Half-time report

proposal/halftime.tex

@@ -81,7 +81,7 @@ Definition of what it means for an object to be a product in a given category.
 
 Definition of what it means for a category to have all products.
 
-\WIP Prove propositionality for being a product and having products.
+\WIP{} Prove propositionality for being a product and having products.
 
 \subsubsection{Exponentials}
 Definition of what it means to be an exponential object.
@@ -132,25 +132,45 @@ Propositionality proofs and equality principle is provided.
 
 \subsubsubsection{Voevodsky's construction}
 
-Provides construction 2.3 as presented in an unpublished paper by the late
+Provides construction 2.3 as presented in an unpublished paper by Vladimir
-Vladimir Voevodsky. This construction is similiar to the equivalence provided
+Voevodsky. This construction is similiar to the equivalence provided for the two
-for the two preceding formulations
+preceding formulations
 \footnote{ TODO: I would like to include in the thesis some motivation for why
   this construction is particularly interesting.}
 
 \subsubsection{Homotopy sets}
 The typical category of sets where the objects are modelled by an Agda set
-(henceforth ``type'') at a given level is not a valid category in this cubical
+(henceforth ``$\Type$'') at a given level is not a valid category in this cubical
-settings, we need to restrict the types to be those that are homotopy sets. Thus the objects of this category are:
+settings, we need to restrict the types to be those that are homotopy sets. Thus
+the objects of this category are:
 %
-$$\Set_\ell \defeq \sum_{A \tp \MCU_\ell} \isSet\ A$$
+$$\hSet_\ell \defeq \sum_{A \tp \MCU_\ell} \isSet\ A$$
 %
-\WIP{} I'm still missing a few details for the proof that this category is
+The definition of univalence for categories I have defined is:
-univalent. Indeed this doesn't not follow immediately from
 %
-$$\mathit{univalence} \tp (A \cong B) \simeq (A \simeq B)$$
+$$\isEquiv\ (\hA \equiv \hB)\ (\hA \cong \hB)\ \idToIso$$
 %
-since $A$ and $B$ are of type $\MCU \neq \Set$.
+Where $\hA and \hB$ denote objects in the category. Note that this is stronger
+than
+%
+$$(\hA \equiv \hB) \simeq (\hA \cong \hB)$$
+%
+Because we require that the equivalence is constructed from the witness to:
+%
+$$\id \comp f \equiv f \x f \comp \id \equiv f$$
+%
+And indeed univalence does not follow immediately from univalence for types:
+%
+$$(A \equiv B) \simeq (A \simeq B)$$
+%
+Because $A\ B \tp \Type$ whereas $\hA\ \hB \tp \hSet$.
+
+For this reason I have shown that this category satisfies the following
+equivalent formulation of being univalent:
+%
+$$\prod_{A \tp hSet} \isContr \left( \sum_{X \tp hSet} A \cong X \right)$$
+%
+But I have not shown that it is indeed equivalent to my former definition.
 \subsubsection{Categories}
 Note that this category does in fact not exist. In stead I provide the
 definition of the ``raw'' category as well as some of the laws.
@@ -169,9 +189,71 @@ the set of presheaf categories.
 \WIP{} I have not shown that the category of functors is univalent.
 
 \subsubsection{Relations}
-The category of relations. \WIP I have not shown that this category is
+The category of relations. \WIP{} I have not shown that this category is
 univalent. Not sure I intend to do so either.
 
 \subsubsection{Free category}
-The free category of a category. \WIP I have not shown that this category is
+The free category of a category. \WIP{} I have not shown that this category is
 univalent.
+
+\subsection{Current Challenges}
+Besides the items marked \WIP{} above I still feel a bit unsure about what to
+include in my report. Most of my work so far has been specifically about
+developing this library. Some ideas:
+%
+\begin{itemize}
+\item
+  Modularity properties
+\item
+  Compare with setoid-approach to solve similiar problems.
+\item
+  How to structure an implementation to best deal with types that have no
+  structure (propositions) and those that do (sets and everything above)
+\end{itemize}
+%
+\subsection{Ideas for future developments}
+\subsubsection{Higher categories}
+I only have a notion of (1-)categories. Perhaps it would be nice to also
+formalize higher categories.
+
+\subsubsection{Hierarchy of concepts related to monads}
+In Haskell the type-class Monad sits in a hierarchy atop the notion of a functor
+and applicative functors. There's probably a similiar notion in the
+category-theoretic approach to developing this.
+
+As I have already defined monads from these two perspectives, it would be
+interesting to take this idea even further and actually show how monads are
+related to applicative functors and functors. I'm not entirely sure how this
+would look in Agda though.
+
+\subsubsection{Use formulation on the standard library}
+I also thought it would be interesting to use this library to show certain
+properties about functors, applicative functors and monads used in the Agda
+Standard library. So I went ahead and tried to show that agda's standard
+library's notion of a functor (along with suitable laws) is equivalent to my
+formulation (in the category of homotopic sets). I ran into two problems here,
+however; the first one is that the standard library's notion of a functor is
+indexed by the object map:
+%
+$$
+\Functor   & \tp (\Type \to \Type) \to \Type
+$$
+%
+Where $\Functor\ F$ has the member:
+%
+$$
+\fmap \tp (A \to B) \to F A \to F B
+$$
+%
+Whereas the object map in my definition is existentially quantified:
+%
+$$
+\Functor   & \tp \Type
+$$
+%
+And $\Functor$ has these members:
+\begin{align*}
+F     & \tp \Type \to \Type \\
+\fmap & \tp (A \to B) \to F A \to F B\}
+\end{align*}
+%

proposal/macros.tex

@@ -19,8 +19,17 @@
 \newcommand{\idFun}{\mathit{id}}
 \newcommand{\Sets}{\mathit{Sets}}
 \newcommand{\Set}{\mathit{Set}}
+\newcommand{\hSet}{\mathit{hSet}}
+\newcommand{\Type}{\mathcal{U}}
+\newcommand{\isEquiv}{\mathit{isEquiv}}
+\newcommand{\idToIso}{\mathit{idToIso}}
 \newcommand{\MCU}{\UU}
 \newcommand{\isSet}{\mathit{isSet}}
-\newcommand{\tp}{\;\mathord{:}\;}
+\newcommand{\isContr}{\mathit{isContr}}
+\newcommand{\id}{\mathit{id}}
+\newcommand{\tp}{\,\mathord{:}\,}
+\newcommand\hA{\mathit{hA}}
+\newcommand\hB{\mathit{hB}}
+\newcommand\Functor{\mathit{Functor}}
 \newcommand{\subsubsubsection}[1]{\textbf{#1}}
-\newcommand{\WIP}[1]{\textbf{[WIP]}}
+\newcommand{\WIP}{\textbf{WIP}}

src/Cat/Category.agda

@@ -113,7 +113,7 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
   module Univalence (isIdentity : IsIdentity 𝟙) where
     -- | The identity isomorphism
     idIso : (A : Object) → A ≅ A
-    idIso A = 𝟙 , (𝟙 , isIdentity)
+    idIso A = 𝟙 , 𝟙 , isIdentity
 
     -- | Extract an isomorphism from an equality
     --