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doc/abstract.tex
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\chapter*{Abstract} \chapter*{Abstract}
The usual notion of propositional equality in intensional type-theory The usual notion of propositional equality in intensional type-theory
is restrictive. For instance it does not admit functional is restrictive. For instance it does not admit functional
extensionality or univalence. This poses a severe limitation on both extensionality nor univalence. This poses a severe limitation on both
what is \emph{provable} and the \emph{re-usability} of proofs. Recent what is \emph{provable} and the \emph{re-usability} of proofs. Recent
developments have, however, resulted in cubical type theory which developments have however resulted in cubical type theory which
permits a constructive proof of these two important notions. The permits a constructive proof of these two important notions. The
programming language Agda has been extended with capabilities for programming language Agda has been extended with capabilities for
working in such a cubical setting. This thesis will explore the working in such a cubical setting. This thesis will explore the
usefulness of this extension in the context of category theory. usefulness of this extension in the context of category theory.
The thesis will motivate and explain why propositional equality in The thesis will motivate the need for univalence and explain why
cubical Agda is more expressive than in standard Agda. Alternative propositional equality in cubical Agda is more expressive than in
approaches to Cubical Agda will be presented and their pros and cons standard Agda. Alternative approaches to Cubical Agda will be
will be explained. It will emphasize why it is useful to have a presented and their pros and cons will be explained. As an example of
constructive interpretation of univalence. As an example of this two the application of univalence two formulations of monads will be
formulations of monads will be presented: Namely monads in the presented: Namely monads in the monoidal form and monads in the
monoidal form and monads in the Kleisli form. Kleisli form and under the univalent interpretation it will be shown
how these are equal.
Finally the thesis will explain the challenges that a developer will Finally the thesis will explain the challenges that a developer will
face when working with cubical Agda and give some techniques to face when working with cubical Agda and give some techniques to

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doc/acknowledgements.tex Normal file
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\chapter*{Acknowledgements}
I would like to thank my supervisor Thierry Coquand for giving me a
chance to work on this interesting topic. I would also like to thank
Andrea Vezzosi for some very long and very insightful meetings during
the project. It is fascinating and almost uncanny how quickly Andrea
can conjure up various proofs. I also want to recognize the support
of Knud Højgaards Fond who graciously sponsored me with a 20.000 DKK
scholarship which helped toward sponsoring the two years I have spent
studying abroad. I would also like to give a warm thanks to my fellow
students Pierre Kraft and Nachiappan Villiappan who have made the time
spent working on the thesis way more enjoyable. Lastly I would like to
give a special thanks to Valentina Méndez who have been a great moral
support throughout the whole process.

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doc/appendix/abstract-funext.tex
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\chapter{Abstract functional extensionality} \chapter{Non-reducing functional extensionality}
\label{app:abstract-funext} \label{app:abstract-funext}
In two places in my formalization was the computational behaviours of In two places in my formalization was the computational behaviours of
functional extensionality used. The reduction behaviour can be functional extensionality used. The reduction behaviour can be

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doc/chalmerstitle.sty
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{\Huge\@title}\\[.5cm] {\Huge\@title}\\[.5cm]
{\Large A formalization of category theory in Cubical Agda}\\[6cm] {\Large A formalization of category theory in Cubical Agda}\\[6cm]
\begin{center} \begin{center}
\includegraphics[width=\linewidth,keepaspectratio]{isomorphism.png} \includegraphics[width=\linewidth,keepaspectratio]{isomorphism.pdf}
%% \includepdf{isomorphism.pdf}
\end{center} \end{center}
% Cover text % Cover text
\vfill \vfill

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doc/conclusion.tex
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@ -8,40 +8,46 @@ with an extension to Agda's type system called Cubical Agda. With
Cubical Agda both functional extensionality and univalence are Cubical Agda both functional extensionality and univalence are
admissible. Cubical Agda is more expressive, but there are certain admissible. Cubical Agda is more expressive, but there are certain
issues that arise that are not present in standard Agda. For one thing issues that arise that are not present in standard Agda. For one thing
ITT and standard Agda enjoys Uniqueness of Identity Proofs (UIP). This Agda enjoys Uniqueness of Identity Proofs (UIP) though a flag exists
is not the case in Cubical Agda. In stead there exists a hierarchy of to turn this off, which is the case in Cubical Agda. In stead
types with increasing \nomen{homotopical structure}{homotopy levels}. there exists a hierarchy of types with increasing \nomen{homotopical
It turns out to be useful to built the formalization with this structure}{homotopy levels}. It turns out to be useful to built the
hierarchy in mind as it can simplify proofs considerably. Another formalization with this hierarchy in mind as it can simplify proofs
issue one must overcome in Cubical Agda is when a type has a field considerably. Another issue one must overcome in Cubical Agda is when
whose type depends on a previous field. In this case paths between a type has a field whose type depends on a previous field. In this
such types will be heterogeneous paths. This problem is related to case paths between such types will be heterogeneous paths. This
Cubical Agda not having the K-rule. In practice it turns out to be problem is related to Cubical Agda not having the K-rule. In practice
considerably more difficult to work heterogeneous paths than with it turns out to be considerably more difficult to work heterogeneous
homogeneous paths. The thesis demonstrated some techniques to overcome paths than with homogeneous paths. The thesis demonstrated some
these difficulties, such as based path-induction. techniques to overcome these difficulties, such as based
path-induction.
This thesis formalized some of the core concepts from category theory including; This thesis formalized some of the core concepts from category theory
categories, functors, products, exponentials, Cartesian closed categories, including; categories, functors, products, exponentials, Cartesian
natural transformations, the yoneda embedding, monads and more. Category theory closed categories, natural transformations, the yoneda embedding,
is an interesting case-study for the application of Cubical Agda for two reasons monads and more. Category theory is an interesting case-study for the
in particular: Because category theory is the study of abstract algebra of application of Cubical Agda for two reasons in particular: Because
functions, meaning that functional extensionality is particularly relevant. category theory is the study of abstract algebra of functions, meaning
Another reason is that in category theory it is commonplace to identify that functional extensionality is particularly relevant. Another
isomorphic structures and univalence allows for making this notion precise. This reason is that in category theory it is commonplace to identify
thesis also demonstrated another technique that is common in category theory; isomorphic structures and univalence allows for making this notion
namely to define categories to prove properties of other structures. precise. This thesis also demonstrated another technique that is
Specifically a category was defined to demonstrate that any two product objects common in category theory; namely to define categories to prove
in a category are isomorphic. Furthermore the thesis showed two formulations of properties of other structures. Specifically a category was defined
monads and proved that they indeed are equivalent: Namely monads in the to demonstrate that any two product objects in a category are
monoidal- and Kleisli- form. The monoidal formulation is more typical to isomorphic. Furthermore the thesis showed two formulations of monads
category theoretic formulations and the Kleisli formulation will be more and proved that they indeed are equivalent: Namely monads in the
familiar to functional programmers. In the formulation we also saw how paths can monoidal- and Kleisli- form. The monoidal formulation is more typical
be used to extract functions. A path between two types induce an isomorphism to category theoretic formulations and the Kleisli formulation will be
between the two types. This e.g. permits developers to write a monad instance more familiar to functional programmers. It would have been very
for a given type using the Kleisli formulation. By transporting along the path difficult to make a similar proof with setoids. In the formulation we
between the monoidal- and Kleisli- formulation one can reuse all the operations also saw how paths can be used to extract functions. A path between
and results shown for monoidal- monads in the context of kleisli monads. two types induce an isomorphism between the two types. This
e.g. permits developers to write a monad instance for a given type
using the Kleisli formulation. By transporting along the path between
the monoidal- and Kleisli- formulation one can reuse all the
operations and results shown for monoidal- monads in the context of
kleisli monads.
%% %%
%% problem with inductive type %% problem with inductive type
%% overcome with cubical %% overcome with cubical

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doc/cubical.tex
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@ -1,29 +1,31 @@
\chapter{Cubical Agda} \chapter{Cubical Agda}
\section{Propositional equality} \section{Propositional equality}
Judgmental equality in Agda is a feature of the type system. Its something that Judgmental equality in Agda is a feature of the type system. It is
can be checked automatically by the type checker: In the example from the something that can be checked automatically by the type checker: In
introduction $n + 0$ can be judged to be equal to $n$ simply by expanding the the example from the introduction $n + 0$ can be judged to be equal to
definition of $+$. $n$ simply by expanding the definition of $+$.
On the other hand, propositional equality is something defined within the On the other hand, propositional equality is something defined within
language itself. Propositional equality cannot be derived automatically. The the language itself. Propositional equality cannot be derived
normal definition of judgmental equality is an inductive data type. Cubical Agda automatically. The normal definition of judgmental equality is an
discards this type in favor of a new primitives that has certain computational inductive data type. Cubical Agda discards this type in favor of some
properties exclusive to it. new primitives.
The source code can be browsed online and is linked in the beginning Most of the source code related with this section is implemented in
of \S\ref{ch:implementation}. \cite{cubical-demo} it can be browsed in hyperlinked and syntax
highlighted HTML online. The links can be found in the beginning of
section \S\ref{ch:implementation}.
\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
two points of $A$; $a_0, a_1 \tp A$ we can form the type: \MCU$ and two points hereof $a_0, a_1 \tp A$ we can form the type:
% %
\begin{align} \begin{align}
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
for any $a \tp A$: constructor $\refl$ that for any $a \tp A$ gives:
% %
\begin{align} \begin{align}
\refl \tp a \equiv a \refl \tp a \equiv a
@ -37,8 +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 likewise has the single constructor $\refl$ that for any $a \tp
for any $a \tp A$: A$ gives:
% %
\begin{align} \begin{align}
\refl \tp a \cong a \refl \tp a \cong a
@ -51,28 +53,28 @@ heterogeneous paths respectively.
Judgmental equality in Cubical Agda is encapsulated with the type: Judgmental equality in Cubical Agda is encapsulated with the type:
% %
\begin{equation} \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} \end{equation}
% %
$I$ is a special data type called the index set. $I$ can be thought of The special type $\I$ is called the index set. The index set can be
simply as the interval on the real numbers from $0$ to $1$. $P$ is a thought of simply as the interval on the real numbers from $0$ to $1$
family of types over the index set $I$. I will sometimes refer to $P$ (both inclusive). The family $P$ over $\I$ will be referred to as the
as the \nomenindex{path space} of some path $p \tp \Path\ P\ a\ b$. By \nomenindex{path space} given some path $p \tp \Path\ P\ a\ b$. By
this token $P\ 0$ then corresponds to the type at the left-endpoint that token $P\ 0$ corresponds to the type at the left endpoint of $p$.
and $P\ 1$ as the type at the right-endpoint. The type is called Likewise $P\ 1$ is the type at the right endpoint. The type is called
$\Path$ because it is connected with paths in homotopy theory. The $\Path$ because the idea has roots in homotopy theory. The intuition
intuition behind this is that $\Path$ describes paths in $\MCU$ -- is that $\Path$ describes\linebreak[1] paths in $\MCU$. I.e.\ paths
i.e.\ between types. For a path $p$ for the point $p\ i$ the index $i$ between types. For a path $p$ the expression $p\ i$ can be thought of
describes how far along the path one has moved. An inhabitant of as a \emph{point} on this path. The index $i$ describes how far along
$\Path\ P\ a_0\ a_1$ is a (dependent-) function, $p$, from the the path one has moved. An inhabitant of $\Path\ P\ a_0\ a_1$ is a
index-space to the path space: (dependent) function from the index set to the path space:
% %
$$ $$
p \tp \prod_{i \tp I} 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 $a_0$ at the
endpoints. I.e.: left endpoint and equal to $a_1$ at the other end. I.e.:
% %
\begin{align*} \begin{align*}
p\ 0 & = a_0 \\ p\ 0 & = a_0 \\
@ -80,80 +82,77 @@ endpoints. I.e.:
\end{align*} \end{align*}
% %
The notion of \nomenindex{homogeneous equalities} is recovered when $P$ does not The notion of \nomenindex{homogeneous equalities} is recovered when $P$ does not
depend on its argument. That is for $A \tp \MCU$, $a_0, a_1 \tp A$ the depend on its argument. That is for $A \tp \MCU$ and $a_0, a_1 \tp A$ the
homogenous equality between $a_0$ and $a_1$ is the type: homogenous equality between $a_0$ and $a_1$ is the type:
% %
$$ $$
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
$$ $$
% %
I will generally prefer to use the notation I will generally prefer to use the notation $a \equiv b$ when talking
$a \equiv b$ when talking about non-dependent paths and use the notation about non-dependent paths and use the notation $\Path\ (\lambda\; i
$\Path\ (\lambda i \to P\ i)\ a\ b$ when the path space is of particular \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 $A \tp
\MCU$ and $a \tp A$: \MCU$ and $a \tp A$:
% %
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
\refl & \tp \Path (\lambda i \to A)\ a\ a \\ \refl & \tp a \equiv a \\
\refl & \defeq \lambda i \to a \refl & \defeq \lambda\; i \to a
\end{aligned} \end{aligned}
\end{equation} \end{equation}
% %
Here the path space is $P \defeq \lambda i \to A$ and it satsifies $P\ i = A$ Here the path space is $P \defeq \lambda\; i \to A$ and it satsifies
definitionally. So to inhabit it, is to give a path $I \to A$ which is $P\ i = A$ definitionally. So to inhabit it, is to give a path $\I \to
judgmentally $a$ at either endpoint. This is satisfied by the constant path; A$ which is judgmentally $a$ at either endpoint. This is satisfied by
i.e.\ the path that stays at $a$ at any index $i$. the constant path; i.e.\ the path that is constantly $a$ at any index
$i \tp \I$.
It is also surpisingly easy to show functional extensionality with which we can It is also surprisingly easy to show functional extensionality.
construct a path between $f$ and $g$ -- the functions defined in the Functional extensionality is the proposition that given a type $A \tp
introduction (section \S\ref{sec:functional-extensionality}). Functional \MCU$, a family of types $B \tp A \to \MCU$ and functions $f, g \tp
extensionality is the proposition, given a type $A \tp \MCU$, a family of types \prod_{a \tp A} B\ a$ gives:
$B \tp A \to \MCU$ and functions $f, g \tp \prod_{a \tp A} B\ a$:
% %
\begin{equation} \begin{equation}
\label{eq:funExt} \label{eq:funExt}
\funExt \tp \prod_{a \tp A} f\ a \equiv g\ a \to f \equiv g \funExt \tp \left(\prod_{a \tp A} f\ a \equiv g\ a \right) \to f \equiv g
\end{equation} \end{equation}
% %
%% p = λ i a → p a i %% 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 So given $η \tp \prod_{a \tp A} f\ a \equiv g\ a$ 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. 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 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 $\phi\ 0 \equiv f\ a$ 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 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 apply $a$ to $η$ and get the path $η\ a \tp f\ a \equiv g\ a$. And this exactly
satisfied the conditions for $\phi$. In conclustion \ref{eq:funExt} is inhabited satisfies the conditions for $\phi$. In conclustion \ref{eq:funExt} is inhabited
by the term: by the term:
% %
\begin{equation} \begin{equation*}
\label{eq:funExt} \funExt\ η \defeq λ\; i\ a → η\ a\ i
\funExt\ p \defeq λ i\ a → p\ a\ i \end{equation*}
\end{equation}
% %
With this we can now prove the desired equality $f \equiv g$ from section With $\funExt$ in place we can now construct a path between
\S\ref{sec:functional-extensionality}: $\var{zeroLeft}$ and $\var{zeroRight}$ -- the functions defined in the
introduction \S\ref{sec:functional-extensionality}:
% %
\begin{align*} \begin{align*}
p & \tp f \equiv g \\ p & \tp \var{zeroLeft} \equiv \var{zeroRight} \\
p & \defeq \funExt\ \phi p & \defeq \funExt\ \var{zrn}
\end{align*} \end{align*}
% %
Here $\phi \tp \prod_{n \tp \bN} \var{zeroLeft}\ n \equiv Here $\var{zrn}$ is the proof from \ref{eq:zrn}.
\var{zeroRight} n$. 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
in some sense, any two inhabitants of $a \equiv b$ are ``equally good'' -- they means that, in some sense, any two inhabitants of $a \equiv b$ are
do not have any interesting structure. This is referred to as Uniqueness of ``equally good''. They do not have any interesting structure. This is
Identity Proofs (UIP). Unfortunately it is not possible to have a type theory referred to as Uniqueness of Identity Proofs (UIP). Unfortunately it
with both univalence and UIP. In stead we have a hierarchy of types with an is not possible to have a type theory with both univalence and UIP. In
increasing amount of homotopic structure. At the bottom of this hierarchy we stead in cubical Agda we have a hierarchy of types with an increasing
have the set of contractible types: amount of homotopic structure. At the bottom of this hierarchy is the
set of contractible types:
% %
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
@ -168,8 +167,9 @@ 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 $A$''. And indeed $\top$ is be thought of as ``the true proposition $A$''. And indeed $\top$ is
contractible: contractible:
%
\begin{equation*} \begin{equation*}
\var{tt} , \lambda x \to \refl \tp \isContr\ \top (\var{tt} , \lambda\; x \to \refl) \tp \isContr\ \top
\end{equation*} \end{equation*}
% %
It is a theorem that if a type is contractible, then it is isomorphic to the It is a theorem that if a type is contractible, then it is isomorphic to the
@ -188,19 +188,19 @@ One can think of $\isProp\ A$ as the set of true and false propositions. And
indeed both $\top$ and $\bot$ are propositions: indeed both $\top$ and $\bot$ are propositions:
% %
\begin{align*} \begin{align*}
λ \var{tt}\ \var{tt} → refl & \tp \isProp\ ⊤ \\ (λ\; \var{tt}, \var{tt} → refl) & \tp \isProp\ ⊤ \\
λ\varnothing\ \varnothing & \tp \isProp\ ⊥ λ\;\varnothing\ \varnothing & \tp \isProp\ ⊥
\end{align*} \end{align*}
% %
The term $\varnothing$ is used here to denote an impossible pattern. It is a The term $\varnothing$ is used here to denote an impossible pattern. It is a
theorem that if a mere proposition $A$ is inhabited, then so is it contractible. 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 If it is not inhabited it is equivalent to the empty-type (or false
proposition).\TODO{Cite} proposition).
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 $\isProp\ A$.
Then comes the set of homotopical sets: The next step in the hierarchy is the set of homotopical sets:
% %
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
@ -209,13 +209,14 @@ Then comes the set of homotopical sets:
\end{aligned} \end{aligned}
\end{equation} \end{equation}
% %
I will not give an example of a set at this point. It turns out that proving I will not give an example of a set at this point. It turns out that
e.g.\ $\isProp\ \bN$ is not so straight-forward (see \cite[\S3.1.4]{hott-2013}). proving e.g.\ $\isProp\ \bN$ directly is not so straightforward (see
There will be examples of sets later in this report. At this point it should be \cite[\S3.1.4]{hott-2013}). Hedberg's theorem states that any type
noted that the term ``set'' is somewhat conflated; there is the notion of sets with decidable equality is a set. There will be examples of sets later
from set-theory, in Agda types are denoted \texttt{Set}. I will use it in this report. At this point it should be noted that the term ``set''
consistently to refer to a type $A$ as a set exactly if $\isSet\ A$ is a is somewhat conflated; there is the notion of sets from set-theory, in
proposition. 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.
As the reader may have guessed the next step in the hierarchy is the type: As the reader may have guessed the next step in the hierarchy is the type:
% %
@ -239,34 +240,40 @@ exposition:
Proposition: Homotopy levels are cumulative. That is, if $A \tp \MCU$ has Proposition: Homotopy levels are cumulative. That is, if $A \tp \MCU$ has
homotopy level $n$ then so does it have $n + 1$. homotopy level $n$ then so does it have $n + 1$.
Let $\left\Vert A \right\Vert = n$ denote that the level of $A$ is $n$. For any level $n$ it is the case that to be of level $n$ is a mere proposition.
Proposition: For any homotopic level $n$ this is a mere proposition.
% %
\section{A few lemmas} \section{A few lemmas}
Rather than getting into the nitty-gritty details of Agda I venture to Rather than getting into the nitty-gritty details of Agda I venture to
take a more ``combinator-based'' approach. That is, I will use take a more ``combinator-based'' approach. That is I will use
theorems about paths already that have already been formalized. theorems about paths that have already been formalized.
Specifically the results come from the Agda library \texttt{cubical} Specifically the results come from the Agda library \texttt{cubical}
(\cite{cubical-demo}). I have used a handful of results from this (\cite{cubical-demo}). I have used a handful of results from this
library as well as contributed a few lemmas myself.\footnote{The library as well as contributed a few lemmas myself%
module \texttt{Cat.Prelude} lists the upstream dependencies. As well \footnote{The module \texttt{Cat.Prelude} lists the upstream
my contribution to \texttt{cubical} can be found in the git logs dependencies. As well my contribution to \texttt{cubical} can be
which are available at found in the git logs which are available at
\hrefsymb{https://github.com/Saizan/cubical-demo}{\texttt{https://github.com/Saizan/cubical-demo}}.} \hrefsymb{https://github.com/Saizan/cubical-demo}{\texttt{https://github.com/Saizan/cubical-demo}}.
}.
These theorems are all purely related to homotopy theory and cubical Agda and as These theorems are all purely related to homotopy type theory and as
such not specific to the formalization of Category Theory. I will present a few such not specific to the formalization of Category Theory. I will
of these theorems here, as they will be used later in chapter present a few of these theorems here as they will be used throughout
\ref{ch:implementation} throughout. chapter \ref{ch:implementation}. They should also give the reader some
intuition about the path type.
\subsection{Path induction} \subsection{Path induction}
\label{sec:pathJ} \label{sec:pathJ}
The induction principle for paths intuitively gives us a way to reason about a The induction principle for paths intuitively gives us a way to reason
type family indexed by a path by only considering if said path is $\refl$ (the about a type family indexed by a path by only considering if said path
\nomen{base case}{path induction}). For \emph{based path induction}, that equality is \emph{based} is $\refl$ (the \nomen{base case}{path induction}). For \emph{based
at some element $a \tp A$. path induction}, that equality is \emph{based} at some element $a
\tp A$.
Let a type $A \tp \MCU$ and an element of the type $a \tp A$ be given. $a$ is said to be the base of the induction. Given a family of types: \pagebreak[3]
\begin{samepage}
Let a type $A \tp \MCU$ and an element of the type $a \tp A$ be
given. $a$ is said to be the base of the induction.\linebreak[3] Given
a family of types:
% %
$$ $$
D \tp \prod_{b \tp A} \prod_{p \tp a ≡ b} \MCU D \tp \prod_{b \tp A} \prod_{p \tp a ≡ b} \MCU
@ -283,7 +290,8 @@ We have the function:
\begin{equation} \begin{equation}
\pathJ\ D\ d \tp \prod_{b \tp A} \prod_{p \tp a ≡ b} D\ b\ p \pathJ\ D\ d \tp \prod_{b \tp A} \prod_{p \tp a ≡ b} D\ b\ p
\end{equation} \end{equation}
% \end{samepage}%
A simple application of $\pathJ$ is for proving that $\var{sym}$ is an A simple application of $\pathJ$ is for proving that $\var{sym}$ is an
involution. Namely for any set $A \tp \MCU$, points $a, b \tp A$ and a path involution. Namely for any set $A \tp \MCU$, points $a, b \tp A$ and a path
between them $p \tp a \equiv b$: between them $p \tp a \equiv b$:
@ -293,8 +301,8 @@ between them $p \tp a \equiv b$:
\var{sym}\ (\var{sym}\ p) ≡ p \var{sym}\ (\var{sym}\ p) ≡ p
\end{equation} \end{equation}
% %
The proof will be by induction on $p$ and will be based at $a$. That is, $D$ The proof will be by induction on $p$ and will be based at $a$. That
will be the family: is $D$ will be the family:
% %
\begin{align*} \begin{align*}
D & \tp \prod_{b' \tp A} \prod_{p \tp a ≡ b'} \MCU \\ D & \tp \prod_{b' \tp A} \prod_{p \tp a ≡ b'} \MCU \\
@ -319,7 +327,7 @@ definitionally. In summary \ref{eq:sym-invol} is inhabited by the term:
% %
Another application of $\pathJ$ is for proving associativity of $\trans$. That Another application of $\pathJ$ is for proving associativity of $\trans$. That
is, given a type $A \tp \MCU$, elements of $A$, $a, b, c, d \tp A$ and paths is, given a type $A \tp \MCU$, elements of $A$, $a, b, c, d \tp A$ and paths
between them, $p \tp a \equiv b$, $q \tp b \equiv c$ and $r \tp c \equiv d$ we between them $p \tp a \equiv b$, $q \tp b \equiv c$ and $r \tp c \equiv d$ we
have the following: have the following:
% %
\begin{equation} \begin{equation}
@ -351,49 +359,52 @@ conclusion \ref{eq:cum-trans} is inhabited by the term:
\pathJ\ T\ t\ d\ r \pathJ\ T\ t\ d\ r
\end{align*} \end{align*}
% %
We shall see another application on path induction in \ref{eq:pathJ-example}. We shall see another application of path induction 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 $\lemPropF$: Given a type $A \tp
\MCU$ and a type family on $A$; $D \tp A \to \MCU$. Let $\var{propD}
To `promote' this to a dependent path we can use another useful combinator; \tp \prod_{x \tp A} \isProp\ (D\ x)$ be the proof that $D$ is a mere
$\lemPropF$. Given a type $A \tp \MCU$ and a type family on $A$; $P \tp A \to proposition for all elements of $A$. Furthermore say we have a path
\MCU$. Let $\var{propP} \tp \prod_{x \tp A} \isProp\ (P\ x)$ be the proof that between some two elements in $A$; $p \tp a_0 \equiv a_1$ then we can
$P$ is a mere proposition for all elements of $A$. Furthermore say we have a built a heterogeneous path between any two elements of $d_0 \tp
path between some two elements in $A$; $p \tp a_0 \equiv a_1$ then we can built D\ a_0$ and $d_1 \tp D\ a_1$.
a heterogeneous path between any two elements of $p_0 \tp P\ a_0$ and $p_1 \tp
P\ a_1$:
% %
$$ $$
\lemPropF\ \var{propP}\ p \tp \Path\ (\lambda\; i \mto P\ (p\ i))\ p_0\ p_1 \lemPropF\ \var{propD}\ p \tp \Path\ (\lambda\; i \mto D\ (p\ i))\ d_0\ d_1
$$ $$
% %
This is quite a mouthful. So let me try to show how this is a very general and Note that $d_0$ and $d_1$, though points of the same family, have
useful result. different types. This is quite a mouthful. So let me try to show how
this is a very general and useful result.
Often when proving equalities between elements of some dependent types Often when proving equalities between elements of some dependent types
$\lemPropF$ can be used to boil this complexity down to showing that the $\lemPropF$ can be used to boil this complexity down to showing that
dependent parts of the type are mere propositions. For instance, saw we have a type: the dependent parts of the type are mere propositions. For instance
say we have a type:
% %
$$ $$
T \defeq \sum_{a \tp A} P\ a T \defeq \sum_{a \tp A} D\ a
$$ $$
% %
For some proposition $P \tp A \to \MCU$. If we want to prove $t_0 \equiv t_1$ For some proposition $D \tp A \to \MCU$. That is we have $\var{propD}
for two elements $t_0, t_1 \tp T$ then this will be a pair of paths: \tp \prod_{a \tp A} \isProp\ (D\ a)$. If we want to prove $t_0 \equiv
t_1$ for two elements $t_0, t_1 \tp T$ then this will be a pair of
paths:
% %
% %
\begin{align*} \begin{align*}
p \tp & \fst\ t_0 \equiv \fst\ t_1 \\ p \tp & \fst\ t_0 \equiv \fst\ t_1 \\
& \Path\ (\lambda i \to P\ (p\ i))\ \snd\ t_0 \equiv \snd\ t_1 & \Path\ (\lambda\; i \to D\ (p\ i))\ (\snd\ t_0)\ (\snd\ t_1)
\end{align*} \end{align*}
% %
Here $\lemPropF$ directly allow us to prove the latter of these: Here $\lemPropF$ directly allow us to prove the latter of these given
that we have already provided $p$.
% %
$$ $$
\lemPropF\ \var{propP}\ p \lemPropF\ \var{propD}\ p
\tp \Path\ (\lambda i \to P\ (p\ i))\ \snd\ t_0 \equiv \snd\ t_1 \tp \Path\ (\lambda\; i \to D\ (p\ i))\ (\snd\ t_0)\ (\snd\ t_1)
$$ $$
% %
\subsection{Functions over propositions} \subsection{Functions over propositions}
@ -407,9 +418,9 @@ $$
\subsection{Pairs over propositions} \subsection{Pairs over propositions}
\label{sec:propSig} \label{sec:propSig}
% %
$\sum$-types preserve propositionality whenever its first component is a $\sum$-types preserve propositionality whenever its first component is
proposition, and its second component is a proposition for all points of in the a proposition, and its second component is a proposition for all
left type. points of the left type.
% %
$$ $$
\mathit{propSig} \tp \isProp\ A \to \left(\prod_{a \tp A} \isProp\ (P\ a)\right) \to \isProp\ \left(\sum_{a \tp A} P\ a\right) \mathit{propSig} \tp \isProp\ A \to \left(\prod_{a \tp A} \isProp\ (P\ a)\right) \to \isProp\ \left(\sum_{a \tp A} P\ a\right)

13
doc/discussion.tex
View file

@ -90,7 +90,7 @@ been put towards proving things that would not have been needed in
classical Agda. The proofs that some given type is a proposition were classical Agda. The proofs that some given type is a proposition were
provided as a strategy to simplify some otherwise very complicated provided as a strategy to simplify some otherwise very complicated
proofs (e.g. \ref{eq:proof-prop-IsPreCategory} proofs (e.g. \ref{eq:proof-prop-IsPreCategory}
and \label{eq:productPath}). Often these proofs would not be this and \ref{eq:productPath}). Often these proofs would not be this
complicated. If the J-rule holds definitionally the proof-assistant complicated. If the J-rule holds definitionally the proof-assistant
can help simplify these goals considerably. The lack of the J-rule has can help simplify these goals considerably. The lack of the J-rule has
a significant impact on the complexity of these kinds of proofs. a significant impact on the complexity of these kinds of proofs.
@ -112,11 +112,6 @@ very simple example this is of course not a big problem, but there are
examples in the source code where this gets more involved. examples in the source code where this gets more involved.
\section{Future work} \section{Future work}
\subsection{Agda \texttt{Prop}}
Jesper Cockx' work extending the universe-level-laws for Agda and the
\texttt{Prop}-type.
\TODO{Do I want to include this?}
\subsection{Compiling Cubical Agda} \subsection{Compiling Cubical Agda}
\label{sec:compiling-cubical-agda} \label{sec:compiling-cubical-agda}
Compilation of program written in Cubical Agda is currently not Compilation of program written in Cubical Agda is currently not
@ -137,3 +132,9 @@ in the context of Category Theory.
A fellow student here at Chalmers, Andreas Källberg, is currently A fellow student here at Chalmers, Andreas Källberg, is currently
working on proving the initiality conjecture\TODO{Citation}. He will working on proving the initiality conjecture\TODO{Citation}. He will
be using this library to do so. be using this library to do so.
\subsection{Proving laws of programs}
Another interesting thing would be to use the Kleisli formulation of
monads to prove properties of functional programs. The existence of
univalence will make it possible to re-use proofs stated in terms of
the monoidal formulation in this setting.

205
doc/implementation.tex
View file

@ -1,7 +1,7 @@
\chapter{Category Theory} \chapter{Category Theory}
\label{ch:implementation} \label{ch:implementation}
This implementaiton, including this report, is available as open The source code for this formalization, including this report, is
source software at: available as open source software at:
% %
\begin{center} \begin{center}
\gitlink \gitlink
@ -17,7 +17,7 @@ link\footnote{%
\begin{center} \begin{center}
\doclink \doclink
\end{center} \end{center}
This implementation formalizes the following concepts: The concepts formalized in this development are:
% %
\begin{center} \begin{center}
\begin{tabular}{ l l } \begin{tabular}{ l l }
@ -41,6 +41,7 @@ Span category & \sourcelink{Cat.Categories.Span} \\
\end{tabular} \end{tabular}
\end{center} \end{center}
% %
\begin{samepage}
Furthermore the following items have been partly formalized: Furthermore the following items have been partly formalized:
% %
\begin{center} \begin{center}
@ -54,70 +55,75 @@ Free category & \sourcelink{Cat.Categories.Free} \\
Monoids & \sourcelink{Cat.Category.Monoid} \\ Monoids & \sourcelink{Cat.Category.Monoid} \\
\end{tabular} \end{tabular}
\end{center} \end{center}
\end{samepage}%
% %
As well as a range of various results about these. E.g.\ I have shown As well as a range of various results about these. E.g.\ I have shown
that the category of sets has products. In the following I aim to that the category of sets has products. In the following I aim to
demonstrate some of the techniques employed in this formalization and demonstrate some of the techniques employed in this formalization and
in the interest of brevity I will not detail all the things I have in the interest of brevity I will not detail all the things I have
formalized. In stead, I have selected parts of this formalization that formalized. In stead I have selected parts of this formalization that
highlight some interesting proof techniques relevant to doing proofs highlight some interesting proof techniques relevant to doing proofs
in Cubical Agda. This chapter will focus on the definition of in Cubical Agda. This chapter will focus on the definition of
\emph{categories}, \emph{equivalences}, the \emph{opposite category}, \emph{categories}, \emph{equivalences}, the \emph{opposite category},
the \emph{category of sets}, \emph{products}, the \emph{span category} the \emph{category of sets}, \emph{products}, the \emph{span category}
and the two formulations of \emph{monads}. and the two formulations of \emph{monads}.
One such technique that is pervasive to this formalization is the idea of One technique employed throughout this formalization is the idea of
distinguishing types with more or less homotopical structure. To do this I have distinguishing types with more or less homotopical structure. To do
followed the following design-principle: I have split concepts up into things this I have followed the following design-principle: I have split
that represent ``data'' and ``laws'' about this data. The idea is that we can concepts up into things that represent \emph{data} and \emph{laws}
provide a proof that the laws are mere propositions. As an example a category is about this data. The idea is that we can provide a proof that the laws
defined to have two members: `raw` which is a collection of the data and are mere propositions. As an example a category is defined to have two
`isCategory` which asserts some laws about that data. members: $\var{raw}$ which is a collection of the data and
$\var{isCategory}$ which asserts some laws about that data.
This allows me to reason about things in a more ``standard mathematical way'', This allows me to reason about things in a more ``standard
where one can reason about two categories by simply focusing on the data. This mathematical way'', where one can reason about two categories by
is achieved by creating a function embodying the ``equality principle'' for a simply focusing on the data. This is achieved by creating a function
given type. embodying the equality principle for a given type.
For the rest of this chapter I will present some of these results. For 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 didactic reasons no source-code has been included in this chapter. To
see the formal definitions the reader is referred to the see the formal definitions the reader is referred to the
implementation which is linked in \S\ref{ch:implementation}. implementation which is linked in the tables above.
\section{Categories} \section{Categories}
\label{sec: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
the sort of arrows; an identity arrow and a composition operation for arrows. type for the sort of arrows; an identity arrow and a composition
Another record encapsulates some laws about this data: associativity of operation for arrows. Another record encapsulates some laws about
composition, identity law for the identity morphism. These are standard this data: associativity of composition, identity law for the identity
constituents of a category and can be found in typical mathematical expositions morphism. These are standard constituents of a category and can be
on the topic. We, however, impose one further requirement on what it means to be found in typical mathematical expositions on the topic. We shall
a category, namely that the type of arrows form a set. impose one further requirement on what it means to be a category,
namely that the type of arrows form a set.
Such categories are called \nomen{1-categories}{1-category}. It is possible to relax Such categories are called \nomen{1-categories}{1-category}. It is
this requirement. This would lead to the notion of higher categories (\cite[p. possible to relax this requirement. This would lead to the notion of
307]{hott-2013}). For the purpose of this project, however, this report will higher categories (\cite[p. 307]{hott-2013}). For the purpose of this
restrict itself to 1-categories\index{1-category}. Generalizing this work to thesis however, this report will restrict itself to
higher categories would be a very natural possible extension of this work. 1-categories\index{1-category}. Generalizing this work to higher
categories would be a very natural extension of this work.
Raw categories satisfying all of the above requirements are called a Raw categories satisfying all of the above requirements are called a
\nomenindex{pre categories}. As a further requirement to be a proper category we \nomenindex{pre-categories}. As a further requirement to be a proper category we
require it to be univalent. Before we can define this, I must introduce two more require it to be univalent. Before we can define this, I must introduce two more
definitions: If we let $p$ be a witness to the identity law, which formally is: definitions: If we let $p$ be a witness to the identity law, which formally is:
% %
\begin{equation} \begin{equation}
\label{eq:identity} \label{eq:identity}
\var{IsIdentity} \defeq \var{IsIdentity} \defeq
\prod_{A\ B \tp \Object} \prod_{f \tp \Arrow\ A\ B} \prod_{A, B \tp \Object} \prod_{f \tp \Arrow\ A\ B}
\left(\id \lll f \equiv f\right) \x \left(f \lll \id \equiv f\right) \left(\id \lll f \equiv f\right) \x \left(f \lll \id \equiv f\right)
\end{equation} \end{equation}
% %
Then we can construct the identity isomorphism $\idIso \tp \identity, Then we can construct the identity isomorphism $\idIso \tp (\identity,
\identity, p \tp A \approxeq A$ for any object $A$. Here $\approxeq$ denotes \identity, p) \tp A \approxeq A$ for any object $A$. Here $\approxeq$
isomorphism on objects (whereas $\cong$ denotes isomorphism of types). This will denotes isomorphism on objects (whereas $\cong$ denotes isomorphism on
be elaborated further on in sections \S\ref{sec:equiv} and types). This will be elaborated further on in sections
\S\ref{sec:univalence}. Moreover, due to substitution for paths we can construct \S\ref{sec:equiv} and \S\ref{sec:univalence}. Moreover due to
an isomorphism from \emph{any} path: substitution for paths we can construct an isomorphism from \emph{any}
path:
% %
\begin{equation} \begin{equation}
\idToIso \tp A ≡ B → A ≊ B \idToIso \tp A ≡ B → A ≊ B
@ -140,12 +146,13 @@ Note that \ref{eq:cat-univ} is \emph{not} the same as:
(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
former simply by ``forgetting'' that $\idToIso$ plays the role of the from the former simply by ``forgetting'' that $\idToIso$ plays the
equivalence. The other direction is more involved and will be discussed in role of the equivalence. The other direction is more involved and will
section \S\ref{sec:univalence}. be discussed in 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:
% %
\begin{align} \begin{align}
\Object & \tp \Type \\ \Object & \tp \Type \\
@ -180,19 +187,20 @@ composition (left-to-right, diagrammatic order) is denoted $\rrr$. The objects
($A$, $B$ and $C$) and arrow ($f$, $g$, $h$) are implicitly universally ($A$, $B$ and $C$) and arrow ($f$, $g$, $h$) are implicitly universally
quantified. quantified.
With all this in place it is now possible to prove that all the laws are indeed With all this in place it is now possible to prove that all the laws
mere propositions. Most of the proofs simply use the fact that the type of are indeed mere propositions. Most of the proofs simply use the fact
arrows are sets. This is because most of the laws are a collection of equations that the type of arrows are sets. This is because most of the laws are
between arrows in the category. And since such a proof does not have any content a collection of equations between arrows in the category. And since
exactly because the type of arrows form a set, two witnesses must be the same. such a proof does not have any content exactly because the type of
All the proofs are really quite mechanical. Lets have a look at one of them. arrows form a set, two witnesses must be the same. All the proofs are
Proving that \ref{eq:identity} is a mere proposition: really quite mechanical. Let us have a look at one of them: Proving
that \ref{eq:identity} is a mere proposition:
% %
\begin{equation} \begin{equation}
\isProp\ \var{IsIdentity} \isProp\ \var{IsIdentity}
\end{equation} \end{equation}
% %
There are multiple ways to prove this. Perhaps one of the more intuitive proofs There are multiple ways to do 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
\S\ref{sec:propPi} and \S\ref{sec:propSig}: \S\ref{sec:propPi} and \S\ref{sec:propSig}:
% %
@ -202,25 +210,26 @@ is by way of the `combinators' $\propPi$ and $\propSig$ presented in sections
\propSig & \tp \isProp\ A \to \left(\prod_{a \tp A} \isProp\ (P\ a)\right) \to \isProp\ \left(\sum_{a \tp A} P\ a\right) \propSig & \tp \isProp\ A \to \left(\prod_{a \tp A} \isProp\ (P\ a)\right) \to \isProp\ \left(\sum_{a \tp A} P\ a\right)
\end{align*} \end{align*}
% %
So the proof goes like this: We `eliminate' the 3 function abstractions by The proof goes like this: We `eliminate' the 3 function abstractions
applying $\propPi$ three times. So our proof obligation becomes: by applying $\propPi$ three times. So our proof obligation becomes:
% %
$$ $$
\isProp\ \left( \left( \id \comp f \equiv f \right) \x \left( f \comp \id \equiv f \right) \right) \isProp\ \left( \left( \id \comp f \equiv f \right) \x \left( f \comp \id \equiv f \right) \right)
$$ $$
% %
Then we eliminate the (non-dependent) sigma-type by applying $\propSig$ giving Then we eliminate the (non-dependent) sigma-type by applying $\propSig$ giving
us the two obligations: $\isProp\ (\id \comp f \equiv f)$ and $\isProp\ (f \comp us the two obligations $\isProp\ (\id \comp f \equiv f)$ and $\isProp\ (f \comp
\id \equiv f)$ which follows from the type of arrows being a \id \equiv f)$ which follows from the type of arrows being a
set. set.
This example illustrates nicely how we can use these combinators to reason about This example illustrates nicely how we can use these combinators to
`canonical' types like $\sum$ and $\prod$. Similar combinators can be defined at reason about `canonical' types like $\sum$ and $\prod$. Similar
the other homotopic levels. These combinators are however not applicable in combinators can be defined at the other homotopic levels. These
situations where we want to reason about other types - e.g.\ types we have combinators are however not applicable in situations where we want to
defined ourselves. For instance, after we have proven that all the projections reason about other types e.g.\ types we have defined ourselves. For
of pre categories are propositions, then we would like to bundle this up to show instance, after we have proven that all the projections of
that the type of pre categories is also a proposition. Formally: pre-categories are propositions, we would like to bundle this up to
show that the type of pre-categories is also a proposition. Formally:
% %
\begin{equation} \begin{equation}
\label{eq:propIsPreCategory} \label{eq:propIsPreCategory}
@ -235,19 +244,20 @@ Where The definition of $\IsPreCategory$ is the triple:
\var{arrowsAreSets} & \tp \var{ArrowsAreSets} \var{arrowsAreSets} & \tp \var{ArrowsAreSets}
\end{align*} \end{align*}
% %
Each corresponding to the first three laws for categories. Note that since Each corresponding to the first three laws for categories. Note that
$\IsPreCategory$ is not formulated with a chain of sigma-types we wont have any since $\IsPreCategory$ is not formulated with a chain of sigma-types
combinators available to help us here. In stead the paths must be used directly. we will not have any combinators available to help us here. In stead
the path type must be used directly.
\ref{eq:propIsPreCategory} is judgmentally the same as: The type \ref{eq:propIsPreCategory} is judgmentally the same as:
% %
$$ $$
\prod_{a\ b \tp \IsPreCategory} a \equiv b \prod_{a, b \tp \IsPreCategory} a \equiv b
$$ $$
% %
So to prove the proposition let $a\ b \tp \IsPreCategory$ be given. To So to prove the proposition let $a, b \tp \IsPreCategory$ be given. To
prove the equality $a \equiv b$ is to give a continuous path from the prove the equality $a \equiv b$ is to give a continuous path from the
index-type into the path-space. I.e.\ a function $I \to index-type into the path-space. I.e.\ a function $\I \to
\IsPreCategory$. This path must satisfy being being judgmentally the \IsPreCategory$. This path must satisfy being being judgmentally the
same as $a$ at the left endpoint and $b$ at the right endpoint. We same as $a$ at the left endpoint and $b$ at the right endpoint. We
know we can form a continuous path between all projections of $a$ and know we can form a continuous path between all projections of $a$ and
@ -261,8 +271,8 @@ $$
a.\isIdentity \equiv b.\isIdentity a.\isIdentity \equiv b.\isIdentity
$$ $$
% %
So to give the continuous function $I \to \IsPreCategory$, which is our goal, we So to give the continuous function $\I \to \IsPreCategory$, which is our goal, we
introduce $i \tp I$ and proceed by constructing an element of $\IsPreCategory$ introduce $i \tp \I$ and proceed by constructing an element of $\IsPreCategory$
by using the fact that all the projections are propositions to generate paths by using the fact that all the projections are propositions to generate paths
between all projections. Once we have such a path e.g.\ $p \tp a.\isIdentity between all projections. Once we have such a path e.g.\ $p \tp a.\isIdentity
\equiv b.\isIdentity$ we can eliminate it with $i$ and thus obtain $p\ i \tp \equiv b.\isIdentity$ we can eliminate it with $i$ and thus obtain $p\ i \tp
@ -299,9 +309,9 @@ unique.
The situation is a bit more complicated when we have a dependent type. The situation is a bit more complicated when we have a dependent type.
For instance, when we want to show that $\IsCategory$ is a mere For instance, when we want to show that $\IsCategory$ is a mere
proposition. The type $\IsCategory$ is a record with two fields, a proposition. The type $\IsCategory$ is a record with two fields, a
witness to being a pre category and the univalence condition. Recall witness to being a pre-category and the univalence condition. Recall
that the univalence condition is indexed by the identity-proof. So to that the univalence condition is indexed by the identity-proof. So to
follow the same recipe as above, let $a\ b \tp \IsCategory$ be given, follow the same recipe as above, let $a, b \tp \IsCategory$ be given,
to show them equal, we now need to give two paths. One homogeneous: to show them equal, we now need to give two paths. One homogeneous:
% %
$$ $$
@ -463,7 +473,7 @@ equalities and isomorphisms (on arrows). It is worthwhile to dwell on this for a
few seconds. This type looks very similar to univalence for types and is few seconds. This type looks very similar to univalence for types and is
therefore perhaps a bit more intuitive to grasp the implications of. Of course therefore perhaps a bit more intuitive to grasp the implications of. Of course
univalence for types (which is a proposition -- i.e.\ provable) does not imply univalence for types (which is a proposition -- i.e.\ provable) does not imply
univalence of all pre category since morphisms in a category are not regular univalence of all pre-category since morphisms in a category are not regular
functions -- in stead they can be thought of as a generalization hereof. The univalence criterion therefore is simply a way of restricting arrows functions -- in stead they can be thought of as a generalization hereof. The univalence criterion therefore is simply a way of restricting arrows
to behave similarly to maps. to behave similarly to maps.
@ -581,7 +591,7 @@ over-bar. So e.g.\ $\idToIso$ is a function in the underlying category and the
corresponding thing is denoted $\wideoverbar{\idToIso}$ in the opposite corresponding thing is denoted $\wideoverbar{\idToIso}$ in the opposite
category. category.
Showing that this forms a pre category is rather straightforward. Showing that this forms a pre-category is rather straightforward.
% %
$$ $$
h \rrr (g \rrr f) \equiv h \rrr g \rrr f h \rrr (g \rrr f) \equiv h \rrr g \rrr f
@ -600,7 +610,7 @@ Finally, that the arrows form sets just follows by flipping the order of the
arguments. Or in other words; since $\Arrow\ A\ B$ is a set for all $A\;B \tp 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 straightforward. Luckily
section \S\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)$.
@ -624,7 +634,7 @@ follows:
\wideoverbar{\isoToId} \comp \wideoverbar{\idToIso} & = \wideoverbar{\isoToId} \comp \wideoverbar{\idToIso} & =
\isoToId \comp \shufflef \comp \wideoverbar{\idToIso} \isoToId \comp \shufflef \comp \wideoverbar{\idToIso}
\\ \\
%% ≡⟨ cong (λ φ → φ x) (cong (λ φ → η ⊙ shuffle ⊙ φ) (funExt lem)) ⟩ \\ %% ≡⟨ cong (λ \; φ → φ x) (cong (λ \; φ → η ⊙ shuffle ⊙ φ) (funExt lem)) ⟩ \\
% %
& \equiv & \equiv
\isoToId \comp \shufflef \comp \inv{\shufflef} \comp \idToIso \isoToId \comp \shufflef \comp \inv{\shufflef} \comp \idToIso
@ -641,7 +651,7 @@ follows:
The other direction is analogous. The other direction is analogous.
The lemma used in step 2 of this proof states that $\wideoverbar{idToIso} \equiv The lemma used in step 2 of this proof states that $\wideoverbar{idToIso} \equiv
\inv{\shufflef} \comp \idToIso$. This is a rather straight-forward proof \inv{\shufflef} \comp \idToIso$. This is a rather straightforward proof
since being-an-inverse-of is a proposition, so it suffices to show that their since being-an-inverse-of is a proposition, so it suffices to show that their
first components are equal, but this holds judgmentally. first components are equal, but this holds judgmentally.
@ -673,7 +683,7 @@ homotopic sets. This is encapsulated in Agda with the following type:
% %
$$\Set \defeq \sum_{A \tp \MCU} \isSet\ A$$ $$\Set \defeq \sum_{A \tp \MCU} \isSet\ A$$
% %
The more straight-forward notion of a category where the objects are types is The more straightforward notion of a category where the objects are types is
not a valid \mbox{(1-)category}. This stems from the fact that types in cubical not a valid \mbox{(1-)category}. This stems from the fact that types in cubical
Agda types can have higher homotopic structure. Agda types can have higher homotopic structure.
@ -681,7 +691,7 @@ Univalence does not follow immediately from univalence for types:
% %
$$(A \equiv B) \simeq (A \simeq B)$$ $$(A \equiv B) \simeq (A \simeq B)$$
% %
Because here $A\ B \tp \Type$ whereas the objects in this category have the type Because here $A, B \tp \Type$ whereas the objects in this category have the type
$\Set$ so we cannot form the type $\var{hA} \simeq \var{hB}$ for objects $\Set$ so we cannot form the type $\var{hA} \simeq \var{hB}$ for objects
$\var{hA}\;\var{hB} \tp \Set$. In stead I show that this category $\var{hA}\;\var{hB} \tp \Set$. In stead I show that this category
satisfies: satisfies:
@ -956,7 +966,7 @@ underlying category. Given that $p$ is the chosen proof of
% %
\begin{align} \begin{align}
\label{eq:productPath} \label{eq:productPath}
λ\ i → d_{\pairA} \lll p\ i ≡ 2 a_{\pairA} × d_{\pairB} \lll p\ i ≡ a_{\pairB} λ \; i → d_{\pairA} \lll p\ i ≡ 2 a_{\pairA} × d_{\pairB} \lll p\ i ≡ a_{\pairB}
\end{align} \end{align}
% %
And these paths are in the type of the hom-set of the underlying category, so And these paths are in the type of the hom-set of the underlying category, so
@ -1006,7 +1016,7 @@ $$
This we get from \ref{eq:productEqPrinc} and the fact that homotopical structure This we get from \ref{eq:productEqPrinc} and the fact that homotopical structure
is cumulative. is cumulative.
This finishes the proof that this is a valid pre category. This finishes the proof that this is a valid pre-category.
\subsubsection{Univalence} \subsubsection{Univalence}
To prove that this is a proper category it must be shown that it is univalent. To prove that this is a proper category it must be shown that it is univalent.
@ -1032,8 +1042,8 @@ The next types will be the triple:
\label{eq:univ-1} \label{eq:univ-1}
\begin{split} \begin{split}
p \tp & X \equiv Y \\ p \tp & X \equiv Y \\
& \Path\ (λ i → \Arrow\ (p\ i)\ \mathcal{A})\ x_{\mathcal{A}}\ y_{\mathcal{A}} \\ & \Path\ (λ \; i → \Arrow\ (p\ i)\ \mathcal{A})\ x_{\mathcal{A}}\ y_{\mathcal{A}} \\
& \Path\ (λ i → \Arrow\ (p\ i)\ \mathcal{B})\ x_{\mathcal{B}}\ y_{\mathcal{B}} & \Path\ (λ \; i → \Arrow\ (p\ i)\ \mathcal{B})\ x_{\mathcal{B}}\ y_{\mathcal{B}}
\end{split} \end{split}
%% \end{split} %% \end{split}
\end{align} \end{align}
@ -1045,8 +1055,8 @@ isomorphism, and create a path from this:
\label{eq:univ-2} \label{eq:univ-2}
\begin{split} \begin{split}
\var{iso} \tp & X \approxeq Y \\ \var{iso} \tp & X \approxeq Y \\
& \Path\ (λ i → \Arrow\ (\widetilde{p}\ i)\ \mathcal{A})\ x_{\mathcal{A}}\ y_{\mathcal{A}} \\ & \Path\ (λ \; i → \Arrow\ (\widetilde{p}\ i)\ \mathcal{A})\ x_{\mathcal{A}}\ y_{\mathcal{A}} \\
& \Path\ (λ i → \Arrow\ (\widetilde{p}\ i)\ \mathcal{B})\ x_{\mathcal{B}}\ y_{\mathcal{B}} & \Path\ (λ \; i → \Arrow\ (\widetilde{p}\ i)\ \mathcal{B})\ x_{\mathcal{B}}\ y_{\mathcal{B}}
\end{split} \end{split}
\end{align} \end{align}
% %
@ -1074,7 +1084,7 @@ This proof of this has been omitted but can be found in the module:
\emph{Proposition} \ref{eq:univ-2} is isomorphic to \ref{eq:univ-3}: For this I \emph{Proposition} \ref{eq:univ-2} is isomorphic to \ref{eq:univ-3}: For this I
will show two corollaries of \ref{eq:coeCod}: For an isomorphism $(\iota, will show two corollaries of \ref{eq:coeCod}: For an isomorphism $(\iota,
\inv{\iota}, \var{inv}) \tp A \cong B$, arrows $f \tp \Arrow\ A\ X$, $g \tp \inv{\iota}, \var{inv}) \tp A \cong B$, arrows $f \tp \Arrow\ A\ X$, $g \tp
\Arrow\ B\ X$ and a heterogeneous path between them, $q \tp \Path\ (\lambda i \Arrow\ B\ X$ and a heterogeneous path between them, $q \tp \Path\ (\lambda\; i
\to p_{\var{dom}}\ i)\ f\ g$, where $p_{\var{dom}} \tp \Arrow\ A\ X \equiv \to p_{\var{dom}}\ i)\ f\ g$, where $p_{\var{dom}} \tp \Arrow\ A\ X \equiv
\Arrow\ B\ X$ is a path induced by $\var{iso}$, we have the following two \Arrow\ B\ X$ is a path induced by $\var{iso}$, we have the following two
results results
@ -1095,9 +1105,9 @@ Now we can prove the equivalence in the following way: Given $(f, \inv{f},
\var{inv}_f) \tp X \cong Y$ and two heterogeneous paths \var{inv}_f) \tp X \cong Y$ and two heterogeneous paths
% %
\begin{align*} \begin{align*}
p_{\mathcal{A}} & \tp \Path\ (\lambda i \to p_{\var{dom}}\ i)\ x_{\mathcal{A}}\ y_{\mathcal{A}}\\ p_{\mathcal{A}} & \tp \Path\ (\lambda\; i \to p_{\var{dom}}\ i)\ x_{\mathcal{A}}\ y_{\mathcal{A}}\\
% %
q_{\mathcal{B}} & \tp \Path\ (\lambda i \to p_{\var{dom}}\ i)\ x_{\mathcal{B}}\ y_{\mathcal{B}} q_{\mathcal{B}} & \tp \Path\ (\lambda\; i \to p_{\var{dom}}\ i)\ x_{\mathcal{B}}\ y_{\mathcal{B}}
\end{align*} \end{align*}
% %
all as in \ref{eq:univ-2}. I use $p_{\var{dom}}$ here again to mean the path all as in \ref{eq:univ-2}. I use $p_{\var{dom}}$ here again to mean the path
@ -1163,8 +1173,8 @@ It then remains to construct the two paths:
\begin{align} \begin{align}
\begin{split} \begin{split}
\label{eq:product-paths} \label{eq:product-paths}
& \Path\ (λ i → \widetilde{p}_{\mathcal{A}}\ i)\ x_{\mathcal{A}}\ y_{\mathcal{A}}\\ & \Path\ (λ \; i → \widetilde{p}_{\mathcal{A}}\ i)\ x_{\mathcal{A}}\ y_{\mathcal{A}}\\
& \Path\ (λ i → \widetilde{p}_{\mathcal{B}}\ i)\ x_{\mathcal{B}}\ y_{\mathcal{B}} & \Path\ (λ \; i → \widetilde{p}_{\mathcal{B}}\ i)\ x_{\mathcal{B}}\ y_{\mathcal{B}}
\end{split} \end{split}
\end{align} \end{align}
% %
@ -1172,7 +1182,7 @@ This is achieved with the following lemma:
% %
\begin{align} \begin{align}
\prod_{a \tp A} \prod_{b \tp B} \prod_{q \tp A \equiv B} \var{coe}\ q\ a ≡ b → \prod_{a \tp A} \prod_{b \tp B} \prod_{q \tp A \equiv B} \var{coe}\ q\ a ≡ b →
\Path\ (λ i → q\ i)\ a\ b \Path\ (λ \; i → q\ i)\ a\ b
\end{align} \end{align}
% %
Which is used without proof. See the implementation for the details. Which is used without proof. See the implementation for the details.
@ -1201,12 +1211,17 @@ the proof uses the fact that isomorphism-of is propositional and that arrows (in
both categories) are sets. The reader is referred to the implementation for the both categories) are sets. The reader is referred to the implementation for the
gory details. gory details.
% %
\subsection{Propositionality of products} \subsection{Products are propositions}
% %
Now that we have constructed the span category\index{span category} I Now that we have constructed the span category\index{span category} I
will demonstrate how to use this to prove that products are will demonstrate how to use this to prove that products are
propositional. I will do this by showing that terminal objects in this propositions. On the face of it this may seem surprising. Products
category are equivalent to products: look like they are a structure on categories. After all it consist of
a select element and two arrows given some two objects. If formulated
in set theory this would be the case but in the present setting
univalence of categories give us that products are properties. I will
show this by showing that terminal objects in the span category are
equivalent to products:
% %
\begin{align} \begin{align}
\var{Terminal} ≃ \var{Product}\ ℂ\ \mathcal{A}\ \mathcal{B} \var{Terminal} ≃ \var{Product}\ ℂ\ \mathcal{A}\ \mathcal{B}
@ -1248,9 +1263,9 @@ is the proof that $f$ satisfies \ref{eq:pairCondRev}). The proof will be a pair
of proofs: of proofs:
% %
\begin{alignat}{3} \begin{alignat}{3}
p \tp & \Path\ (\lambda i \to \Arrow\ X\ Y)\quad p \tp & \Path\ (\lambda\; i \to \Arrow\ X\ Y)\quad
&& f\quad && y_𝒜 \x y_ℬ \\ && f\quad && y_𝒜 \x y_ℬ \\
& \Path\ (\lambda i \to \Phi\ (p\ i))\quad & \Path\ (\lambda\; i \to \Phi\ (p\ i))\quad
&& \phi_f\quad && \phi_{y_𝒜 \x y_ℬ} && \phi_f\quad && \phi_{y_𝒜 \x y_ℬ}
\end{alignat} \end{alignat}
% %
@ -1283,7 +1298,7 @@ preserve homotopic levels along with \ref{eq:termProp} we get our final result.
That in any category: That in any category:
% %
\begin{align} \begin{align}
\prod_{A\ B \tp \Object} \isProp\ (\var{Product}\ \bC\ A\ B) \prod_{A, B \tp \Object} \isProp\ (\var{Product}\ \bC\ A\ B)
\end{align} \end{align}
% %
\section{Functors and natural transformations} \section{Functors and natural transformations}

229
doc/introduction.tex
View file

@ -6,22 +6,27 @@ of equality: \nomenindex{judgmental equality} and
\nomenindex{propositional equality}. Judgmental equality is a property \nomenindex{propositional equality}. Judgmental equality is a property
of the type system. Judgmental equality on the other hand is usually of the type system. Judgmental equality on the other hand is usually
defined \emph{within} the system. When introducing definitions this defined \emph{within} the system. When introducing definitions this
report will use the notation $\defeq$. Judgmental equalities written report will use the symbol $\defeq$. Judgmental equalities will be
$=$. For propositional equalities the notation $\equiv$ is used. denoted with $=$ and for propositional equalities the notation
$\equiv$ is used.
For judgmental equality there are some properties that it must The rules of judgmental equality are related with $β$- and
satisfy. \nomenindex{sound}, enjoy \nomenindex{canonicity} and be a $η$-reduction which gives a notion of computation in a given type
\nomen{congruence relation}. Soundness means that things judged to be theory.
equal are equal with respects to the model of the theory (the meta %
theory). It must be a congruence relation because otherwise the There are some properties that one usually want judgmental equality to
relation certainly does not adhere to our notion of equality. One satisfy. It must be \nomenindex{sound}, enjoy \nomenindex{canonicity}
would be able to conclude things like: $x \equiv y \rightarrow f\ x and be a \nomenindex{congruence relation}. Soundness means that things
\nequiv f\ y$. Canonicity means that any well typed term evaluates to judged to be equal are equal with respects to the \nomenindex{model}
a \emph{canonical} form. For example for a closed term $e \tp \bN$ it of the theory or the \emph{meta theory}. It must be a congruence
will be the case that $e$ reduces to $n$ applications of relation because otherwise the relation certainly does not adhere to
$\mathit{suc}$ to $0$ for some $n$; $e = \mathit{suc}^n\ 0$. Without our notion of equality. One would be able to conclude things like: $x
canonicity terms in the language can get ``stuck'' -- meaning that \equiv y \rightarrow f\ x \nequiv f\ y$. Canonicity means that any
they do not reduce to a canonical form. 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 To work as a programming languages it is necessary for judgmental
equality to be \nomenindex{decidable}. Being decidable simply means equality to be \nomenindex{decidable}. Being decidable simply means
@ -34,27 +39,27 @@ is not in general possible to decide the correctness of logical
propositions (cf.\ Hilbert's \emph{entscheidigungsproblem}). propositions (cf.\ Hilbert's \emph{entscheidigungsproblem}).
There are two flavors of type-theory. \emph{Intensional-} and There are two flavors of type-theory. \emph{Intensional-} and
\emph{extensional-} type theory. Identity types in extensional type \emph{extensional-} type theory (ITT and ETT respectively). Identity
theory are required to be \nomen{propositions}{proposition}. That is, types in extensional type theory are required to be
a type with at most one inhabitant. In extensional type thoery the \nomen{propositions}{proposition}. That is, a type with at most one
principle of reflection inhabitant. In extensional type theory the principle of reflection
% %
$$a ≡ b → a = b$$ $$a ≡ b → a = b$$
% %
is enough to make type checking undecidable. This report focuses on 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 Agda which at a glance can be thought of as a version of intensional
theory. Pattern-matching in regular Agda let's one prove type theory. Pattern-matching in regular Agda lets one prove
\nomenindex{axiom K}. Axiom K states that any two identity proofs are \nomenindex{Uniqueness of Identity Proofs} (UIP). UIP states that any
propositionally identical. two identity proofs are propositionally identical.
The usual notion of propositional equality in \nomenindex{Intensional The usual notion of propositional equality in ITT is quite
Type Theory} (ITT) is quite restrictive. In the next section a few restrictive. In the next section a few motivating examples will
motivating examples will highlight this. There exist techniques to highlight this. There exist techniques to circumvent these problems,
circumvent these problems, as we shall see. This thesis will explore as we shall see. This thesis will explore an extension to Agda that
an extension to Agda that redefines the notion of propositional redefines the notion of propositional equality and as such is an
equality and as such is an alternative to these other techniques. The alternative to these other techniques. The extension is called cubical
extension is called cubical Agda. Cubical Agda drops Axiom K as this Agda. Cubical Agda drops UIP as this does not permit
does not permit \nomenindex{functional extensionality} and \nomenindex{functional extensionality} and
\nomenindex{univalence}. What makes this extension particularly \nomenindex{univalence}. What makes this extension particularly
interesting is that it gives a \emph{constructive} interpretation of interesting is that it gives a \emph{constructive} interpretation of
univalence. What all this means will be elaborated in the following univalence. What all this means will be elaborated in the following
@ -69,74 +74,53 @@ some limitations inherent in ITT and -- by extension -- Agda.
\label{sec:functional-extensionality}% \label{sec:functional-extensionality}%
Consider the functions: Consider the functions:
% %
\begin{multicols}{2} \begin{align*}%
\noindent% \var{zeroLeft} & \defeq \lambda\; (n \tp \bN) \to (0 + n \tp \bN) \\
\begin{equation*}% \var{zeroRight} & \defeq \lambda\; (n \tp \bN) \to (n + 0 \tp \bN)
f \defeq \lambda\ (n \tp \bN) \to (0 + n \tp \bN) \end{align*}%
\end{equation*}%
\begin{equation*}%
g \defeq \lambda\ (n \tp \bN) \to (n + 0 \tp \bN)
\end{equation*}%
\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 \nomenindex{judgmental
write as $n + 0 = n$. This is also called equality}. We call it definitional equality because the
\nomenindex{judgmental equality}. \emph{equality} arises from the \emph{definition} of $+$ which is:
We call it definitional equality because the \emph{equality} arises
from the \emph{definition} of $+$ which is:
% %
\begin{align*} \begin{align*}
+ & \tp \bN \to \bN \to \bN \\ + & \tp \bN \to \bN \to \bN \\
n + 0 & \defeq n \\ n + 0 & \defeq n \\
n + (\suc{m}) & \defeq \suc{(n + m)} n + (\suc{m}) & \defeq \suc{(n + m)}
\end{align*} \end{align*}
% %
Note that $0 + n$ is \emph{not} definitionally equal to $n$. $0 + n$ Note that $0 + n$ is \emph{not} definitionally equal to $n$. This is
is in normal form. I.e.; there is no rule for $+$ whose left hand side because $0 + n$ is in normal form. I.e.\ there is no rule for $+$
matches this expression. We \emph{do}, however, have that they are whose left hand side matches this expression. We do however have that
\nomen{propositionally}{propositional equality} equal, which we write they are \nomen{propositionally}{propositional equality} equal, which
as $n + 0 \equiv n$. Propositional equality means that there is a we write as $n \equiv n + 0$. Propositional equality means that there
proof that exhibits this relation. Since equality is a transitive is a proof that exhibits this relation. We can do induction over $n$
relation we have that $n + 0 \equiv 0 + n$. to prove this:
%
Unfortunately we don't have $f \equiv g$. There is no way to construct \begin{align}
a proof asserting the obvious equivalence of $f$ and $g$. Actually \label{eq:zrn}
showing this is outside the scope of this text. Essentially it would \begin{split}
involve giving a model for our type theory that validates all our \var{zrn}\ & \tp ∀ n → n ≡ \var{zeroRight}\ n \\
axioms but where $f \equiv g$ is not true. We cannot show that they \var{zrn}\ \var{zero} & \defeq \var{refl} \\
are equal, even though we can prove them equal for all points. For \var{zrn}\ (\var{suc}\ n) & \defeq \var{cong}\ \var{suc}\ (\var{zrn}\ n)
functions this is exactly the notion of equality that we are \end{split}
interested in: Functions are considered equal when they are equal for \end{align}
all inputs. This is called \nomenindex{point wise equality}, where the %
\emph{points} of a function refer to its arguments. This show that zero is a right neutral element hence the name $\var{zrn}$.
Since equality is a transitive relation we have that $\forall n \to
%% In the context of category theory functional extensionality is e.g. \var{zeroLeft}\ n \equiv \var{zeroRight}\ n$. Unfortunately we don't
%% needed to show that representable functors are indeed functors. The have $\var{zeroLeft} \equiv \var{zeroRight}$. There is no way to
%% representable functor is defined for a fixed category $\bC$ and an construct a proof asserting the obvious equivalence of
%% object $X \in \bC$. It's map on objects is defined thus: $\var{zeroLeft}$ and $\var{zeroRight}$. Actually showing this is
%% % outside the scope of this text. Essentially it would involve giving a
%% \begin{align*} model for our type theory that validates all our axioms but where
%% \lambda\ A \to \Arrow\ X\ A $\var{zeroLeft} \equiv \var{zeroRight}$ is not true. We cannot show
%% \end{align*} 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
%% That is, it maps objects to arrows. So, it's map on arrows must map an arrow $\Arrow\ A\ B$ to an are interested in: Functions are considered equal when they are equal
%% The map on objects is defined thus: for all inputs. This is called \nomenindex{pointwise equality}, where
%% % the \emph{points} of a function refer to its arguments.
%% \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} \subsection{Equality of isomorphic types}
% %
@ -155,7 +139,6 @@ without thinking twice. Unfortunately such an identification can not
be performed in ITT. 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 \nomenindex{equivalent} types. I will return to the definition of
equivalence later in section \S\ref{sec:equiv}, but for now it is 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. sufficient to think of an equivalence as a one-to-one correspondence.
@ -165,27 +148,30 @@ The principle of univalence says that:
$$\mathit{univalence} \tp (A \simeq B) \simeq (A \equiv B)$$ $$\mathit{univalence} \tp (A \simeq B) \simeq (A \equiv B)$$
% %
In particular this allows us to construct an equality from an equivalence 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. %
$$\mathit{ua} \tp (A \simeq B) \to (A \equiv B)$$
%
and vice versa.
\section{Formalizing Category Theory} \section{Formalizing Category Theory}
% %
The above examples serve to illustrate a limitation of ITT. One case where these The above examples serve to illustrate a limitation of ITT. One case
limitations are particularly prohibitive is in the study of Category Theory. At where these limitations are particularly prohibitive is in the study
a glance category theory can be described as ``the mathematical study of of Category Theory. At a glance category theory can be described as
(abstract) algebras of functions'' (\cite{awodey-2006}). By that token ``the mathematical study of (abstract) algebras of functions''
functional extensionality is particularly useful for formulating Category (\cite{awodey-2006}). By that token functional extensionality is
Theory. In Category theory it is also common to identify isomorphic structures particularly useful for formulating Category Theory. In Category
and univalence gives us a way to make this notion precise. In fact we can theory it is also commonplace to identify isomorphic structures and
formulate this requirement within our formulation of categories by requiring the univalence gives us a way to make this notion precise. In fact we can
\emph{categories} themselves to be univalent as we shall see. formulate this requirement within our formulation of categories by
requiring the \emph{categories} themselves to be univalent as we shall
see in \S\ref{sec:univalence}.
\section{Context} \section{Context}
\label{sec:context} \label{sec:context}
% %
The idea of formalizing Category Theory in proof assistants is not new. There 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 are a multitude of these available online. Notably:
question on Math Overflow: \cite{mo-formalizations}. Notably these
implementations of category theory in Agda:
% %
\begin{itemize} \begin{itemize}
\item \item
@ -217,12 +203,12 @@ other shortcomings, e.g. you lose \nomenindex{canonicity}
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 \nomenindex{extensional sets} $(X, \sim)$, that is a type $X \tp 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 \MCU$ and an equivalence relation $\sim\ \tp X \to X \to \MCU$ on that
type. Under the setoid interpretation the equivalence relation serve type. Under the setoid interpretation the equivalence relation serve
as a sort of ``local'' propositional equality. Since the developer as a sort of ``local'' propositional equality. Since the developer
gets to pick this relation it is not guaranteed to be a congruence gets to pick this relation it is not a\~priori a congruence
relation a priori. So this must be verified manually by the developer. relation. So this must be verified manually by the developer.
Furthermore, functions between different setoids must be shown to be Furthermore, functions between different setoids must be shown to be
setoid homomorphism, that is; they preserve the relation. setoid homomorphism, that is; they preserve the relation.
@ -237,12 +223,14 @@ makes it very cumbersome to work with in practice (\cite[p.
4]{huber-2016}). 4]{huber-2016}).
\section{Conventions} \section{Conventions}
In the remainder of this paper I will use the term In the remainder of this paper I will use the term \nomenindex{Type}
\nomenindex{Type} to describe -- to describe -- well -- types. Thereby departing from the notation in
well, types. Thereby diverging from the notation in Agda where the keyword Agda where the keyword \texttt{Set} refers to types. \nomenindex{Set}
\texttt{Set} refers to types. on the other hand shall refer to the homotopical notion of a set. I
\nomenindex{Set} on the other hand shall refer to the will also leave all universe levels implicit. This of course does not
homotopical notion of a set. I will also leave all universe levels implicit. mean that a statement such as $\MCU \tp \MCU$ means that we have
type-in-type but rather that the arguments to the universes are
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,
@ -253,11 +241,13 @@ whereas the terms
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.
$\defeq$ will be used for introducing definitions. $=$ will be used to for As already noted $\defeq$ will be used for introducing definitions $=$
judgmental equality and $\equiv$ will be used for propositional equality. will be used to for judgmental equality and $\equiv$ will be used for
propositional equality.
All this is summarized in the following table: All this is summarized in the following table:
%
\begin{samepage}
\begin{center} \begin{center}
\begin{tabular}{ c c c } \begin{tabular}{ c c c }
Name & Agda & Notation \\ Name & Agda & Notation \\
@ -277,3 +267,4 @@ Judgmental equality & \null & $̱=$ \\
Propositional equality & \null & $̱\equiv$ Propositional equality & \null & $̱\equiv$
\end{tabular} \end{tabular}
\end{center} \end{center}
\end{samepage}

1
doc/macros.tex
View file

@ -127,3 +127,4 @@
\newcommand\coe{\varindex{coe}} \newcommand\coe{\varindex{coe}}
\newcommand\Monoidal{\varindex{Monoidal}} \newcommand\Monoidal{\varindex{Monoidal}}
\newcommand\Kleisli{\varindex{Kleisli}} \newcommand\Kleisli{\varindex{Kleisli}}
\newcommand\I{\mathds{I}}

2
doc/main.tex
View file

@ -1,4 +1,5 @@
\documentclass[a4paper]{report} \documentclass[a4paper]{report}
%% \documentclass[tightpage]{preview}
%% \documentclass[compact,a4paper]{article} %% \documentclass[compact,a4paper]{article}
\input{packages.tex} \input{packages.tex}
@ -49,6 +50,7 @@
\myfrontmatter \myfrontmatter
\maketitle \maketitle
\input{abstract.tex} \input{abstract.tex}
\input{acknowledgements.tex}
\tableofcontents \tableofcontents
\mainmatter \mainmatter
% %

5
doc/packages.tex
View file

@ -16,13 +16,14 @@
%% \hypersetup{allbordercolors={darkorange}} %% \hypersetup{allbordercolors={darkorange}}
\hypersetup{hidelinks} \hypersetup{hidelinks}
\usepackage{graphicx} \usepackage{graphicx}
%% \usepackage[active,tightpage]{preview}
\usepackage{parskip} \usepackage{parskip}
\usepackage{multicol} \usepackage{multicol}
\usepackage{amssymb,amsmath,amsthm,stmaryrd,mathrsfs,wasysym} \usepackage{amssymb,amsmath,amsthm,stmaryrd,mathrsfs,wasysym}
\usepackage[toc,page]{appendix} \usepackage[toc,page]{appendix}
\usepackage{xspace} \usepackage{xspace}
\usepackage[a4paper,top=3cm,bottom=3cm]{geometry} \usepackage[paper=a4paper,top=3cm,bottom=3cm]{geometry}
\usepackage{makeidx} \usepackage{makeidx}
\makeindex \makeindex
% \setlength{\parskip}{10pt} % \setlength{\parskip}{10pt}
@ -83,7 +84,7 @@
\newunicodechar{∨}{\textfallback{∨}} \newunicodechar{∨}{\textfallback{∨}}
\newunicodechar{∧}{\textfallback{∧}} \newunicodechar{∧}{\textfallback{∧}}
\newunicodechar{⊔}{\textfallback{⊔}} \newunicodechar{⊔}{\textfallback{⊔}}
\newunicodechar{≊}{\textfallback{≊}} %% \newunicodechar{≊}{\textfallback{≊}}
\newunicodechar{∈}{\textfallback{∈}} \newunicodechar{∈}{\textfallback{∈}}
\newunicodechar{ℂ}{\textfallback{ℂ}} \newunicodechar{ℂ}{\textfallback{ℂ}}
\newunicodechar{∘}{\textfallback{∘}} \newunicodechar{∘}{\textfallback{∘}}

206
doc/presentation.tex
View file

@ -1,4 +1,5 @@
\documentclass[a4paper,handout]{beamer} \documentclass[a4paper,handout]{beamer}
\usetheme{metropolis}
\beamertemplatenavigationsymbolsempty \beamertemplatenavigationsymbolsempty
%% \usecolortheme[named=seagull]{structure} %% \usecolortheme[named=seagull]{structure}
@ -21,28 +22,28 @@
\framesubtitle{Functional extensionality} \framesubtitle{Functional extensionality}
Consider the functions Consider the functions
\begin{align*} \begin{align*}
\var{zeroLeft} & \defeq \lambda (n \tp \bN) \mto (0 + n \tp \bN) \\ \var{zeroLeft} & ≜ \lambda (n \tp \bN) \mto (0 + n \tp \bN) \\
\var{zeroRight} & \defeq \lambda (n \tp \bN) \mto (n + 0 \tp \bN) \var{zeroRight} & ≜ \lambda (n \tp \bN) \mto (n + 0 \tp \bN)
\end{align*} \end{align*}
\pause \pause
We have We have
% %
$$ $$
\prod_{n \tp \bN} \var{zeroLeft}\ n \equiv \var{zeroRight}\ n ∏_{n \tp \bN} \var{zeroLeft}\ n ≡ \var{zeroRight}\ n
$$ $$
% %
\pause \pause
But not But not
% %
$$ $$
\var{zeroLeft} \equiv \var{zeroRight} \var{zeroLeft} ≡ \var{zeroRight}
$$ $$
% %
\pause \pause
We need We need
% %
$$ $$
\funExt \tp \prod_{a \tp A} f\ a \equiv g\ a \to f \equiv g \funExt \tp ∏_{a \tp A} f\ a ≡ g\ a → f ≡ g
$$ $$
\end{frame} \end{frame}
\begin{frame} \begin{frame}
@ -52,20 +53,20 @@
$\{x \mid \phi\ x \land \psi\ x\}$ $\{x \mid \phi\ x \land \psi\ x\}$
\pause \pause
If we show $\forall x . \psi\ x \equiv \top$ If we show $∀ x . \psi\ x ≡ \top$
then we want to conclude then we want to conclude
$\{x \mid \phi\ x \land \psi\ x\} \equiv \{x \mid \phi\ x\}$ $\{x \mid \phi\ x \land \psi\ x\} ≡ \{x \mid \phi\ x\}$
\pause \pause
We need univalence: We need univalence:
$$(A \simeq B) \simeq (A \equiv B)$$ $$(A ≃ B) ≃ (A ≡ B)$$
\pause \pause
% %
We will return to $\simeq$, but for now think of it as an We will return to $≃$, but for now think of it as an
isomorphism, so it induces maps: isomorphism, so it induces maps:
\begin{align*} \begin{align*}
\var{toPath} & \tp (A \simeq B) \to (A \equiv B) \\ \var{toPath} & \tp (A ≃ B) → (A ≡ B) \\
\var{toEquiv} & \tp (A \equiv B) \to (A \simeq B) \var{toEquiv} & \tp (A ≡ B) → (A ≃ B)
\end{align*} \end{align*}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
@ -76,11 +77,11 @@
\Path \tp (P \tp I → \MCU) → P\ 0 → P\ 1 → \MCU \Path \tp (P \tp I → \MCU) → P\ 0 → P\ 1 → \MCU
\end{equation*} \end{equation*}
\pause \pause
For $P \tp I \to \MCU$, $A \tp \MCU$ and $a_0, a_1 \tp A$ For $P \tp I → \MCU$, $A \tp \MCU$ and $a_0, a_1 \tp A$
inhabitants of $\Path\ P\ a_0\ a_1$ are like functions inhabitants of $\Path\ P\ a_0\ a_1$ are like functions
% %
$$ $$
p \tp \prod_{i \tp I} P\ i p \tp ∏_{i \tp I} P\ i
$$ $$
% %
Which satisfy $p\ 0 & = a_0$ and $p\ 1 & = a_1$ Which satisfy $p\ 0 & = a_0$ and $p\ 1 & = a_1$
@ -88,46 +89,46 @@
Homogenous paths Homogenous paths
$$ $$
a_0 \equiv a_1 \defeq \Path\ (\var{const}\ A)\ a_0\ a_1 a_0 ≡ a_1 ≜ \Path\ (\var{const}\ A)\ a_0\ a_1
$$ $$
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Paths} \frametitle{Paths}
\framesubtitle{Functional extenstionality} \framesubtitle{Functional extenstionality}
$$ $$
\funExt & \tp \prod_{a \tp A} f\ a \equiv g\ a \to f \equiv g \funExt & \tp ∏_{a \tp A} f\ a ≡ g\ a → f ≡ g
$$ $$
\pause \pause
$$ $$
\funExt\ p \defeq λ i\ a → p\ a\ i \funExt\ p ≜ λ i\ a → p\ a\ i
$$ $$
\pause \pause
$$ $$
\funExt\ (\var{const}\ \refl) \funExt\ (\var{const}\ \refl)
\tp \tp
\var{zeroLeft} \equiv \var{zeroRight} \var{zeroLeft} ≡ \var{zeroRight}
$$ $$
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Paths} \frametitle{Paths}
\framesubtitle{Homotopy levels} \framesubtitle{Homotopy levels}
\begin{align*} \begin{align*}
& \isContr && \tp \MCU \to \MCU \\ & \isContr && \tp \MCU → \MCU \\
& \isContr\ A && \defeq \sum_{c \tp A} \prod_{a \tp A} a \equiv c & \isContr\ A && ≜ ∑_{c \tp A} ∏_{a \tp A} a ≡ c
\end{align*} \end{align*}
\pause \pause
\begin{align*} \begin{align*}
& \isProp && \tp \MCU \to \MCU \\ & \isProp && \tp \MCU → \MCU \\
& \isProp\ A && \defeq \prod_{a_0, a_1 \tp A} a_0 \equiv a_1 & \isProp\ A && ≜ ∏_{a_0, a_1 \tp A} a_0 ≡ a_1
\end{align*} \end{align*}
\pause \pause
\begin{align*} \begin{align*}
& \isSet && \tp \MCU \to \MCU \\ & \isSet && \tp \MCU → \MCU \\
& \isSet\ A && \defeq \prod_{a_0, a_1 \tp A} \isProp\ (a_0 \equiv a_1) & \isSet\ A && ≜ ∏_{a_0, a_1 \tp A} \isProp\ (a_0 ≡ a_1)
\end{align*} \end{align*}
\begin{align*} \begin{align*}
& \isGroupoid && \tp \MCU \to \MCU \\ & \isGroupoid && \tp \MCU → \MCU \\
& \isGroupoid\ A && \defeq \prod_{a_0, a_1 \tp A} \isSet\ (a_0 \equiv a_1) & \isGroupoid\ A && ≜ ∏_{a_0, a_1 \tp A} \isSet\ (a_0 ≡ a_1)
\end{align*} \end{align*}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
@ -135,7 +136,7 @@
\framesubtitle{A few lemmas} \framesubtitle{A few lemmas}
Let $D$ be a type-family: Let $D$ be a type-family:
$$ $$
D \tp \prod_{b \tp A} \prod_{p \tp a ≡ b} \MCU D \tp ∏_{b \tp A} ∏_{p \tp a ≡ b} \MCU
$$ $$
% %
\pause \pause
@ -149,7 +150,7 @@
We then have the function: We then have the function:
% %
$$ $$
\pathJ\ D\ d \tp \prod_{b \tp A} \prod_{p \tp a ≡ b} D\ b\ p \pathJ\ D\ d \tp ∏_{b \tp A} ∏_{p \tp a ≡ b} D\ b\ p
$$ $$
\end{frame} \end{frame}
\begin{frame} \begin{frame}
@ -158,9 +159,9 @@
Given Given
\begin{align*} \begin{align*}
A & \tp \MCU \\ A & \tp \MCU \\
P & \tp A \to \MCU \\ P & \tp A → \MCU \\
\var{propP} & \tp \prod_{x \tp A} \isProp\ (P\ x) \\ \var{propP} & \tp ∏_{x \tp A} \isProp\ (P\ x) \\
p & \tp a_0 \equiv a_1 \\ p & \tp a_0 ≡ a_1 \\
p_0 & \tp P\ a_0 \\ p_0 & \tp P\ a_0 \\
p_1 & \tp P\ a_1 p_1 & \tp P\ a_1
\end{align*} \end{align*}
@ -176,17 +177,17 @@
\begin{frame} \begin{frame}
\frametitle{Paths} \frametitle{Paths}
\framesubtitle{A few lemmas} \framesubtitle{A few lemmas}
$\prod$ preserves $\isProp$: $∏$ preserves $\isProp$:
$$ $$
\mathit{propPi} \mathit{propPi}
\tp \tp
\left(\prod_{a \tp A} \isProp\ (P\ a)\right) \left(∏_{a \tp A} \isProp\ (P\ a)\right)
\to \isProp\ \left(\prod_{a \tp A} P\ a\right) → \isProp\ \left(∏_{a \tp A} P\ a\right)
$$ $$
\pause \pause
$\sum$ preserves $\isProp$: $∑$ preserves $\isProp$:
$$ $$
\mathit{propSig} \tp \isProp\ A \to \left(\prod_{a \tp A} \isProp\ (P\ a)\right) \to \isProp\ \left(\sum_{a \tp A} P\ a\right) \mathit{propSig} \tp \isProp\ A → \left(∏_{a \tp A} \isProp\ (P\ a)\right) → \isProp\ \left(∑_{a \tp A} P\ a\right)
$$ $$
\end{frame} \end{frame}
\begin{frame} \begin{frame}
@ -195,9 +196,9 @@
Data: Data:
\begin{align*} \begin{align*}
\Object & \tp \Type \\ \Object & \tp \Type \\
\Arrow & \tp \Object \to \Object \to \Type \\ \Arrow & \tp \Object → \Object → \Type \\
\identity & \tp \Arrow\ A\ A \\ \identity & \tp \Arrow\ A\ A \\
\lll & \tp \Arrow\ B\ C \to \Arrow\ A\ B \to \Arrow\ A\ C \lll & \tp \Arrow\ B\ C → \Arrow\ A\ B → \Arrow\ A\ C
\end{align*} \end{align*}
% %
\pause \pause
@ -208,7 +209,7 @@
$$ $$
$$ $$
(\identity \lll f ≡ f) (\identity \lll f ≡ f)
\x ×
(f \lll \identity ≡ f) (f \lll \identity ≡ f)
$$ $$
\pause \pause
@ -221,7 +222,7 @@
\frametitle{Pre categories} \frametitle{Pre categories}
\framesubtitle{Propositionality} \framesubtitle{Propositionality}
$$ $$
\isProp\ \left( (\identity \comp f \equiv f) \x (f \comp \identity \equiv f) \right) \isProp\ \left( (\identity \comp f ≡ f) × (f \comp \identity ≡ f) \right)
$$ $$
\pause \pause
\begin{align*} \begin{align*}
@ -247,14 +248,17 @@
\frametitle{Categories} \frametitle{Categories}
\framesubtitle{Univalence} \framesubtitle{Univalence}
\begin{align*} \begin{align*}
\var{IsIdentity} & \defeq \var{IsIdentity} & ≜
\prod_{A\ B \tp \Object} \prod_{f \tp \Arrow\ A\ B} \phi\ f ∏_{A\ B \tp \Object} ∏_{f \tp \Arrow\ A\ B} \phi\ f
%% \\ %% \\
%% & \mathrel{\ } \identity \lll f \equiv f \x f \lll \identity \equiv f %% & \mathrel{\ } \identity \lll f ≡ f × f \lll \identity ≡ f
\end{align*} \end{align*}
where where
$$ $$
\phi\ f \defeq \identity \lll f \equiv f \x f \lll \identity \equiv f \phi\ f ≜ \identity
( \lll f ≡ f )
×
( f \lll \identity ≡ f)
$$ $$
\pause \pause
Let $\approxeq$ denote ismorphism of objects. We can then construct Let $\approxeq$ denote ismorphism of objects. We can then construct
@ -271,29 +275,29 @@
For a category to be univalent we require this to be an equivalence: For a category to be univalent we require this to be an equivalence:
% %
$$ $$
\isEquiv\ (A \equiv B)\ (A \approxeq B)\ \idToIso \isEquiv\ (A ≡ B)\ (A \approxeq B)\ \idToIso
$$ $$
% %
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Categories} \frametitle{Categories}
\framesubtitle{Univalence, cont'd} \framesubtitle{Univalence, cont'd}
$$\isEquiv\ (A \equiv B)\ (A \approxeq B)\ \idToIso$$ $$\isEquiv\ (A ≡ B)\ (A \approxeq B)\ \idToIso$$
\pause% \pause%
$$(A \equiv B) \simeq (A \approxeq B)$$ $$(A ≡ B) ≃ (A \approxeq B)$$
\pause% \pause%
$$(A \equiv B) \cong (A \approxeq B)$$ $$(A ≡ B) ≅ (A \approxeq B)$$
\pause% \pause%
Name the above maps: Name the above maps:
$$\idToIso \tp A ≡ B → A ≊ B$$ $$\idToIso \tp A ≡ B → A ≊ B$$
% %
$$\isoToId \tp (A \approxeq B) \to (A \equiv B)$$ $$\isoToId \tp (A \approxeq B) → (A ≡ B)$$
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Categories} \frametitle{Categories}
\framesubtitle{Propositionality} \framesubtitle{Propositionality}
$$ $$
\isProp\ \IsCategory = \prod_{a, b \tp \IsCategory} a \equiv b \isProp\ \IsCategory = ∏_{a, b \tp \IsCategory} a ≡ b
$$ $$
\pause \pause
So, for So, for
@ -303,8 +307,8 @@
the proof obligation is the pair: the proof obligation is the pair:
% %
\begin{align*} \begin{align*}
p & \tp a.\isPreCategory \equiv b.\isPreCategory \\ p & \tp a.\isPreCategory ≡ b.\isPreCategory \\
& \mathrel{\ } \Path\ (\lambda\; i \to (p\ i).Univalent)\ a.\isPreCategory\ b.\isPreCategory & \mathrel{\ } \Path\ (\lambda\; i → (p\ i).Univalent)\ a.\isPreCategory\ b.\isPreCategory
\end{align*} \end{align*}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
@ -313,17 +317,17 @@
First path given by: First path given by:
$$ $$
p p
\defeq ≜
\var{propIsPreCategory}\ a\ b \var{propIsPreCategory}\ a\ b
\tp \tp
a.\isPreCategory \equiv b.\isPreCategory a.\isPreCategory ≡ b.\isPreCategory
$$ $$
\pause \pause
Use $\lemPropF$ for the latter. Use $\lemPropF$ for the latter.
\pause \pause
% %
Univalence is indexed by an identity proof. So $A \defeq Univalence is indexed by an identity proof. So $A ≜
IsIdentity\ identity$ and $B \defeq \var{Univalent}$. IsIdentity\ identity$ and $B ≜ \var{Univalent}$.
\pause \pause
% %
$$ $$
@ -342,16 +346,16 @@
The isomorphism induces the path The isomorphism induces the path
% %
$$ $$
p \defeq \idToIso\ (\iota, \inv{\iota}) \tp A \equiv B p ≜ \idToIso\ (\iota, \inv{\iota}) \tp A ≡ B
$$ $$
% %
\pause \pause
and consequently an arrow: and consequently an arrow:
% %
$$ $$
p_{\var{dom}} \defeq \congruence\ (λ x → \Arrow\ x\ X)\ p p_{\var{dom}} ≜ \congruence\ (λ x → \Arrow\ x\ X)\ p
\tp \tp
\Arrow\ A\ X \equiv \Arrow\ B\ X \Arrow\ A\ X ≡ \Arrow\ B\ X
$$ $$
% %
\pause \pause
@ -360,8 +364,8 @@
\begin{align} \begin{align}
\label{eq:coeDom} \label{eq:coeDom}
\tag{$\var{coeDom}$} \tag{$\var{coeDom}$}
\prod_{f \tp A \to X} ∏_{f \tp A → X}
\var{coe}\ p_{\var{dom}}\ f \equiv f \lll \inv{\iota} \var{coe}\ p_{\var{dom}}\ f ≡ f \lll \inv{\iota}
\end{align} \end{align}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
@ -369,23 +373,23 @@
\framesubtitle{A theorem, proof} \framesubtitle{A theorem, proof}
\begin{align*} \begin{align*}
\var{coe}\ p_{\var{dom}}\ f \var{coe}\ p_{\var{dom}}\ f
& \equiv f \lll \inv{(\idToIso\ p)} && \text{By path-induction} \\ & ≡ f \lll \inv{(\idToIso\ p)} && \text{By path-induction} \\
& \equiv f \lll \inv{\iota} & ≡ f \lll \inv{\iota}
&& \text{$\idToIso$ and $\isoToId$ are inverses}\\ && \text{$\idToIso$ and $\isoToId$ are inverses}\\
\end{align*} \end{align*}
\pause \pause
% %
Induction will be based at $A$. Let $\widetilde{B}$ and $\widetilde{p} Induction will be based at $A$. Let $\widetilde{B}$ and $\widetilde{p}
\tp A \equiv \widetilde{B}$ be given. \tp A ≡ \widetilde{B}$ be given.
% %
\pause \pause
% %
Define the family: Define the family:
% %
$$ $$
D\ \widetilde{B}\ \widetilde{p} \defeq D\ \widetilde{B}\ \widetilde{p} ≜
\var{coe}\ \widetilde{p}_{\var{dom}}\ f \var{coe}\ \widetilde{p}_{\var{dom}}\ f
\equiv ≡
f \lll \inv{(\idToIso\ \widetilde{p})} f \lll \inv{(\idToIso\ \widetilde{p})}
$$ $$
\pause \pause
@ -393,7 +397,7 @@
The base-case becomes: The base-case becomes:
$$ $$
d \tp D\ A\ \refl = d \tp D\ A\ \refl =
\var{coe}\ \refl_{\var{dom}}\ f \equiv f \lll \inv{(\idToIso\ \refl)} \var{coe}\ \refl_{\var{dom}}\ f ≡ f \lll \inv{(\idToIso\ \refl)}
$$ $$
\end{frame} \end{frame}
\begin{frame} \begin{frame}
@ -401,17 +405,17 @@
\framesubtitle{A theorem, proof, cont'd} \framesubtitle{A theorem, proof, cont'd}
$$ $$
d \tp d \tp
\var{coe}\ \refl_{\var{dom}}\ f \equiv f \lll \inv{(\idToIso\ \refl)} \var{coe}\ \refl_{\var{dom}}\ f ≡ f \lll \inv{(\idToIso\ \refl)}
$$ $$
\pause \pause
\begin{align*} \begin{align*}
\var{coe}\ \refl^*\ f \var{coe}\ \refl^*\ f
& \equiv f & ≡ f
&& \text{$\refl$ is a neutral element for $\var{coe}$}\\ && \text{$\refl$ is a neutral element for $\var{coe}$}\\
& \equiv f \lll \identity \\ & ≡ f \lll \identity \\
& \equiv f \lll \var{subst}\ \refl\ \identity & ≡ f \lll \var{subst}\ \refl\ \identity
&& \text{$\refl$ is a neutral element for $\var{subst}$}\\ && \text{$\refl$ is a neutral element for $\var{subst}$}\\
& \equiv f \lll \inv{(\idToIso\ \refl)} & ≡ f \lll \inv{(\idToIso\ \refl)}
&& \text{By definition of $\idToIso$}\\ && \text{By definition of $\idToIso$}\\
\end{align*} \end{align*}
\pause \pause
@ -429,7 +433,7 @@
\pause \pause
Objects: Objects:
$$ $$
\sum_{X \tp Object} \Arrow\ X\ \pairA × \Arrow\ X\ \pairB ∑_{X \tp Object} \Arrow\ X\ \pairA × \Arrow\ X\ \pairB
$$ $$
\pause \pause
% %
@ -437,9 +441,9 @@
$B ,\ b_{\pairA} ,\ b_{\pairB}$: $B ,\ b_{\pairA} ,\ b_{\pairB}$:
% %
$$ $$
\sum_{f \tp \Arrow\ A\ B} ∑_{f \tp \Arrow\ A\ B}
b_{\pairA} \lll f \equiv a_{\pairA} \x b_{\pairA} \lll f ≡ a_{\pairA} ×
b_{\pairB} \lll f \equiv a_{\pairB} b_{\pairB} \lll f ≡ a_{\pairB}
$$ $$
\end{frame} \end{frame}
\begin{frame} \begin{frame}
@ -447,25 +451,25 @@
\framesubtitle{Univalence} \framesubtitle{Univalence}
\begin{align*} \begin{align*}
\label{eq:univ-0} \label{eq:univ-0}
(X , x_{\mathcal{A}} , x_{\mathcal{B}}) ≡ (Y , y_{\mathcal{A}} , y_{\mathcal{B}}) (X , x_{𝒜} , x_{ℬ}) ≡ (Y , y_{𝒜} , y_{ℬ})
\end{align*} \end{align*}
\begin{align*} \begin{align*}
\label{eq:univ-1} \label{eq:univ-1}
\begin{split} \begin{split}
p \tp & X \equiv Y \\ p \tp & X ≡ Y \\
& \Path\ (λ i → \Arrow\ (p\ i)\ \mathcal{A})\ x_{\mathcal{A}}\ y_{\mathcal{A}} \\ & \Path\ (λ i → \Arrow\ (p\ i)\ 𝒜)\ x_{𝒜}\ y_{𝒜} \\
& \Path\ (λ i → \Arrow\ (p\ i)\ \mathcal{B})\ x_{\mathcal{B}}\ y_{\mathcal{B}} & \Path\ (λ i → \Arrow\ (p\ i)\ ℬ)\ x_{ℬ}\ y_{ℬ}
\end{split} \end{split}
\end{align*} \end{align*}
\begin{align*} \begin{align*}
\begin{split} \begin{split}
\var{iso} \tp & X \approxeq Y \\ \var{iso} \tp & X \approxeq Y \\
& \Path\ (λ i → \Arrow\ (\widetilde{p}\ i)\ \mathcal{A})\ x_{\mathcal{A}}\ y_{\mathcal{A}} \\ & \Path\ (λ i → \Arrow\ (\widetilde{p}\ i)\ 𝒜)\ x_{𝒜}\ y_{𝒜} \\
& \Path\ (λ i → \Arrow\ (\widetilde{p}\ i)\ \mathcal{B})\ x_{\mathcal{B}}\ y_{\mathcal{B}} & \Path\ (λ i → \Arrow\ (\widetilde{p}\ i)\ ℬ)\ x_{ℬ}\ y_{ℬ}
\end{split} \end{split}
\end{align*} \end{align*}
\begin{align*} \begin{align*}
(X , x_{\mathcal{A}} , x_{\mathcal{B}}) ≊ (Y , y_{\mathcal{A}} , y_{\mathcal{B}}) (X , x_{𝒜} , x_{ℬ}) ≊ (Y , y_{𝒜} , y_{ℬ})
\end{align*} \end{align*}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
@ -475,12 +479,12 @@
\begin{align*} \begin{align*}
%% (f, \inv{f}, \var{inv}_f, \var{inv}_{\inv{f}}) %% (f, \inv{f}, \var{inv}_f, \var{inv}_{\inv{f}})
%% \tp %% \tp
(X, x_{\mathcal{A}}, x_{\mathcal{B}}) \approxeq (Y, y_{\mathcal{A}}, y_{\mathcal{B}}) (X, x_{𝒜}, x_{ℬ}) \approxeq (Y, y_{𝒜}, y_{ℬ})
\to \to
\begin{split} \begin{split}
\var{iso} \tp & X \approxeq Y \\ \var{iso} \tp & X \approxeq Y \\
& \Path\ (λ i → \Arrow\ (\widetilde{p}\ i)\ \mathcal{A})\ x_{\mathcal{A}}\ y_{\mathcal{A}} \\ & \Path\ (λ i → \Arrow\ (\widetilde{p}\ i)\ 𝒜)\ x_{𝒜}\ y_{𝒜} \\
& \Path\ (λ i → \Arrow\ (\widetilde{p}\ i)\ \mathcal{B})\ x_{\mathcal{B}}\ y_{\mathcal{B}} & \Path\ (λ i → \Arrow\ (\widetilde{p}\ i)\ ℬ)\ x_{ℬ}\ y_{ℬ}
\end{split} \end{split}
\end{align*} \end{align*}
\pause \pause
@ -503,8 +507,8 @@
% %
\begin{align*} \begin{align*}
\begin{split} \begin{split}
\widetilde{p} & \tp X \equiv Y \\ \widetilde{p} & \tp X ≡ Y \\
\widetilde{p}_{\mathcal{A}} & \tp \Arrow\ X\ \mathcal{A} \equiv \Arrow\ Y\ \mathcal{A} \\ \widetilde{p}_{𝒜} & \tp \Arrow\ X\ 𝒜 ≡ \Arrow\ Y\ 𝒜 \\
\end{split} \end{split}
\end{align*} \end{align*}
% %
@ -517,7 +521,7 @@
\begin{align*} \begin{align*}
\begin{split} \begin{split}
\label{eq:product-paths} \label{eq:product-paths}
& \Path\ (λ i → \widetilde{p}_{\mathcal{A}}\ i)\ x_{\mathcal{A}}\ y_{\mathcal{A}} & \Path\ (λ i → \widetilde{p}_{𝒜}\ i)\ x_{𝒜}\ y_{𝒜}
\end{split} \end{split}
\end{align*} \end{align*}
\pause \pause
@ -525,9 +529,9 @@
This is achieved with the following lemma: This is achieved with the following lemma:
% %
\begin{align*} \begin{align*}
\prod_{q \tp A \equiv B} \var{coe}\ q\ x_{\mathcal{A}} ≡ y_{\mathcal{A}} ∏_{q \tp A ≡ B} \var{coe}\ q\ x_{𝒜} ≡ y_{𝒜}
→ →
\Path\ (λ i → q\ i)\ x_{\mathcal{A}}\ y_{\mathcal{A}} \Path\ (λ i → q\ i)\ x_{𝒜}\ y_{𝒜}
\end{align*} \end{align*}
% %
Which is used without proof.\pause Which is used without proof.\pause
@ -535,15 +539,15 @@
So the construction reduces to: So the construction reduces to:
% %
\begin{align*} \begin{align*}
\var{coe}\ \widetilde{p}_{\mathcal{A}}\ x_{\mathcal{A}} ≡ y_{\mathcal{A}} \var{coe}\ \widetilde{p}_{𝒜}\ x_{𝒜} ≡ y_{𝒜}
\end{align*}% \end{align*}%
\pause% \pause%
This is proven with: This is proven with:
% %
\begin{align*} \begin{align*}
\var{coe}\ \widetilde{p}_{\mathcal{A}}\ x_{\mathcal{A}} \var{coe}\ \widetilde{p}_{𝒜}\ x_{𝒜}
& ≡ x_{\mathcal{A}} \lll \fst\ \inv{f} && \text{\ref{eq:coeDom}} \\ & ≡ x_{𝒜} \lll \fst\ \inv{f} && \text{\ref{eq:coeDom}} \\
& ≡ y_{\mathcal{A}} && \text{Property of span category} & ≡ y_{𝒜} && \text{Property of span category}
\end{align*} \end{align*}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
@ -556,13 +560,13 @@
% %
We can show: We can show:
\begin{align*} \begin{align*}
\var{Terminal} ≃ \var{Product}\ ℂ\ \mathcal{A}\ \mathcal{B} \var{Terminal} ≃ \var{Product}\ ℂ\ 𝒜\ ℬ
\end{align*} \end{align*}
\pause \pause
And since equivalences preserve homotopy levels we get: And since equivalences preserve homotopy levels we get:
% %
$$ $$
\isProp\ \left(\var{Product}\ \bC\ \mathcal{A}\ \mathcal{B}\right) \isProp\ \left(\var{Product}\ \bC\ 𝒜\ ℬ\right)
$$ $$
\end{frame} \end{frame}
\begin{frame} \begin{frame}
@ -595,7 +599,7 @@
% %
\begin{align*} \begin{align*}
\omapR & \tp \Object → \Object \\ \omapR & \tp \Object → \Object \\
\pure & \tp % \prod_{X \tp Object} \pure & \tp % ∏_{X \tp Object}
\Arrow\ X\ (\omapR\ X) \\ \Arrow\ X\ (\omapR\ X) \\
\bind & \tp \bind & \tp
\Arrow\ X\ (\omapR\ Y) \Arrow\ X\ (\omapR\ Y)
@ -610,7 +614,7 @@
\Arrow\ B\ (\omapR\ C) \Arrow\ B\ (\omapR\ C)
→ →
\Arrow\ A\ (\omapR\ C) \\ \Arrow\ A\ (\omapR\ C) \\
f \fish g & \defeq f \rrr (\bind\ g) f \fish g & ≜ f \rrr (\bind\ g)
\end{align*} \end{align*}
\pause \pause
% %
@ -629,20 +633,20 @@
In the monoidal formulation we can define $\bind$: In the monoidal formulation we can define $\bind$:
% %
$$ $$
\bind\ f \defeq \join \lll \fmap\ f \bind\ f ≜ \join \lll \fmap\ f
$$ $$
\pause \pause
% %
And likewise in the Kleisli formulation we can define $\join$: And likewise in the Kleisli formulation we can define $\join$:
% %
$$ $$
\join \defeq \bind\ \identity \join ≜ \bind\ \identity
$$ $$
\pause \pause
The laws are logically equivalent. So we get: The laws are logically equivalent. So we get:
% %
$$ $$
\var{Monoidal} \simeq \var{Kleisli} \var{Monoidal} ≃ \var{Kleisli}
$$ $$
% %
\end{frame} \end{frame}