Final touch-up on report and acknowledgments

doc/acknowledgement.tex

@@ -0,0 +1 @@
+\chapter*{Acknowledgements}

doc/conclusion.tex

@@ -3,34 +3,34 @@ 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
+what is provable and the reusability of proofs.  This issue is
-with an extension to Agda's type system called Cubical Agda. With
+overcome with an extension to Agda's type system called Cubical Agda.
-Cubical Agda both functional extensionality and univalence are
+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
+issues that arise that are not present in standard Agda.  For one
-Agda enjoys Uniqueness of Identity Proofs (UIP) though a flag exists
+thing Agda enjoys Uniqueness of Identity Proofs (UIP) though a flag
-to turn this off, which is the case in Cubical Agda. In stead
+exists to turn this off.  This feature is not present in Cubical Agda.
-there exists a hierarchy of types with increasing \nomen{homotopical
+Rather than having unique identity proofs cubical Agda gives rise to 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
+case paths between such types will be heterogeneous paths.  In
-problem is related to Cubical Agda not having the K-rule. In practice
+practice it turns out to be considerably more difficult to work with
-it turns out to be considerably more difficult to work heterogeneous
+heterogeneous paths than with homogeneous paths.  The thesis
-paths than with homogeneous paths. The thesis demonstrated some
+demonstrated the application of some techniques to overcome these
-techniques to overcome these difficulties, such as based
+difficulties, such as based path induction.
-path-induction.
 
-This thesis formalized some of the core concepts from category theory
+This thesis formalizes 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
+monads and more.  Category theory is an interesting case study for the
-application of Cubical Agda for two reasons in particular: Because
+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
+isomorphic structures.  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
@@ -40,14 +40,14 @@ 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
+difficult to make a similar proof with setoids and the proof would be
-also saw how paths can be used to extract functions. A path between
+very difficult to read.  In the formulation we also saw how paths can
-two types induce an isomorphism between the two types. This
+be used to extract functions.  A path between two types induce an
-e.g. permits developers to write a monad instance for a given type
+isomorphism between the two types.  This e.g.\ permits developers to
-using the Kleisli formulation. By transporting along the path between
+write a monad instance for a given type using the Kleisli formulation.
-the monoidal- and Kleisli- formulation one can reuse all the
+By transporting along the path between the monoidal- and Kleisli-
-operations and results shown for monoidal- monads in the context of
+formulation one can reuse all the operations and results shown for
-kleisli monads.
+monoidal- monads in the context of kleisli monads.
 %%
 %% problem with inductive type
 %% overcome with cubical

doc/discussion.tex

@@ -16,41 +16,43 @@ interesting result of this development is how much this influenced the
 development.  In particular having a functional extensionality that
 ``computes'' should simplify some proofs.
 
-I have tested this theory by using a feature of Agda where one can
+I have tested this by using a feature of Agda where one can mark
-mark certain bindings as being \emph{abstract}. This means that the
+certain bindings as being \emph{abstract}.  This means that the
-type-checker will not try to reduce that term further when
+type-checker will not try to reduce that term further during type
-type-checking is performed. I tried making univalence and functional
+checking.  I tried making univalence and functional extensionality
-extensionality abstract. It turns out that the conversion behaviour of
+abstract.  It turns out that the conversion behaviour of univalence is
-univalence is not used anywhere. For functional extensionality there
+not used anywhere.  For functional extensionality there are two places
-are two places in the whole solution where the reduction behaviour is
+in the whole solution where the reduction behaviour is used to
-used to simplify some proofs. This is in showing that the maps between
+simplify some proofs.  This is in showing that the maps between the
-the two formulations of monads are inverses. See the notes in this
+two formulations of monads are inverses.  See the notes in this
 module:
 %
 \begin{center}
 \sourcelink{Cat.Category.Monad.Voevodsky}
 \end{center}
 %
-I've also put this in a source listing in \ref{app:abstract-funext}. I
-will not reproduce it in full here as the type is quite involved. The
-method used to find in what places the computational behaviour of
-these proofs are needed has the caveat of only working for places that
-directly or transitively uses these two proofs. Fortunately though the
-code is structured in such a way that this should be the case.
-Nonetheless it is quite surprising that this computational behaviours
-is not used more widely in the formalization.
 
-Barring this, however, the computational behaviour of paths can still
+I will not reproduce it in full here as the type is quite involved. In
-be useful. E.g. if a programmer want's to reuse functions that operate
+stead I have put this in a source listing in \ref{app:abstract-funext}.
-on a monoidal monads to work with a monad in the Kleisli form that
+The method used to find in what places the computational behaviour of
-this programmer has specified. To make this idea concrete, say we are
+these proofs are needed has the caveat of only working for places that
+directly or transitively uses these two proofs.  Fortunately though
+the code is structured in such a way that this is the case. So in
+conclusion the way I have structured these proofs means that the
+computational behaviour of functional extensionality and univalence
+has not been so relevant.
+
+Barring this the computational behaviour of paths can still be useful.
+E.g.\ if a programmer wants to reuse functions that operate on a
+monoidal monads to work with a monad in the Kleisli form that the
+programmer has specified.  To make this idea concrete, say we are
 given some function $f \tp \Kleisli \to T$ having a path between $p
 \tp \Monoidal \equiv \Kleisli$ induces a map $\coe\ p \tp \Monoidal
 \to \Kleisli$.  We can compose $f$ with this map to get $f \comp
 \coe\ p \tp \Monoidal \to T$.  Of course, since that map was
 constructed with an isomorphism these maps already exist and could be
-used directly. So this is arguably only interesting when one wants to
+used directly.  So this is arguably only interesting when one also
-prove properties of such functions.
+wants to prove properties of applying such functions.
 
 \subsection{Reusability of proofs}
 The previous example illustrate how univalence unifies two otherwise
@@ -68,18 +70,18 @@ course generalizes to any family $P \tp 𝒰 → 𝒰$ where $P$ is inhabited
 at either formulation (i.e.\ either $P\ \Monoidal$ or $P\ \Kleisli$
 holds).
 
-The introduction (section \S\ref{sec:context}) mentioned an often
+The introduction (section \S\ref{sec:context}) mentioned that a
-employed-technique for enabling extensional equalities is to use the
+typical way of getting access to functional extensionality is to work
-setoid-interpretation. Nowhere in this formalization has this been
+with setoids.  Nowhere in this formalization has this been necessary,
-necessary, $\Path$ has been used globally in the project as
+$\Path$ has been used globally in the project for propositional
-propositional equality. One interesting place where this becomes
+equality.  One interesting place where this becomes apparent is in
-apparent is in interfacing with the Agda standard library. Multiple
+interfacing with the Agda standard library.  Multiple definitions in
-definitions in the Agda standard library have been designed with the
+the Agda standard library have been designed with the
-setoid-interpretation in mind. E.g. the notion of ``unique
+setoid-interpretation in mind.  E.g.\ the notion of \emph{unique
-existential'' is indexed by a relation that should play the role of
+  existential} is indexed by a relation that should play the role of
-propositional equality. Likewise for equivalence relations, they are
+propositional equality.  Equivalence relations are likewise indexed,
-indexed, not only by the actual equivalence relation, but also by
+not only by the actual equivalence relation but also by another
-another relation that serve as propositional equality.
+relation that serve as propositional equality.
 %% Unfortunately we cannot use the definition of equivalences found in
 %% the standard library to do equational reasoning directly.  The
 %% reason for this is that the equivalence relation defined there must
@@ -89,14 +91,12 @@ In the formalization at present a significant amount of energy has
 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}
+proofs (e.g.\ \ref{eq:proof-prop-IsPreCategory}
 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.
 
-\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
@@ -121,20 +121,19 @@ This means that even though the path-type gives us a computational
 interpretation of functional extensionality, univalence, transport,
 etc., we do not have a way of actually using this to compile our
 programs that use these primitives.  It would be interesting to see
-practical applications of this. The path between monads that this
+practical applications of this.
-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.
 
 \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.
+
+%% \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 at Chalmers, Andreas Källberg, is currently working
+on proving the initiality conjecture.  He will be using this library
+to do so.

doc/introduction.tex

@@ -207,13 +207,13 @@ 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 a\~priori a congruence
+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.
 
 This approach has other drawbacks; it does not satisfy all
-propositional equalities of type theory a priori. That is, the
+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 ∼