2016}) and makes
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equational proofs less reusable since equational proofs $a \sim_{X} b$ are
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inherently `local' to the extensional set $(X , \sim)$.
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\section{Conventions}
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\TODO{Talk a bit about terminology. Find a good place to stuff this little
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section.}
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In the remainder of this paper I will use the term
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\nomenindex{Type} to describe --
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well, types. Thereby diverging from the notation in Agda where the keyword
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\texttt{Set} refers to types.
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\nomenindex{Set} on the other hand shall refer to the
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homotopical notion of a set. I will also leave all universe levels implicit.
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And I use the term
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\nomenindex{arrow} to refer to morphisms in a category,
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whereas the terms
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\nomenindex{morphism},
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\nomenindex{map} or
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\nomenindex{function}
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shall be reserved for talking about type theoretic functions; i.e.
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functions in Agda.
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$\defeq$ will be used for introducing definitions. $=$ will be used to for
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judgmental equality and $\equiv$ will be used for propositional equality.
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All this is summarized in the following table:
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\begin{center}
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\begin{tabular}{ c c c }
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Name & Agda & Notation \\
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\hline
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\varindex{Type} & \texttt{Set} & $\Type$ \\
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\varindex{Set} & \texttt{Σ Set IsSet} & $\Set$ \\
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Function, morphism, map & \texttt{A → B} & $A → B$ \\
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Dependent- ditto & \texttt{(a : A) → B} & $∏_{a \tp A} B$ \\
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\varindex{Arrow} & \texttt{Arrow A B} & $\Arrow\ A\ B$ \\
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\varindex{Object} & \texttt{C.Object} & $̱ℂ.Object$ \\
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Definition & \texttt{=} & $̱\defeq$ \\
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Judgmental equality & \null & $̱=$ \\
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Propositional equality & \null & $̱\equiv$
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\end{tabular}
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\end{center}
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