fredefox/cat
1
0
Fork 0
Code Issues Pull requests Releases Wiki Activity

Add introduction

Browse source
This commit is contained in:
Frederik Hanghøj Iversen 2018-05-15 18:34:25 +02:00
parent 8a0ea9f4a5
commit d33c814e78
1 changed files with 95 additions and 55 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

150
doc/introduction.tex
View file

@ -1,8 +1,44 @@
\chapter{Introduction} \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, it is a property that is automatically checked by
a type checker. As such there are some properties judgmental
equalities must crucially have. It must be \nomenindex{decidable},
\nomenindex{sound}, enjoy \nomenindex{canonicity} and be a
\nomen{congruence relation}. 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. Soundness means that things judged to be equal actually
\emph{are} considered equal. 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: $x \nequiv y
\rightarrow f\ x \equiv f\ y$. Canonicity will be explained later in
this introduction after we've seen an example of judgmental- and
propositional equality at play for a simple example.\TODO{How to
motivate canonicity for equality}.
For propositional equality the decidability requirement is relaxed. It
is not in general possible to decide the correctness of logical
propositions (cf. Hilbert's \nomen{entscheidigungsproblem}).
Propositional equality are provided by the developer. When introducing
definitions this report will use the notation $\defeq$. Judgmental
equalities written $=$. For propositional equalities the notation
$\equiv$ is used.
The usual notion of propositional equality in \nomen{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.
%
\section{Motivating examples} \section{Motivating examples}
% %
In the following two sections I present two examples that illustrate some In the following two sections I present two examples that illustrate
limitations inherent in ITT and -- by extension -- Agda. some limitations inherent in ITT and -- by extension -- Agda.
% %
\subsection{Functional extensionality} \subsection{Functional extensionality}
\label{sec:functional-extensionality}% \label{sec:functional-extensionality}%
@ -18,9 +54,9 @@ Consider the functions:
\end{equation*}% \end{equation*}%
\end{multicols}% \end{multicols}%
% %
The term $n + 0$ is The term $n + 0$ is
\nomenindex{definitionally} equal to $n$, which we \nomenindex{definitionally} equal to $n$, which we
write as $n + 0 = n$. This is also called write as $n + 0 = n$. This is also called
\nomenindex{judgmental equality}. \nomenindex{judgmental equality}.
We call it definitional equality because the \emph{equality} arises We call it definitional equality because the \emph{equality} arises
from the \emph{definition} of $+$ which is: from the \emph{definition} of $+$ which is:
@ -39,35 +75,43 @@ as $n + 0 \equiv n$. Propositional equality means that there is a
proof that exhibits this relation. Since equality is a transitive proof that exhibits this relation. Since equality is a transitive
relation we have that $n + 0 \equiv 0 + n$. relation we have that $n + 0 \equiv 0 + n$.
Unfortunately we don't have $f \equiv g$.\footnote{Actually showing this is Unfortunately we don't have $f \equiv g$. There is no way to construct
outside the scope of this text. Essentially it would involve giving a model a proof asserting the obvious equivalence of $f$ and $g$. Actually
for our type theory that validates all our axioms but where $f \equiv g$ is showing this is outside the scope of this text. Essentially it would
not true.} There is no way to construct a proof asserting the obvious involve giving a model for our type theory that validates all our
equivalence of $f$ and $g$ -- even though we can prove them equal for all axioms but where $f \equiv g$ is not true. We cannot show that they
points. This is exactly the notion of equality of functions that we are are equal, even though we can prove them equal for all points. For
interested in; that they are equal for all inputs. We call this 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.
\nomenindex{point-wise equality}, where the \emph{points} of a function refers %% In the context of category theory functional extensionality is e.g.
to its arguments. %% needed to show that representable functors are indeed functors. The
%% representable functor is defined for a fixed category $\bC$ and an
In the context of category theory functional extensionality is e.g. needed to %% object $X \in \bC$. It's map on objects is defined thus:
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*}
% %% \lambda\ A \to \Arrow\ X\ A
\begin{align*} %% \end{align*}
\fmap \defeq \lambda\ X \to \Hom_{\bC}(A, X) %% %
\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:
The proof obligation that this satisfies the identity law of functors %% %
($\fmap\ \idFun \equiv \idFun$) thus becomes: %% \begin{align*}
% %% \lambda f \to
\begin{align*} %% \end{align*}
\Hom(A, \idFun_{\bX}) = (\lambda\ g \to \idFun \comp g) \equiv \idFun_{\Sets} %% %
\end{align*} %% The proof obligation that this satisfies the identity law of functors
% %% ($\fmap\ \idFun \equiv \idFun$) thus becomes:
One needs functional extensionality to ``go under'' the function arrow and apply %% %
the (left) identity law of the underlying category to prove $\idFun \comp g %% \begin{align*}
\equiv g$ and thus close the goal. %% \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} \subsection{Equality of isomorphic types}
% %
@ -75,7 +119,7 @@ Let $\top$ denote the unit type -- a type with a single constructor.
In the propositions as types interpretation of type theory $\top$ is 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 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 an element for each $a \tp A$. So in a sense they have the same shape
(Greek; (Greek;
\nomenindex{isomorphic}). The second element of the pair does not \nomenindex{isomorphic}). The second element of the pair does not
add any ``interesting information''. It can be useful to identify such add any ``interesting information''. It can be useful to identify such
types. In fact, it is quite commonplace in mathematics. Say we look at types. In fact, it is quite commonplace in mathematics. Say we look at
@ -87,10 +131,11 @@ be performed in ITT.
More specifically what we are interested in is a way of identifying More specifically what we are interested in is a way of identifying
\nomenindex{equivalent} types. I will return to the definition of equivalence later \nomenindex{equivalent} types. I will return to the definition of
in section \S\ref{sec:equiv}, but for now it is sufficient to think of an equivalence later in section \S\ref{sec:equiv}, but for now it is
equivalence as a one-to-one correspondence. We write $A \simeq B$ to assert that sufficient to think of an equivalence as a one-to-one correspondence.
$A$ and $B$ are equivalent types. The principle of univalence says that: We write $A \simeq B$ to assert that $A$ and $B$ are equivalent types.
The principle of univalence says that:
% %
$$\mathit{univalence} \tp (A \simeq B) \simeq (A \equiv B)$$ $$\mathit{univalence} \tp (A \simeq B) \simeq (A \equiv B)$$
% %
@ -119,25 +164,20 @@ implementations of category theory in Agda:
% %
\begin{itemize} \begin{itemize}
\item \item
A formalization in Agda using the setoid approach:
\url{https://github.com/copumpkin/categories} \url{https://github.com/copumpkin/categories}
A formalization in Agda using the setoid approach
\item \item
A formalization in Agda with univalence and functional
extensionality as postulates:
\url{https://github.com/pcapriotti/agda-categories} \url{https://github.com/pcapriotti/agda-categories}
A formalization in Agda with univalence and functional extensionality as
postulates.
\item \item
A formalization in Coq in the homotopic setting:
\url{https://github.com/HoTT/HoTT/tree/master/theories/Categories} \url{https://github.com/HoTT/HoTT/tree/master/theories/Categories}
A formalization in Coq in the homotopic setting
\item \item
\url{https://github.com/mortberg/cubicaltt}
A formalization in CubicalTT - a language designed for cubical type theory. A formalization in CubicalTT - a language designed for cubical type theory.
Formalizes many different things, but only a few concepts from category Formalizes many different things, but only a few concepts from category
theory. theory:
\url{https://github.com/mortberg/cubicaltt}
\end{itemize} \end{itemize}
% %
The contribution of this thesis is to explore how working in a cubical setting The contribution of this thesis is to explore how working in a cubical setting
@ -158,7 +198,7 @@ canonical form.
Another approach is to use the \emph{setoid interpretation} of type Another approach is to use the \emph{setoid interpretation} of type
theory (\cite{hofmann-1995,huber-2016}). With this approach one works theory (\cite{hofmann-1995,huber-2016}). With this approach one works
with with
\nomenindex{extensional sets} $(X, \sim)$, that is a type $X \tp \MCU$ \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. and an equivalence relation $\sim\ \tp X \to X \to \MCU$ on that type.
Under the setoid interpretation the equivalence relation serve as a Under the setoid interpretation the equivalence relation serve as a
@ -178,18 +218,18 @@ inherently `local' to the extensional set $(X , \sim)$.
\TODO{Talk a bit about terminology. Find a good place to stuff this little \TODO{Talk a bit about terminology. Find a good place to stuff this little
section.} section.}
In the remainder of this paper I will use the term In the remainder of this paper I will use the term
\nomenindex{Type} to describe -- \nomenindex{Type} to describe --
well, types. Thereby diverging from the notation in Agda where the keyword well, types. Thereby diverging from the notation in Agda where the keyword
\texttt{Set} refers to types. \texttt{Set} refers to types.
\nomenindex{Set} on the other hand shall refer to the \nomenindex{Set} on the other hand shall refer to the
homotopical notion of a set. I will also leave all universe levels implicit. homotopical notion of a set. I will also leave all universe levels implicit.
And I use the term And I use the term
\nomenindex{arrow} to refer to morphisms in a category, \nomenindex{arrow} to refer to morphisms in a category,
whereas the terms whereas the terms
\nomenindex{morphism}, \nomenindex{morphism},
\nomenindex{map} or \nomenindex{map} or
\nomenindex{function} \nomenindex{function}
shall be reserved for talking about type theoretic functions; i.e. shall be reserved for talking about type theoretic functions; i.e.
functions in Agda. functions in Agda.