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doc/cubical.tex
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@ -1,17 +1,18 @@
\chapter{Cubical Agda} \chapter{Cubical Agda}
\section{Propositional equality} \section{Propositional equality}
In Agda judgmental equality is a feature of the type-system. It's a property of Judgmental equality in Agda is a feature of the type-system. It's something that
types that can be checked by computational means. In the example from the can be checked automatically by the type-checker: In the example from the
introduction $n + 0$ can be judged to be equal to $n$ simply by expanding the introduction $n + 0$ can be judged to be equal to $n$ simply by expanding the
definition of $+$. definition of $+$.
Propositional equality on the other hand is defined within the language itself. On the other hand, propositional equality is something defined within the
Cubical Agda extends the underlying type system (\TODO{Cite someone smarter than language itself. Propositional equality cannot be derived automatically. The
me with a good resource on this}) but introduces a data-type within the normal definition of judgmental equality is an inductive data-type. Cubical Agda
languages. discards this type in favor of a new primitives that has certain computational
properties exclusive to it.
Exceprts of the source code relevant to this section can be found in appendix Exceprts of the source code relevant to this section can be found in appendix
\ref{sec:app-cubical}. \S\ref{sec:app-cubical}.
\subsection{The equality type} \subsection{The equality type}
The usual notion of judgmental equality says that given a type $A \tp \MCU$ and The usual notion of judgmental equality says that given a type $A \tp \MCU$ and
@ -21,7 +22,8 @@ two points of $A$; $a_0, a_1 \tp A$ we can form the type:
a_0 \equiv a_1 \tp \MCU a_0 \equiv a_1 \tp \MCU
\end{align} \end{align}
% %
In Agda this is defined as an inductive data-type with the single constructor: In Agda this is defined as an inductive data-type with the single constructor
for any $a \tp A$:
% %
\begin{align} \begin{align}
\refl \tp a \equiv a \refl \tp a \equiv a
@ -29,7 +31,7 @@ In Agda this is defined as an inductive data-type with the single constructor:
% %
For any $a \tp A$. For any $a \tp A$.
There also exist a related notion of \emph{heterogeneous} equality where allows There also exist a related notion of \emph{heterogeneous} equality which allows
for equating points of different types. In this case given two types $A, B \tp for equating points of different types. In this case given two types $A, B \tp
\MCU$ and two points $a \tp A$, $b \tp B$ we can construct the type: \MCU$ and two points $a \tp A$, $b \tp B$ we can construct the type:
% %
@ -37,7 +39,8 @@ for equating points of different types. In this case given two types $A, B \tp
a \cong b \tp \MCU a \cong b \tp \MCU
\end{align} \end{align}
% %
This is likewise defined as an inductive data-type with a single constructors: This is likewise defined as an inductive data-type with a single constructors
for any $a \tp A$:
% %
\begin{align} \begin{align}
\refl \tp a \cong a \refl \tp a \cong a
@ -47,11 +50,11 @@ In Cubical Agda these two notions are paralleled with homogeneous- and
heterogeneous paths respectively. heterogeneous paths respectively.
% %
\subsection{The path type} \subsection{The path type}
In Cubical Agda judgmental equality is encapsulated with the type: Judgmental equality in Cubical Agda is encapsulated with the type:
% %
$$ \begin{equation}
\Path \tp (P \tp I → \MCU) → P\ 0 → P\ 1 → \MCU \Path \tp (P \tp I → \MCU) → P\ 0 → P\ 1 → \MCU
$$ \end{equation}
% %
$I$ is a special data-type (\TODO{that also has special computational properties $I$ is a special data-type (\TODO{that also has special computational properties
AFAIK}) called the index set. $I$ can be thought of simply as the interval on AFAIK}) called the index set. $I$ can be thought of simply as the interval on
@ -62,11 +65,11 @@ left-endpoint and $P\ 1$ as the type at the right-endpoint. The type is called
$\Path$ because it is connected with paths in homotopy theory. The intuition $\Path$ because it is connected with paths in homotopy theory. The intuition
behind this is that $\Path$ describes paths in $\MCU$ -- i.e. between types. For behind this is that $\Path$ describes paths in $\MCU$ -- i.e. between types. For
a path $p$ for the point $p\ i$ the index $i$ describes how far along the path a path $p$ for the point $p\ i$ the index $i$ describes how far along the path
one has moved. An inhabitant of $\Path\ P\ a_0\ a_1$ is a (dependent-) one has moved. An inhabitant of $\Path\ P\ a_0\ a_1$ is a (dependent-) function,
function, $p$, from the index-space to the path-space: $p$, from the index-space to the path-space:
% %
$$ $$
p \tp I \to P\ i p \tp \prod_{i \tp I} P\ i
$$ $$
% %
Which must satisfy being judgmentally equal to $a_0$ (respectively $a_1$) at the Which must satisfy being judgmentally equal to $a_0$ (respectively $a_1$) at the
@ -77,16 +80,16 @@ endpoints. I.e.:
p\ 1 & = a_1 p\ 1 & = a_1
\end{align*} \end{align*}
% %
The notion of ``homogeneous equalities'' can be recovered by not letting the The notion of ``homogeneous equalities'' is recovered when $P$ does not depend
path-space $P$ depend on it's argument: on it's argument:
% %
$$ $$
a_0 \equiv a_1 \defeq \Path\ (\lambda i \to A)\ a_0\ a_1 a_0 \equiv a_1 \defeq \Path\ (\lambda i \to A)\ a_0\ a_1
$$ $$
% %
For $A \tp \MCU$, $a_0, a_1 \tp A$. I will generally prefer to use the notation For $A \tp \MCU$, $a_0, a_1 \tp A$. I will generally prefer to use the notation
$a_0 \equiv a_1$ when talking about non-dependent paths and use the notation $a \equiv b$ when talking about non-dependent paths and use the notation
$\Path\ (\lambda i \to A)\ a_0\ a_1$ when the path-space is of particular $\Path\ (\lambda i \to P\ i)\ a\ b$ when the path-space is of particular
interest. interest.
With this definition we can also recover reflexivity. That is, for any $A \tp With this definition we can also recover reflexivity. That is, for any $A \tp
@ -99,28 +102,63 @@ With this definition we can also recover reflexivity. That is, for any $A \tp
\end{aligned} \end{aligned}
\end{equation} \end{equation}
% %
Or, in other terms; reflexivity is the path in $A$ that is $a$ at the left Here the path-space is $P \defeq \lambda i \to A$ and it satsifies $P\ i = A$
endpoint as well as at the right endpoint. It is inhabited by the path which definitionally. So to inhabit it, is to give a path $I \to A$ which is
stays constantly at $a$ at any index $i$. judgmentally $a$ at either endpoint. This is satisfied by the constant path;
i.e. the path that stays at $a$ at any index $i$.
Paths have some other important properties, but they are not the focus of this It's also surpisingly easy to show functional extensionality with which we can
thesis. \TODO{Refer the reader somewhere for more info.} construct a path between $f$ and $g$ -- the function defined in the introduction
(section \S\ref{sec:functional-extensionality}).
%% module _ {ℓa ℓb} {A : Set ℓa} {B : A → Set ℓb} where
%% funExt : {f g : (x : A) → B x} → ((x : A) → f x ≡ g x) → f ≡ g
Functional extensionality is the proposition, given a type $A \tp \MCU$, a
family of types $B \tp A \to \MCU$ and functions $f, g \tp \prod_{a \tp A}
B\ a$:
% %
\begin{equation}
\label{eq:funExt}
\var{funExt} \tp \prod_{a \tp A} f\ a \equiv g\ a \to f \equiv g
\end{equation}
%
%% p = λ i a → p a i
So given $p \tp \prod_{a \tp A} f\ a \equiv g\ a$ we must give a path $f \equiv
g$. That is a function $I \to \prod_{a \tp A} B\ a$. So let $i \tp I$ be given.
We must now give an expression $\phi \tp \prod_{a \tp A} B\ a$ satisfying
$\phi\ 0 \equiv f\ a$ and $\phi\ 1 \equiv g\ a$. This neccesitates that the
expression must be a lambda-abstraction, so let $a \tp A$ be given. Now we can
apply $a$ to $p$ and get the path $p\ a \tp f\ a \equiv g\ a$. And this exactly
satisfied the conditions for $\phi$. In conclustion \ref{eq:funExt} is inhabited
by the term:
%
\begin{equation}
\label{eq:funExt}
\var{funExt}\ p \defeq λ i\ a → p\ a\ i
\end{equation}
%
With this we can now prove the desired equality $f \equiv g$ from section
\S\ref{sec:functional-extensionality}:
%
\begin{align*}
p & \tp f \equiv g \\
p & \defeq \var{funExt}\ \lambda n \to \refl
\end{align*}
%
Paths have some other important properties, but they are not the focus of
this thesis. \TODO{Refer the reader somewhere for more info.}
\section{Homotopy levels} \section{Homotopy levels}
In ITT all equality proofs are identical (in a closed context). This means that, In ITT all equality proofs are identical (in a closed context). This means that,
in some sense, any two inhabitants of $a \equiv b$ are ``equally good'' -- they in some sense, any two inhabitants of $a \equiv b$ are ``equally good'' -- they
don't have any interesting structure. This is referred to as uniqueness of don't have any interesting structure. This is referred to as Uniqueness of
identity proofs. Unfortunately this is orthogonal to univalence that only makes Identity Proofs (UIP). Unfortunately it is not possible to have a type-theory
sense in the absence of UIP. with both univalence and UIP. In stead we have a hierarchy of types with an
increasing amount of homotopic structure. At the bottom of this hierarchy we
In homotopy type theory we have a hierarchy of types based on their ``internal have the set of contractible types:
structure''. At the bottom of this hierarchy we have the set of contractible
types:
% %
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
%% \begin{split} %% \begin{split}
& \isContr && \tp \MCU \to \MCU \\ & \isContr && \tp \MCU \to \MCU \\
& \isContr\ A && \defeq \sum_{c \tp A} \prod_{a \tp A} a \equiv c & \isContr\ A && \defeq \sum_{c \tp A} \prod_{a \tp A} a \equiv c
%% \end{split} %% \end{split}
\end{aligned} \end{aligned}
@ -128,8 +166,14 @@ types:
% %
The first component of $\isContr\ A$ is called ``the center of contraction''. The first component of $\isContr\ A$ is called ``the center of contraction''.
Under the propositions-as-types interpretation of type-theory $\isContr\ A$ can Under the propositions-as-types interpretation of type-theory $\isContr\ A$ can
be thought of as ``the true proposition $A$''. It is a theorem that if a type is be thought of as ``the true proposition $A$''. And indeed $\top$ is
contractible, then it is isomorphic to the unit-type $\top$. contractible:
\begin{equation*}
\var{tt} , \lambda x \to \refl \tp \isContr\ \top
\end{equation*}
%
It is a theorem that if a type is contractible, then it is isomorphic to the
unit-type.
The next step in the hierarchy is the set of mere propositions: The next step in the hierarchy is the set of mere propositions:
% %
@ -140,10 +184,18 @@ The next step in the hierarchy is the set of mere propositions:
\end{aligned} \end{aligned}
\end{equation} \end{equation}
% %
$\isProp\ A$ can be thought of as the set of true and false propositions. It is $\isProp\ A$ can be thought of as the set of true and false propositions. And
a result that if a mere proposition $A$ is inhabited, then so is it indeed both $\top$ and $\bot$ are propositions:
contractible. If it is not inhabited it is equivalent to the empty-type (or %
false proposition).\TODO{Cite!!} \begin{align*}
λ \var{tt}\ \var{tt} → refl & \tp \isProp\ ⊤ \\
λ\varnothing\ \varnothing & \tp \isProp\ ⊥
\end{align*}
%
I've used $\varnothing$ here to denote an impossible pattern. It is a theorem
that if a mere proposition $A$ is inhabited, then so is it contractible. If it
is not inhabited it is equivalent to the empty-type (or false
proposition).\TODO{Cite!!}
I will refer to a type $A \tp \MCU$ as a \emph{mere} proposition if I want to I will refer to a type $A \tp \MCU$ as a \emph{mere} proposition if I want to
stress that we have $\isProp\ A$. stress that we have $\isProp\ A$.
@ -157,10 +209,13 @@ Then comes the set of homotopical sets:
\end{aligned} \end{aligned}
\end{equation} \end{equation}
% %
At this point it should be noted that the term ``set'' is somewhat conflated; I won't give an example of a set at this point. It turns out that proving e.g.
there is the notion of sets from set-theory, in Agda types are denoted $\isProp\ \bN$ is not so straight-forward (see \S3.1.4 in \ref{hott-2013}).
\texttt{Set}. I will use it consistently to refer to a type $A$ as a set exactly There will be examples of sets later in this report. At this point it should be
if $\isSet\ A$ is inhabited. noted that the term ``set'' is somewhat conflated; there is the notion of sets
from set-theory, in Agda types are denoted \texttt{Set}. I will use it
consistently to refer to a type $A$ as a set exactly if $\isSet\ A$ is
a proposition.
The next step in the hierarchy is, as the reader might've guessed, the type: The next step in the hierarchy is, as the reader might've guessed, the type:
% %
@ -219,10 +274,13 @@ $$
% %
We have the function: We have the function:
% %
$$ \begin{equation}
\pathJ\ P\ p \tp \prod_{a' \tp A} \prod_{p \tp a ≡ a'} P\ a\ p \pathJ\ P\ p \tp \prod_{a' \tp A} \prod_{p \tp a ≡ a'} P\ a\ p
$$ \end{equation}
% %
I will not give an example of using $\pathJ$ here. But we'll see an application
of it in \ref{eq:pathJ-example}.
\subsection{Paths over propositions} \subsection{Paths over propositions}
\label{sec:lemPropF} \label{sec:lemPropF}
Another very useful combinator is $\lemPropF$: Another very useful combinator is $\lemPropF$:

2
doc/discussion.tex
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@ -26,7 +26,7 @@ disparate areas: The category-theoretic study of monads; and monads as in
functional programming. Univalence thus allows one to reuse proofs. You could functional programming. Univalence thus allows one to reuse proofs. You could
say that univalence gives the developer two proofs for the price of one. say that univalence gives the developer two proofs for the price of one.
The introduction (section \ref{sec:context}) mentioned an often The introduction (section \S\ref{sec:context}) mentioned an often
employed-technique for enabling extensional equalities is to use the employed-technique for enabling extensional equalities is to use the
setoid-interpretation. Nowhere in this formalization has this been necessary, setoid-interpretation. Nowhere in this formalization has this been necessary,
$\Path$ has been used globally in the project as propositional equality. One $\Path$ has been used globally in the project as propositional equality. One

84
doc/implementation.tex
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@ -53,9 +53,15 @@ given type.
For the rest of this chapter I will present some of these results. For didactic For the rest of this chapter I will present some of these results. For didactic
reasons no source-code has been included in this chapter. To see the formal reasons no source-code has been included in this chapter. To see the formal
definitions excerpts of the implementation have been included in appendix definitions excerpts of the implementation have been included in appendix
\ref{ch:app-sources}. \S\ref{ch:app-sources}:
Appendix
\S\ref{sec:app-categories} corresponds to section \S\ref{sec:categories},
appendix \S\ref{sec:app-products} to section \S\ref{sec:products}
and appendix \S\ref{sec:app-monads} to section \S\ref{sec:monads}.
\section{Categories} \section{Categories}
\label{sec:categories}
The data for a category consist of a type for the sort of objects; a type for The data for a category consist of a type for the sort of objects; a type for
the sort of arrows; an identity arrow and a composition operation for arrows. the sort of arrows; an identity arrow and a composition operation for arrows.
Another record encapsulates some laws about this data: associativity of Another record encapsulates some laws about this data: associativity of
@ -85,9 +91,9 @@ definitions: If we let $p$ be a witness to the identity law, which formally is:
Then we can construct the identity isomorphism $\var{idIso} \tp \identity, Then we can construct the identity isomorphism $\var{idIso} \tp \identity,
\identity, p \tp A \approxeq A$ for any object $A$. Here $\approxeq$ denotes \identity, p \tp A \approxeq A$ for any object $A$. Here $\approxeq$ denotes
isomorphism on objects (whereas $\cong$ denotes isomorphism of types). This will isomorphism on objects (whereas $\cong$ denotes isomorphism of types). This will
be elaborated further on in sections \ref{sec:equiv} and \ref{sec:univalence}. be elaborated further on in sections \S\ref{sec:equiv} and
Moreover, due to substitution for paths we can construct an isomorphism from \S\ref{sec:univalence}. Moreover, due to substitution for paths we can construct
\emph{any} path: an isomorphism from \emph{any} path:
% %
\begin{equation} \begin{equation}
\var{idToIso} \tp A ≡ B → A ≊ B \var{idToIso} \tp A ≡ B → A ≊ B
@ -106,14 +112,14 @@ Note that \ref{eq:cat-univ} is \emph{not} the same as:
% %
\begin{equation} \begin{equation}
\label{eq:cat-univalence} \label{eq:cat-univalence}
\tag{Univalence, category} %% \tag{Univalence, category}
(A \equiv B) \simeq (A \approxeq B) (A \equiv B) \simeq (A \approxeq B)
\end{equation} \end{equation}
% %
However the two are logically equivalent: One can construct the latter from the However the two are logically equivalent: One can construct the latter from the
former simply by ``forgetting'' that $\idToIso$ plays the role of the former simply by ``forgetting'' that $\idToIso$ plays the role of the
equivalence. The other direction is more involved and will be discussed in equivalence. The other direction is more involved and will be discussed in
section \ref{sec:univalence}. section \S\ref{sec:univalence}.
In summary, the definition of a category is the following collection of data: In summary, the definition of a category is the following collection of data:
% %
@ -127,14 +133,14 @@ In summary, the definition of a category is the following collection of data:
And laws: And laws:
% %
\begin{align} \begin{align}
\tag{associativity} %% \tag{associativity}
h \lll (g \lll f) ≡ (h \lll g) \lll f \\ h \lll (g \lll f) ≡ (h \lll g) \lll f \\
\tag{identity} %% \tag{identity}
\identity \lll f ≡ f \x \identity \lll f ≡ f \x
f \lll \identity ≡ f f \lll \identity ≡ f
\\ \\
\label{eq:arrows-are-sets} \label{eq:arrows-are-sets}
\tag{arrows are sets} %% \tag{arrows are sets}
\isSet\ (\Arrow\ A\ B)\\ \isSet\ (\Arrow\ A\ B)\\
\tag{\ref{eq:cat-univ}} \tag{\ref{eq:cat-univ}}
\isEquiv\ (A \equiv B)\ (A \approxeq B)\ \idToIso \isEquiv\ (A \equiv B)\ (A \approxeq B)\ \idToIso
@ -159,7 +165,7 @@ Proving that \ref{eq:identity} is a mere proposition:
% %
There are multiple ways to prove this. Perhaps one of the more intuitive proofs There are multiple ways to prove this. Perhaps one of the more intuitive proofs
is by way of the `combinators' $\propPi$ and $\propSig$ presented in sections is by way of the `combinators' $\propPi$ and $\propSig$ presented in sections
\ref{sec:propPi} and \ref{sec:propSig}: \S\ref{sec:propPi} and \S\ref{sec:propSig}:
% %
\begin{align*} \begin{align*}
\var{propPi} & \tp \left(\prod_{a \tp A} \isProp\ (P\ a)\right) \to \isProp\ \left(\prod_{a \tp A} P\ a\right) \var{propPi} & \tp \left(\prod_{a \tp A} \isProp\ (P\ a)\right) \to \isProp\ \left(\prod_{a \tp A} P\ a\right)
@ -279,7 +285,7 @@ we have shown, $\IsPreCategory$ is a proposition. However, even though
$\Univalent$ is also a proposition, we cannot use this directly to show the $\Univalent$ is also a proposition, we cannot use this directly to show the
latter. This is because $\isProp$ talks about non-dependent paths. So we need to latter. This is because $\isProp$ talks about non-dependent paths. So we need to
'promote' the result that univalence is a proposition to a heterogeneous path. 'promote' the result that univalence is a proposition to a heterogeneous path.
To this end we can use $\lemPropF$, which was introduced in \ref{sec:lemPropF}. To this end we can use $\lemPropF$, which was introduced in \S\ref{sec:lemPropF}.
In this case $A = \var{IsIdentity}\ \identity$ and $B = \var{Univalent}$. We've In this case $A = \var{IsIdentity}\ \identity$ and $B = \var{Univalent}$. We've
shown that being a category is a proposition, a result that holds for any choice shown that being a category is a proposition, a result that holds for any choice
@ -476,33 +482,43 @@ what we're trying to prove but talks about paths rather than isomorphisms:
\end{equation} \end{equation}
% %
Again $p_{\var{dom}}$ denotes the path $\var{Arrow}\ A\ X \equiv Again $p_{\var{dom}}$ denotes the path $\var{Arrow}\ A\ X \equiv
\var{Arrow}\ B\ X$ induced by $p$. To prove this statement I let $f$ and $p$ \var{Arrow}\ B\ X$ induced by $p$. To prove this statement I let $f$ and $p$ be
be given and then invoke based-path-induction. The induction will be based at $A given and then invoke based-path-induction. The induction will be based at $A
\tp \var{Object}$, so let $\widetilde{B} \tp \Object$ and $\widetilde{p} \tp \tp \var{Object}$ Let $\widetilde{B} \tp \Object$ and $\widetilde{p} \tp A
A \equiv \widetilde{B}$ be given. The family that we perform induction over will \equiv \widetilde{B}$ be given. The family that we perform induction over will
be: be:
% %
$$ \begin{align}
\var{coe}\ {\widetilde{p}}^*\ f D\ \widetilde{B}\ \widetilde{p} \defeq
%% \prod_{\widetilde{B} \tp \Object}
%% \prod_{\widetilde{p} \tp A \equiv \widetilde{B}}
\var{coe}\ {\widetilde{p}}^*\ f
\equiv \equiv
f \lll \inv{(\var{idToIso}\ \widetilde{p})} f \lll \inv{(\var{idToIso}\ \widetilde{p})}
$$ \end{align}
The base-case therefore becomes: The base-case therefore becomes
$d \tp \var{coe}\ \refl^*\ f \equiv f \lll \inv{(\var{idToIso}\ \refl)}$
and is inhabited by:
\begin{align*} \begin{align*}
\var{coe}\ {\widetilde{\refl}}^*\ f \var{coe}\ \refl^*\ f
& \equiv f \\ & \equiv f
&& \text{$\refl$ is a neutral element for $\var{coe}$}\\
& \equiv f \lll \var{identity} \\ & \equiv f \lll \var{identity} \\
& \equiv f \lll \inv{(\var{idToIso}\ \widetilde{\refl})} & \equiv f \lll \var{subst}\ \var{refl}\ \var{identity}
&& \text{$\refl$ is a neutral element for $\var{subst}$}\\
& \equiv f \lll \inv{(\var{idToIso}\ \refl)}
&& \text{By definition of $\var{idToIso}$}\\
\end{align*} \end{align*}
% %
The first step follows because reflexivity is a neutral element for coercion. To close the based-path-induction we must supply the value ``at the other''. In
The second step is the identity law in the category. The last step has to do this case this is simply $B \tp \Object$ and $p \tp A \equiv B$ which we have.
with the fact that $\var{idToIso}$ is constructed by substituting according to In summary the proof of \ref{eq:coeDomIso} is the term:
the supplied path and since reflexivity is also the neutral element for %
substitutions we arrive at the desired expression. To close the \begin{equation}
based-path-induction we must supply the value ``at the other''. In this case \label{eq:pathJ-example}
this is simply $B \tp \Object$ and $p \tp A \equiv B$ which we have. \var{pathJ}\ D\ d\ B\ p
\end{equation}
%
And this finishes the proof of \ref{eq:coeDomIso} and thus \ref{eq:coeDom}. And this finishes the proof of \ref{eq:coeDomIso} and thus \ref{eq:coeDom}.
% %
\section{Categories} \section{Categories}
@ -546,7 +562,7 @@ arguments. Or in other words; since $\Arrow\ A\ B$ is a set for all $A\;B \tp
\Object$ then so is $Arrow\ B\ A$. \Object$ then so is $Arrow\ B\ A$.
Now, to show that this category is univalent is not as straight-forward. Luckily Now, to show that this category is univalent is not as straight-forward. Luckily
section \ref{sec:equiv} gave us some tools to work with equivalences. We saw section \S\ref{sec:equiv} gave us some tools to work with equivalences. We saw
that we can prove this category univalent by giving an inverse to that we can prove this category univalent by giving an inverse to
$\wideoverbar{\idToIso} \tp (A \equiv B) \to (A \wideoverbar{\approxeq} B)$. $\wideoverbar{\idToIso} \tp (A \equiv B) \to (A \wideoverbar{\approxeq} B)$.
From the original category we have that $\idToIso \tp (A \equiv B) \to (A \cong From the original category we have that $\idToIso \tp (A \equiv B) \to (A \cong
@ -635,7 +651,7 @@ $$
(\var{hA} \equiv \var{hB}) \simeq (\var{hA} \approxeq \var{hB}) (\var{hA} \equiv \var{hB}) \simeq (\var{hA} \approxeq \var{hB})
$$ $$
% %
Which, as we saw in section \ref{sec:univalence}, is sufficient to show that the Which, as we saw in section \S\ref{sec:univalence}, is sufficient to show that the
category is univalent. The way that I have shown this is with a three-step category is univalent. The way that I have shown this is with a three-step
process. For objects $(A, s_A)\; (B, s_B) \tp \Set$ I show the following chain process. For objects $(A, s_A)\; (B, s_B) \tp \Set$ I show the following chain
of equivalences: of equivalences:
@ -721,7 +737,7 @@ choose:
& \tp \var{Isomorphism}\ f \to \isEquiv\ f & \tp \var{Isomorphism}\ f \to \isEquiv\ f
\end{align*} \end{align*}
% %
As mentioned in section \ref{sec:equiv}. These maps are not in general inverses As mentioned in section \S\ref{sec:equiv}. These maps are not in general inverses
of each other. In stead, we will use the fact that $A$ and $B$ are sets. The first thing we must prove is: of each other. In stead, we will use the fact that $A$ and $B$ are sets. The first thing we must prove is:
% %
\begin{align*} \begin{align*}
@ -777,6 +793,7 @@ where we have a witness to this being a category. This is useful if this library
is later extended to talk about higher categories. is later extended to talk about higher categories.
\section{Products} \section{Products}
\label{sec:products}
In the following I'll demonstrate a technique for using categories to prove In the following I'll demonstrate a technique for using categories to prove
properties. The goal in this section is to show that products are propositions: properties. The goal in this section is to show that products are propositions:
% %
@ -1214,6 +1231,7 @@ That in any category:
\end{align} \end{align}
% %
\section{Monads} \section{Monads}
\label{sec:monads}
In this section I present two formulations of monads. The two representations In this section I present two formulations of monads. The two representations
are referred to as the monoidal- and Kleisli- representation respectively or are referred to as the monoidal- and Kleisli- representation respectively or
simply monoidal monads and Kleisli monads for short. We then show that the two simply monoidal monads and Kleisli monads for short. We then show that the two

3
doc/introduction.tex
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@ -16,6 +16,7 @@ In the following two sections I present two examples that illustrate some
limitations inherent in ITT and -- by extension -- Agda. limitations inherent in ITT and -- by extension -- Agda.
% %
\subsection{Functional extensionality} \subsection{Functional extensionality}
\label{sec:functional-extensionality}
Consider the functions: Consider the functions:
% %
\begin{multicols}{2} \begin{multicols}{2}
@ -92,7 +93,7 @@ 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
\nomen{equivalent} types. I will return to the definition of equivalence later \nomen{equivalent} types. I will return to the definition of equivalence later
in section \ref{sec:equiv}, but for now it is sufficient to think of an 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 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: $A$ and $B$ are equivalent types. The principle of univalence says that:
% %

3
doc/macros.tex
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@ -31,7 +31,7 @@
\resizebox{\@tempdima}{\height}{=}% \resizebox{\@tempdima}{\height}{=}%
} }
\makeatother \makeatother
\newcommand{\var}[1]{\ensuremath{\mathit{#1}}} \newcommand{\var}[1]{\ensuremath{\mathit{#1}}\index{#1}}
\newcommand{\Hom}{\var{Hom}} \newcommand{\Hom}{\var{Hom}}
\newcommand{\fmap}{\var{fmap}} \newcommand{\fmap}{\var{fmap}}
\newcommand{\bind}{\var{bind}} \newcommand{\bind}{\var{bind}}
@ -86,3 +86,4 @@
\newcommand\NT[2]{\NTsym\ #1\ #2} \newcommand\NT[2]{\NTsym\ #1\ #2}
\newcommand\Endo[1]{\var{Endo}\ #1} \newcommand\Endo[1]{\var{Endo}\ #1}
\newcommand\EndoR{\mathcal{R}} \newcommand\EndoR{\mathcal{R}}
\newcommand\funExt{\var{funExt}}

1
doc/main.tex
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@ -68,6 +68,7 @@
\nocite{coquand-2013} \nocite{coquand-2013}
\bibliography{refs} \bibliography{refs}
%% \printindex
\begin{appendices} \begin{appendices}
\setcounter{page}{1} \setcounter{page}{1}
\pagenumbering{roman} \pagenumbering{roman}

3
doc/packages.tex
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@ -16,7 +16,8 @@
\usepackage[toc,page]{appendix} \usepackage[toc,page]{appendix}
\usepackage{xspace} \usepackage{xspace}
\usepackage[a4paper]{geometry} \usepackage[a4paper]{geometry}
\usepackage{makeidx}
\makeindex
% \setlength{\parskip}{10pt} % \setlength{\parskip}{10pt}
% \usepackage{tikz} % \usepackage{tikz}

1
doc/sources.tex
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@ -10,6 +10,7 @@ those.
\section{Cubical} \section{Cubical}
\label{sec:app-cubical} \label{sec:app-cubical}
\begin{figure}[h] \begin{figure}[h]
\label{fig:path}
\begin{Verbatim} \begin{Verbatim}
postulate postulate
PathP : ∀ {ℓ} (A : I → Set ℓ) → A i0 → A i1 → Set ℓ PathP : ∀ {ℓ} (A : I → Set ℓ) → A i0 → A i1 → Set ℓ