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doc/abstract.tex
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@ -1,21 +1,22 @@
\chapter*{Abstract}
The usual notion of propositional equality in intensional type-theory
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
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
programming language Agda has been extended with capabilities for
working in such a cubical setting. This thesis will explore the
usefulness of this extension in the context of category theory.
The thesis will motivate and explain why propositional equality in
cubical Agda is more expressive than in standard Agda. Alternative
approaches to Cubical Agda will be presented and their pros and cons
will be explained. It will emphasize why it is useful to have a
constructive interpretation of univalence. As an example of this two
formulations of monads will be presented: Namely monads in the
monoidal form and monads in the Kleisli form.
The thesis will motivate the need for univalence and explain why
propositional equality in cubical Agda is more expressive than in
standard Agda. Alternative approaches to Cubical Agda will be
presented and their pros and cons will be explained. As an example of
the application of univalence two formulations of monads will be
presented: Namely monads in the monoidal form and monads in the
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
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}
In two places in my formalization was the computational behaviours of
functional extensionality used. The reduction behaviour can be

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doc/chalmerstitle.sty
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@ -46,7 +46,8 @@
{\Huge\@title}\\[.5cm]
{\Large A formalization of category theory in Cubical Agda}\\[6cm]
\begin{center}
\includegraphics[width=\linewidth,keepaspectratio]{isomorphism.png}
\includegraphics[width=\linewidth,keepaspectratio]{isomorphism.pdf}
%% \includepdf{isomorphism.pdf}
\end{center}
% Cover text
\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
admissible. Cubical Agda is more expressive, but there are certain
issues that arise that are not present in standard Agda. For one thing
ITT and standard Agda enjoys Uniqueness of Identity Proofs (UIP). This
is not the case in Cubical Agda. In stead there exists a hierarchy of
types with increasing \nomen{homotopical structure}{homotopy levels}.
It turns out to be useful to built the formalization with this
hierarchy in mind as it can simplify proofs considerably. Another
issue one must overcome in Cubical Agda is when a type has a field
whose type depends on a previous field. In this case paths between
such types will be heterogeneous paths. This problem is related to
Cubical Agda not having the K-rule. In practice it turns out to be
considerably more difficult to work heterogeneous paths than with
homogeneous paths. The thesis demonstrated some techniques to overcome
these difficulties, such as based path-induction.
Agda enjoys Uniqueness of Identity Proofs (UIP) though a flag exists
to turn this off, which is the case in Cubical Agda. In stead
there exists a hierarchy of types with increasing \nomen{homotopical
structure}{homotopy levels}. It turns out to be useful to built the
formalization with this hierarchy in mind as it can simplify proofs
considerably. Another issue one must overcome in Cubical Agda is when
a type has a field whose type depends on a previous field. In this
case paths between such types will be heterogeneous paths. This
problem is related to Cubical Agda not having the K-rule. In practice
it turns out to be considerably more difficult to work heterogeneous
paths than with homogeneous paths. The thesis demonstrated some
techniques to overcome these difficulties, such as based
path-induction.
This thesis formalized some of the core concepts from category theory including;
categories, functors, products, exponentials, Cartesian closed categories,
natural transformations, the yoneda embedding, monads and more. Category theory
is an interesting case-study for the application of Cubical Agda for two reasons
in particular: Because category theory is the study of abstract algebra of
functions, meaning that functional extensionality is particularly relevant.
Another reason is that in category theory it is commonplace to identify
isomorphic structures and univalence allows for making this notion precise. This
thesis also demonstrated another technique that is common in category theory;
namely to define categories to prove properties of other structures.
Specifically a category was defined to demonstrate that any two product objects
in a category are isomorphic. Furthermore the thesis showed two formulations of
monads and proved that they indeed are equivalent: Namely monads in the
monoidal- and Kleisli- form. The monoidal formulation is more typical to
category theoretic formulations and the Kleisli formulation will be more
familiar to functional programmers. In the formulation we also saw how paths can
be used to extract functions. A path between 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.
This thesis formalized some of the core concepts from category theory
including; categories, functors, products, exponentials, Cartesian
closed categories, natural transformations, the yoneda embedding,
monads and more. Category theory is an interesting case-study for the
application of Cubical Agda for two reasons in particular: Because
category theory is the study of abstract algebra of functions, meaning
that functional extensionality is particularly relevant. Another
reason is that in category theory it is commonplace to identify
isomorphic structures and univalence allows for making this notion
precise. This thesis also demonstrated another technique that is
common in category theory; namely to define categories to prove
properties of other structures. Specifically a category was defined
to demonstrate that any two product objects in a category are
isomorphic. Furthermore the thesis showed two formulations of monads
and proved that they indeed are equivalent: Namely monads in the
monoidal- and Kleisli- form. The monoidal formulation is more typical
to category theoretic formulations and the Kleisli formulation will be
more familiar to functional programmers. It would have been very
difficult to make a similar proof with setoids. In the formulation we
also saw how paths can be used to extract functions. A path between
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
%% overcome with cubical

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

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
provided as a strategy to simplify some otherwise very complicated
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
can help simplify these goals considerably. The lack of the J-rule has
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.
\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}
\label{sec:compiling-cubical-agda}
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
working on proving the initiality conjecture\TODO{Citation}. He will
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}
\label{ch:implementation}
This implementaiton, including this report, is available as open
source software at:
The source code for this formalization, including this report, is
available as open source software at:
%
\begin{center}
\gitlink
@ -17,7 +17,7 @@ link\footnote{%
\begin{center}
\doclink
\end{center}
This implementation formalizes the following concepts:
The concepts formalized in this development are:
%
\begin{center}
\begin{tabular}{ l l }
@ -41,6 +41,7 @@ Span category & \sourcelink{Cat.Categories.Span} \\
\end{tabular}
\end{center}
%
\begin{samepage}
Furthermore the following items have been partly formalized:
%
\begin{center}
@ -54,70 +55,75 @@ Free category & \sourcelink{Cat.Categories.Free} \\
Monoids & \sourcelink{Cat.Category.Monoid} \\
\end{tabular}
\end{center}
\end{samepage}%
%
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
demonstrate some of the techniques employed in this formalization and
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
in Cubical Agda. This chapter will focus on the definition of
\emph{categories}, \emph{equivalences}, the \emph{opposite category},
the \emph{category of sets}, \emph{products}, the \emph{span category}
and the two formulations of \emph{monads}.
One such technique that is pervasive to this formalization is the idea of
distinguishing types with more or less homotopical structure. To do this I have
followed the following design-principle: I have split concepts up into things
that represent ``data'' and ``laws'' about this data. The idea is that we can
provide a proof that the laws are mere propositions. As an example a category is
defined to have two members: `raw` which is a collection of the data and
`isCategory` which asserts some laws about that data.
One technique employed throughout this formalization is the idea of
distinguishing types with more or less homotopical structure. To do
this I have followed the following design-principle: I have split
concepts up into things that represent \emph{data} and \emph{laws}
about this data. The idea is that we can provide a proof that the laws
are mere propositions. As an example a category is defined to have two
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'',
where one can reason about two categories by simply focusing on the data. This
is achieved by creating a function embodying the ``equality principle'' for a
given type.
This allows me to reason about things in a more ``standard
mathematical way'', where one can reason about two categories by
simply focusing on the data. This is achieved by creating a function
embodying the equality principle for a given type.
For the rest of this chapter I will present some of these results. For
didactic reasons no source-code has been included in this chapter. To
see the formal 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}
\label{sec:categories}
The data for a category consist of a type for the sort of objects; a type for
the sort of arrows; an identity arrow and a composition operation for arrows.
Another record encapsulates some laws about this data: associativity of
composition, identity law for the identity morphism. These are standard
constituents of a category and can be found in typical mathematical expositions
on the topic. We, however, impose one further requirement on what it means to be
a category, namely that the type of arrows form a set.
The data for a category consist of a type for the sort of objects; a
type for the sort of arrows; an identity arrow and a composition
operation for arrows. Another record encapsulates some laws about
this data: associativity of composition, identity law for the identity
morphism. These are standard constituents of a category and can be
found in typical mathematical expositions on the topic. We shall
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
this requirement. This would lead to the notion of higher categories (\cite[p.
307]{hott-2013}). For the purpose of this project, however, this report will
restrict itself to 1-categories\index{1-category}. Generalizing this work to
higher categories would be a very natural possible extension of this work.
Such categories are called \nomen{1-categories}{1-category}. It is
possible to relax this requirement. This would lead to the notion of
higher categories (\cite[p. 307]{hott-2013}). For the purpose of this
thesis however, this report will restrict itself to
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
\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
definitions: If we let $p$ be a witness to the identity law, which formally is:
%
\begin{equation}
\label{eq:identity}
\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)
\end{equation}
%
Then we can construct the identity isomorphism $\idIso \tp \identity,
\identity, p \tp A \approxeq A$ for any object $A$. Here $\approxeq$ denotes
isomorphism on objects (whereas $\cong$ denotes isomorphism of types). This will
be elaborated further on in sections \S\ref{sec:equiv} and
\S\ref{sec:univalence}. Moreover, due to substitution for paths we can construct
an isomorphism from \emph{any} path:
Then we can construct the identity isomorphism $\idIso \tp (\identity,
\identity, p) \tp A \approxeq A$ for any object $A$. Here $\approxeq$
denotes isomorphism on objects (whereas $\cong$ denotes isomorphism on
types). This will be elaborated further on in sections
\S\ref{sec:equiv} and \S\ref{sec:univalence}. Moreover due to
substitution for paths we can construct an isomorphism from \emph{any}
path:
%
\begin{equation}
\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)
\end{equation}
%
However the two are logically equivalent: One can construct the latter from the
former simply by ``forgetting'' that $\idToIso$ plays the role of the
equivalence. The other direction is more involved and will be discussed in
section \S\ref{sec:univalence}.
However the two are logically equivalent: One can construct the latter
from the former simply by ``forgetting'' that $\idToIso$ plays the
role of the equivalence. The other direction is more involved and will
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}
\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
quantified.
With all this in place it is now possible to prove that all the laws are indeed
mere propositions. Most of the proofs simply use the fact that the type of
arrows are sets. This is because most of the laws are a collection of equations
between arrows in the category. And since such a proof does not have any content
exactly because the type of arrows form a set, two witnesses must be the same.
All the proofs are really quite mechanical. Lets have a look at one of them.
Proving that \ref{eq:identity} is a mere proposition:
With all this in place it is now possible to prove that all the laws
are indeed mere propositions. Most of the proofs simply use the fact
that the type of arrows are sets. This is because most of the laws are
a collection of equations between arrows in the category. And since
such a proof does not have any content exactly because the type of
arrows form a set, two witnesses must be the same. All the proofs are
really quite mechanical. Let us have a look at one of them: Proving
that \ref{eq:identity} is a mere proposition:
%
\begin{equation}
\isProp\ \var{IsIdentity}
\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
\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)
\end{align*}
%
So the proof goes like this: We `eliminate' the 3 function abstractions by
applying $\propPi$ three times. So our proof obligation becomes:
The proof goes like this: We `eliminate' the 3 function abstractions
by applying $\propPi$ three times. So our proof obligation becomes:
%
$$
\isProp\ \left( \left( \id \comp f \equiv f \right) \x \left( f \comp \id \equiv f \right) \right)
$$
%
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
set.
This example illustrates nicely how we can use these combinators to reason about
`canonical' types like $\sum$ and $\prod$. Similar combinators can be defined at
the other homotopic levels. These combinators are however not applicable in
situations where we want to reason about other types - e.g.\ types we have
defined ourselves. For instance, after we have proven that all the projections
of pre categories are propositions, then we would like to bundle this up to show
that the type of pre categories is also a proposition. Formally:
This example illustrates nicely how we can use these combinators to
reason about `canonical' types like $\sum$ and $\prod$. Similar
combinators can be defined at the other homotopic levels. These
combinators are however not applicable in situations where we want to
reason about other types e.g.\ types we have defined ourselves. For
instance, after we have proven that all the projections of
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}
\label{eq:propIsPreCategory}
@ -235,19 +244,20 @@ Where The definition of $\IsPreCategory$ is the triple:
\var{arrowsAreSets} & \tp \var{ArrowsAreSets}
\end{align*}
%
Each corresponding to the first three laws for categories. Note that since
$\IsPreCategory$ is not formulated with a chain of sigma-types we wont have any
combinators available to help us here. In stead the paths must be used directly.
Each corresponding to the first three laws for categories. Note that
since $\IsPreCategory$ is not formulated with a chain of sigma-types
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
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
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
@ -261,8 +271,8 @@ $$
a.\isIdentity \equiv b.\isIdentity
$$
%
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 $\IsPreCategory$
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 $\IsPreCategory$
by using the fact that all the projections are propositions to generate paths
between all projections. Once we have such a path e.g.\ $p \tp a.\isIdentity
\equiv b.\isIdentity$ 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.
For instance, when we want to show that $\IsCategory$ is a mere
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
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:
%
$$
@ -463,7 +473,7 @@ equalities and isomorphisms (on arrows). It is worthwhile to dwell on this for a
few seconds. This type looks very similar to univalence for types and is
therefore perhaps a bit more intuitive to grasp the implications of. Of course
univalence for types (which is a proposition -- i.e.\ provable) does not imply
univalence of all pre category since morphisms in a category are not regular
univalence of all pre-category since morphisms in a category are not regular
functions -- in stead they can be thought of as a generalization hereof. The univalence criterion therefore is simply a way of restricting arrows
to behave similarly to maps.
@ -581,7 +591,7 @@ over-bar. So e.g.\ $\idToIso$ is a function in the underlying category and the
corresponding thing is denoted $\wideoverbar{\idToIso}$ in the opposite
category.
Showing that this forms a pre category is rather straightforward.
Showing that this forms a pre-category is rather straightforward.
%
$$
h \rrr (g \rrr f) \equiv h \rrr g \rrr f
@ -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 $A\;B \tp
\Object$ 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
that we can prove this category univalent by giving an inverse to
$\wideoverbar{\idToIso} \tp (A \equiv B) \to (A \wideoverbar{\approxeq} B)$.
@ -624,7 +634,7 @@ follows:
\wideoverbar{\isoToId} \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
\isoToId \comp \shufflef \comp \inv{\shufflef} \comp \idToIso
@ -641,7 +651,7 @@ follows:
The other direction is analogous.
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
first components are equal, but this holds judgmentally.
@ -673,7 +683,7 @@ homotopic sets. This is encapsulated in Agda with the following type:
%
$$\Set \defeq \sum_{A \tp \MCU} \isSet\ A$$
%
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
Agda types can have higher homotopic structure.
@ -681,7 +691,7 @@ Univalence does not follow immediately from univalence for types:
%
$$(A \equiv B) \simeq (A \simeq B)$$
%
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
$\var{hA}\;\var{hB} \tp \Set$. In stead I show that this category
satisfies:
@ -956,7 +966,7 @@ underlying category. Given that $p$ is the chosen proof of
%
\begin{align}
\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}
%
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
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}
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}
\begin{split}
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{B})\ x_{\mathcal{B}}\ y_{\mathcal{B}}
& \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}}
\end{split}
%% \end{split}
\end{align}
@ -1045,8 +1055,8 @@ isomorphism, and create a path from this:
\label{eq:univ-2}
\begin{split}
\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{B})\ x_{\mathcal{B}}\ y_{\mathcal{B}}
& \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}}
\end{split}
\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
will show two corollaries of \ref{eq:coeCod}: For an isomorphism $(\iota,
\inv{\iota}, \var{inv}) \tp A \cong B$, arrows $f \tp \Arrow\ A\ X$, $g \tp
\Arrow\ B\ X$ 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 $p_{\var{dom}} \tp \Arrow\ A\ X \equiv
\Arrow\ B\ X$ is a path induced by $\var{iso}$, we have the following two
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
%
\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*}
%
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{split}
\label{eq:product-paths}
& \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{A}}\ i)\ x_{\mathcal{A}}\ y_{\mathcal{A}}\\
& \Path\ (λ \; i → \widetilde{p}_{\mathcal{B}}\ i)\ x_{\mathcal{B}}\ y_{\mathcal{B}}
\end{split}
\end{align}
%
@ -1172,7 +1182,7 @@ This is achieved with the following lemma:
%
\begin{align}
\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}
%
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
gory details.
%
\subsection{Propositionality of products}
\subsection{Products are propositions}
%
Now that we have constructed the span category\index{span category} I
will demonstrate how to use this to prove that products are
propositional. I will do this by showing that terminal objects in this
category are equivalent to products:
propositions. On the face of it this may seem surprising. 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}
\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:
%
\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 \\
& \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}
\end{alignat}
%
@ -1283,7 +1298,7 @@ preserve homotopic levels along with \ref{eq:termProp} we get our final result.
That in any category:
%
\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}
%
\section{Functors and natural transformations}

225
doc/introduction.tex
View file

@ -6,22 +6,27 @@ of equality: \nomenindex{judgmental equality} and
\nomenindex{propositional equality}. Judgmental equality is a property
of the type system. Judgmental equality on the other hand is usually
defined \emph{within} the system. When introducing definitions this
report will use the notation $\defeq$. Judgmental equalities written
$=$. For propositional equalities the notation $\equiv$ is used.
report will use the symbol $\defeq$. Judgmental equalities will be
denoted with $=$ and for propositional equalities the notation
$\equiv$ is used.
For judgmental equality there are some properties that it must
satisfy. \nomenindex{sound}, enjoy \nomenindex{canonicity} and be a
\nomen{congruence relation}. Soundness means that things judged to be
equal are equal with respects to the model of the theory (the meta
theory). It must be a congruence relation because otherwise the
relation certainly does not adhere to our notion of equality. One
would be able to conclude things like: $x \equiv y \rightarrow f\ x
\nequiv f\ y$. Canonicity means that any 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.
The rules of judgmental equality are related with $β$- and
$η$-reduction which gives a notion of computation in a given type
theory.
%
There are some properties that one usually want judgmental equality to
satisfy. It must be \nomenindex{sound}, enjoy \nomenindex{canonicity}
and be a \nomenindex{congruence relation}. Soundness means that things
judged to be equal are equal with respects to the \nomenindex{model}
of the theory or the \emph{meta theory}. It must be a congruence
relation because otherwise the relation certainly does not adhere to
our notion of equality. One would be able to conclude things like: $x
\equiv y \rightarrow f\ x \nequiv f\ y$. Canonicity means that any
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
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}).
There are two flavors of type-theory. \emph{Intensional-} and
\emph{extensional-} type theory. Identity types in extensional type
theory are required to be \nomen{propositions}{proposition}. That is,
a type with at most one inhabitant. In extensional type thoery the
principle of reflection
\emph{extensional-} type theory (ITT and ETT respectively). Identity
types in extensional type theory are required to be
\nomen{propositions}{proposition}. That is, a type with at most one
inhabitant. In extensional type theory the principle of reflection
%
$$a ≡ b → a = b$$
%
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
theory. Pattern-matching in regular Agda let's one prove
\nomenindex{axiom K}. Axiom K states that any two identity proofs are
propositionally identical.
Agda which at a glance can be thought of as a version of intensional
type theory. Pattern-matching in regular Agda lets one prove
\nomenindex{Uniqueness of Identity Proofs} (UIP). UIP states that any
two identity proofs are propositionally identical.
The usual notion of propositional equality in \nomenindex{Intensional
Type Theory} (ITT) is quite restrictive. In the next section a few
motivating examples will highlight this. There exist techniques to
circumvent these problems, as we shall see. This thesis will explore
an extension to Agda that redefines the notion of propositional
equality and as such is an alternative to these other techniques. The
extension is called cubical Agda. Cubical Agda drops Axiom K as this
does not permit \nomenindex{functional extensionality} and
The usual notion of propositional equality in ITT is quite
restrictive. In the next section a few motivating examples will
highlight this. There exist techniques to circumvent these problems,
as we shall see. This thesis will explore an extension to Agda that
redefines the notion of propositional equality and as such is an
alternative to these other techniques. The extension is called cubical
Agda. Cubical Agda drops UIP as this does not permit
\nomenindex{functional extensionality} and
\nomenindex{univalence}. What makes this extension particularly
interesting is that it gives a \emph{constructive} interpretation of
univalence. What all this means will be elaborated in the following
@ -69,22 +74,15 @@ some limitations inherent in ITT and -- by extension -- Agda.
\label{sec:functional-extensionality}%
Consider the functions:
%
\begin{multicols}{2}
\noindent%
\begin{equation*}%
f \defeq \lambda\ (n \tp \bN) \to (0 + n \tp \bN)
\end{equation*}%
\begin{equation*}%
g \defeq \lambda\ (n \tp \bN) \to (n + 0 \tp \bN)
\end{equation*}%
\end{multicols}%
\begin{align*}%
\var{zeroLeft} & \defeq \lambda\; (n \tp \bN) \to (0 + n \tp \bN) \\
\var{zeroRight} & \defeq \lambda\; (n \tp \bN) \to (n + 0 \tp \bN)
\end{align*}%
%
The term $n + 0$ is
\nomenindex{definitionally} equal to $n$, which we
write as $n + 0 = n$. This is also called
\nomenindex{judgmental equality}.
We call it definitional equality because the \emph{equality} arises
from the \emph{definition} of $+$ which is:
The term $n + 0$ is \nomenindex{definitionally} equal to $n$, which we
write as $n + 0 = n$. This is also called \nomenindex{judgmental
equality}. We call it definitional equality because the
\emph{equality} arises from the \emph{definition} of $+$ which is:
%
\begin{align*}
+ & \tp \bN \to \bN \to \bN \\
@ -92,51 +90,37 @@ from the \emph{definition} of $+$ which is:
n + (\suc{m}) & \defeq \suc{(n + m)}
\end{align*}
%
Note that $0 + n$ is \emph{not} definitionally equal to $n$. $0 + n$
is in normal form. I.e.; there is no rule for $+$ whose left hand side
matches this expression. We \emph{do}, however, have that they are
\nomen{propositionally}{propositional equality} equal, which we write
as $n + 0 \equiv n$. Propositional equality means that there is a
proof that exhibits this relation. Since equality is a transitive
relation we have that $n + 0 \equiv 0 + n$.
Unfortunately we don't have $f \equiv g$. There is no way to construct
a proof asserting the obvious equivalence of $f$ and $g$. Actually
showing this is outside the scope of this text. Essentially it would
involve giving a model for our type theory that validates all our
axioms but where $f \equiv g$ is not true. We cannot show that they
are equal, even though we can prove them equal for all points. For
functions this is exactly the notion of equality that we are
interested in: Functions are considered equal when they are equal for
all inputs. This is called \nomenindex{point wise equality}, where the
\emph{points} of a function refer to its arguments.
%% In the context of category theory functional extensionality is e.g.
%% needed to show that representable functors are indeed functors. The
%% representable functor is defined for a fixed category $\bC$ and an
%% object $X \in \bC$. It's map on objects is defined thus:
%% %
%% \begin{align*}
%% \lambda\ A \to \Arrow\ X\ A
%% \end{align*}
%% %
%% That is, it maps objects to arrows. So, it's map on arrows must map an arrow $\Arrow\ A\ B$ to an
%% The map on objects is defined thus:
%% %
%% \begin{align*}
%% \lambda f \to
%% \end{align*}
%% %
%% The proof obligation that this satisfies the identity law of functors
%% ($\fmap\ \idFun \equiv \idFun$) thus becomes:
%% %
%% \begin{align*}
%% \Hom(A, \idFun_{\bX}) = (\lambda\ g \to \idFun \comp g) \equiv \idFun_{\Sets}
%% \end{align*}
%% %
%% One needs functional extensionality to ``go under'' the function arrow and apply
%% the (left) identity law of the underlying category to prove $\idFun \comp g
%% \equiv g$ and thus close the goal.
Note that $0 + n$ is \emph{not} definitionally equal to $n$. This is
because $0 + n$ is in normal form. I.e.\ there is no rule for $+$
whose left hand side matches this expression. We do however have that
they are \nomen{propositionally}{propositional equality} equal, which
we write as $n \equiv n + 0$. Propositional equality means that there
is a proof that exhibits this relation. We can do induction over $n$
to prove this:
%
\begin{align}
\label{eq:zrn}
\begin{split}
\var{zrn}\ & \tp ∀ n → n ≡ \var{zeroRight}\ n \\
\var{zrn}\ \var{zero} & \defeq \var{refl} \\
\var{zrn}\ (\var{suc}\ n) & \defeq \var{cong}\ \var{suc}\ (\var{zrn}\ n)
\end{split}
\end{align}
%
This show that zero is a right neutral element hence the name $\var{zrn}$.
Since equality is a transitive relation we have that $\forall n \to
\var{zeroLeft}\ n \equiv \var{zeroRight}\ n$. Unfortunately we don't
have $\var{zeroLeft} \equiv \var{zeroRight}$. There is no way to
construct a proof asserting the obvious equivalence of
$\var{zeroLeft}$ and $\var{zeroRight}$. Actually showing this is
outside the scope of this text. Essentially it would involve giving a
model for our type theory that validates all our axioms but where
$\var{zeroLeft} \equiv \var{zeroRight}$ is not true. We cannot show
that they are equal even though we can prove them equal for all
points. For functions this is exactly the notion of equality that we
are interested in: Functions are considered equal when they are equal
for all inputs. This is called \nomenindex{pointwise equality}, where
the \emph{points} of a function refer to its arguments.
%
\subsection{Equality of isomorphic types}
%
@ -155,7 +139,6 @@ without thinking twice. Unfortunately such an identification can not
be performed in ITT.
More specifically what we are interested in is a way of identifying
\nomenindex{equivalent} types. I will return to the definition of
equivalence later in section \S\ref{sec:equiv}, but for now it is
sufficient to think of an equivalence as a one-to-one correspondence.
@ -165,27 +148,30 @@ The principle of univalence says that:
$$\mathit{univalence} \tp (A \simeq B) \simeq (A \equiv B)$$
%
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}
%
The above examples serve to illustrate a limitation of ITT. One case where these
limitations are particularly prohibitive is in the study of Category Theory. At
a glance category theory can be described as ``the mathematical study of
(abstract) algebras of functions'' (\cite{awodey-2006}). By that token
functional extensionality is particularly useful for formulating Category
Theory. In Category theory it is also common to identify isomorphic structures
and univalence gives us a way to make this notion precise. In fact we can
formulate this requirement within our formulation of categories by requiring the
\emph{categories} themselves to be univalent as we shall see.
The above examples serve to illustrate a limitation of ITT. One case
where these limitations are particularly prohibitive is in the study
of Category Theory. At a glance category theory can be described as
``the mathematical study of (abstract) algebras of functions''
(\cite{awodey-2006}). By that token functional extensionality is
particularly useful for formulating Category Theory. In Category
theory it is also commonplace to identify isomorphic structures and
univalence gives us a way to make this notion precise. In fact we can
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}
\label{sec:context}
%
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
question on Math Overflow: \cite{mo-formalizations}. Notably these
implementations of category theory in Agda:
are a multitude of these available online. Notably:
%
\begin{itemize}
\item
@ -217,12 +203,12 @@ other shortcomings, e.g. you lose \nomenindex{canonicity}
Another approach is to use the \emph{setoid interpretation} of type
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
type. Under the setoid interpretation the equivalence relation serve
as a sort of ``local'' propositional equality. Since the developer
gets to pick this relation it is not guaranteed to be a congruence
relation a priori. So this must be verified manually by the developer.
gets to pick this relation it is not a\~priori a congruence
relation. So this must be verified manually by the developer.
Furthermore, functions between different setoids must be shown to be
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}).
\section{Conventions}
In the remainder of this paper I will use the term
\nomenindex{Type} to describe --
well, types. Thereby diverging from the notation in Agda where the keyword
\texttt{Set} refers to types.
\nomenindex{Set} on the other hand shall refer to the
homotopical notion of a set. I will also leave all universe levels implicit.
In the remainder of this paper I will use the term \nomenindex{Type}
to describe -- well -- types. Thereby departing from the notation in
Agda where the keyword \texttt{Set} refers to types. \nomenindex{Set}
on the other hand shall refer to the homotopical notion of a set. I
will also leave all universe levels implicit. This of course does not
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
\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.
functions in Agda.
$\defeq$ will be used for introducing definitions. $=$ will be used to for
judgmental equality and $\equiv$ will be used for propositional equality.
As already noted $\defeq$ will be used for introducing definitions $=$
will be used to for judgmental equality and $\equiv$ will be used for
propositional equality.
All this is summarized in the following table:
%
\begin{samepage}
\begin{center}
\begin{tabular}{ c c c }
Name & Agda & Notation \\
@ -277,3 +267,4 @@ Judgmental equality & \null & $̱=$ \\
Propositional equality & \null & $̱\equiv$
\end{tabular}
\end{center}
\end{samepage}

1
doc/macros.tex
View file

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

2
doc/main.tex
View file

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

5
doc/packages.tex
View file

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

206
doc/presentation.tex
View file

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