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Frederik Hanghøj Iversen 2018-05-15 18:34:25 +02:00
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\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}
%
In the following two sections I present two examples that illustrate some
limitations inherent in ITT and -- by extension -- Agda.
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}%
@ -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
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
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.
\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.
%% 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}
%
@ -87,10 +131,11 @@ 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:
\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:
%
$$\mathit{univalence} \tp (A \simeq B) \simeq (A \equiv B)$$
%
@ -119,25 +164,20 @@ implementations of category theory in Agda:
%
\begin{itemize}
\item
A formalization in Agda using the setoid approach:
\url{https://github.com/copumpkin/categories}
A formalization in Agda using the setoid approach
\item
A formalization in Agda with univalence and functional
extensionality as postulates:
\url{https://github.com/pcapriotti/agda-categories}
A formalization in Agda with univalence and functional extensionality as
postulates.
\item
A formalization in Coq in the homotopic setting:
\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.
theory:
\url{https://github.com/mortberg/cubicaltt}
\end{itemize}
%
The contribution of this thesis is to explore how working in a cubical setting