Remove some TODO-notes, add section on motifs

doc/conclusion.tex

@@ -1,22 +1,24 @@
 \chapter{Conclusion}
-This thesis highlighted some issues with the standard inductive definition of
-propositional equality used in Agda. Functional extensionality and univalence
-are examples of two propositions not admissible in Intensional Type Theory
-(ITT). This has a big impact on what is provable and the reusability of proofs.
-This issue is overcome 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 \TODO{Not mentioned anywhere in the report}. 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
+This thesis highlighted some issues with the standard inductive
+definition of propositional equality used in Agda. Functional
+extensionality and univalence are examples of two propositions not
+admissible in Intensional Type Theory (ITT). This has a big impact on
+what is provable and the reusability of proofs. This issue is overcome
+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.
 
 This thesis formalized some of the core concepts from category theory including;

doc/cubical.tex

@@ -54,17 +54,18 @@ Judgmental equality in Cubical Agda is encapsulated with the type:
 \Path \tp (P \tp I → \MCU) → P\ 0 → P\ 1 → \MCU
 \end{equation}
 %
-$I$ is a special data type (\TODO{that also has special computational properties
-  AFAIK}) 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:
+$I$ 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:
 %
 $$
 p \tp \prod_{i \tp I} P\ i
@@ -194,7 +195,7 @@ indeed both $\top$ and $\bot$ are propositions:
 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).\TODO{Cite}
 
 I will refer to a type $A \tp \MCU$ as a \emph{mere} proposition if I want to
 stress that we have $\isProp\ A$.
@@ -242,11 +243,16 @@ 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.
 %
 \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. Specifically the results come from
-the Agda library \texttt{cubical} (\TODO{Cite}). 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 \TODO{Cite}.}
+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.
+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}}.}
 
 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

doc/discussion.tex

@@ -97,10 +97,25 @@ a significant impact on the complexity of these kinds of proofs.
 
 \TODO{Universe levels.}
 
+\subsection{Motifs}
+An oft-used technique in this development is using based path
+induction to prove certain properties. One particular challenge that
+arises when doing so is that Agda is not able to automatically infer
+the family that one wants to do induction over. For instance in the
+proof $\var{sym}\ (\var{sym}\ p) ≡ p$ from \ref{eq:sym-invol} the
+family that we chose to do induction over was $D\ b'\ p' \defeq
+\var{sym}\ (\var{sym}\ p') ≡ p'$. However, if one interactively tries
+to give this hole, all the information that Agda can provide is that
+one must provide an element of $𝒰$. Agda could be more helpful in this
+context, perhaps even infer this family in some situations. In this
+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}
@@ -117,3 +132,8 @@ library exposes could provide one particularly interesting case-study.
 \subsection{Higher inductive types}
 This library has not explored the usefulness of higher inductive types
 in the context of Category Theory.
+
+\subsection{Initiality conjecture}
+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.

doc/implementation.tex

@@ -277,16 +277,19 @@ $i$ becomes the triple:
 \end{aligned}
 \end{equation}
 %
-I have found this to be a general pattern when proving things in homotopy type
-theory, namely that you have to wrap and unwrap equalities at different levels.
-It is worth noting that proving this theorem with the regular inductive equality
-type would already not be possible, since we at least need extensionality (the
-projections are all $\prod$-types). Assuming we had functional extensionality
-available to us as an axiom, we would use functional extensionality \TODO{in
-reverse?} to retrieve the equalities in $a$ and $b$, pattern-match on them to
-see that they are both $\refl$ and then close the proof with $\refl$.
-Of course this theorem is not so interesting in the setting of ITT since we know
-a priori that equality proofs are unique.
+I have found this to be a general pattern when proving things in
+homotopy type theory, namely that you have to wrap and unwrap
+equalities at different levels. It is worth noting that proving this
+theorem with the regular inductive equality type would already not be
+possible, since we at least need functional
+extensionality\index{functional extensionality} (the projections are
+all $\prod$-types). Assuming we had functional extensionality
+available to us as an axiom, we would use functional extensionality
+\TODO{in reverse?} to retrieve the equalities in $a$ and $b$,
+pattern-match on them to see that they are both $\refl$ and then close
+the proof with $\refl$. Of course this theorem is not so interesting
+in the setting of ITT since we know a priori that equality proofs are
+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
@@ -326,8 +329,8 @@ $$
 And this finishes the proof that being-a-category is a mere proposition
 (\ref{eq:propIsPreCategory}).
 
-When we have a proper category we can make precise the notion of ``identifying
-isomorphic types'' \TODO{cite Awodey here}. That is, we can construct the
+When we have a proper category we can make precise the notion of
+``identifying isomorphic types''. That is, we can construct the
 function:
 %
 $$
@@ -345,7 +348,11 @@ terminal objects are propositional:
 It follows from the usual observation that any two terminal objects are
 isomorphic - and since categories are univalent, so are they equal. The proof is
 omitted here, but the curious reader can check the implementation for the
-details. \TODO{The proof is a bit fun, should I include it?}
+details. It is in the module:
+%
+\begin{center}
+\sourcelink{Cat.Category}
+\end{center}
 
 \section{Equivalences}
 \label{sec:equiv}
@@ -744,15 +751,18 @@ the two pairs:
   \left( \fst\ u \equiv \fst\ v \right) \to u \equiv v
 \end{align*}
 %
-The proof that these are indeed inverses has been omitted. \TODO{Do I really
-  want to omit it?}\QED
+The proof that these are indeed inverses has been omitted. The details
+can be found in the module:
+\begin{center}
+\sourcelink{Cat.Categories.Sets}
+\end{center}
 
-Now to prove \ref{eq:equivSig}: Let $e \tp \prod_{a \tp A} \left( P\ a \simeq
-Q\ a \right)$ be given. To prove the equivalence, it suffices to give an
-isomorphism between $\sum_{a \tp A} P\ a$ and $\sum_{a \tp A} Q\ a$, but since
-they have identical first components it suffices to give an isomorphism between
-$P\ a$ and $Q\ a$ for all $a \tp A$. This is exactly what we can get from
-the equivalence $e$.\QED
+Now to prove \ref{eq:equivSig}: Let $e \tp \prod_{a \tp A} \left( P\ a
+\simeq Q\ a \right)$ be given. To prove the equivalence, it suffices
+to give an isomorphism between $\sum_{a \tp A} P\ a$ and $\sum_{a \tp
+  A} Q\ a$, but since they have identical first components it suffices
+to give an isomorphism between $P\ a$ and $Q\ a$ for all $a \tp A$.
+This is exactly what we can get from the equivalence $e$.\QED
 
 Lastly we prove \ref{eq:equivIso}. Let $f \tp A \to B$ be given. For the maps we
 choose:
@@ -1050,7 +1060,10 @@ $b_0, b_1$ corresponds to a pair of paths between $a_0,b_0$ and $a_1,b_1$ (check
 the implementation for the details).
 
 \emph{Proposition} \ref{eq:univ-1} is isomorphic to \ref{eq:univ-2}:
-\TODO{Super complicated}
+This proof of this has been omitted but can be found in the module:
+\begin{center}
+\sourcelink{Cat.Categories.Span}
+\end{center}
 
 \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,
@@ -1067,7 +1080,10 @@ f & \equiv g \lll \iota \\
 g & \equiv f \lll \inv{\iota}
 \end{align}
 %
-Proof: \TODO{\ldots}
+The proof is omitted but can be found in the module:
+\begin{center}
+\sourcelink{Cat.Category}
+\end{center}
 
 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

doc/introduction.tex

@@ -207,14 +207,13 @@ implementations of category theory in Agda:
 %
 The contribution of this thesis is to explore how working in a cubical setting
 will make it possible to prove more things and to reuse proofs and to try and
-compare some aspects of this formalization with the existing ones.\TODO{How can
-  I live up to this?}
+compare some aspects of this formalization with the existing ones.
 
 There are alternative approaches to working in a cubical setting where
 one can still have univalence and functional extensionality. One
 option is to postulate these as axioms. This approach, however, has
 other shortcomings, e.g. you lose \nomenindex{canonicity}
-(\TODO{Pageno!} \cite{huber-2016}).
+(\cite[p. 3]{huber-2016}).
 
 Another approach is to use the \emph{setoid interpretation} of type
 theory (\cite{hofmann-1995,huber-2016}). With this approach one works
@@ -227,16 +226,17 @@ relation a priori. 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.
 
-This approach has other drawbacks; it does not satisfy
-all propositional equalities of type theory (\TODO{Citation needed}), is
-cumbersome to work with in practice (\cite[p. 4]{huber-2016}) and makes
-equational proofs less reusable since equational proofs $a \sim_{X} b$ are
-inherently `local' to the extensional set $(X , \sim)$.
+This approach has other drawbacks; it does not satisfy all
+propositional equalities of type theory a priori. That is, the
+developer must manually show that e.g.\ the relation is a congruence.
+Equational proofs $a \sim_{X} b$ are in some sense `local' to the
+extensional set $(X , \sim)$. To e.g.\ prove that $x ∼ y → f\ x ∼
+f\ y$ for some function $f \tp A → B$ between two extensional sets $A$
+and $B$ it must be shown that $f$ is a groupoid homomorphism. This
+makes it very cumbersome to work with in practice (\cite[p.
+  4]{huber-2016}).
 
 \section{Conventions}
-\TODO{Talk a bit about terminology. Find a good place to stuff this little
-  section.}
-
 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