Stuff about half-time report

proposal/BACKLOG.md

@@ -0,0 +1,22 @@
+Remove stuff about models of type theory
+
+Add references to specific (noteable) implementaitons of category theory:
+* Unimath
+* cubicaltt
+* https://github.com/pcapriotti/agda-categories
+* https://github.com/copumpkin/categories
+* ...
+
+Talk about structure of library:
+===
+
+Propositional- and non-propositional stuff split up
+Providing "equiality principles"
+Provide overview of what has been proven.
+
+What can I say about reusability?
+
+Misc
+====
+
+Propositional content

proposal/halftime.tex

@@ -0,0 +1,177 @@
+\section{Halftime report}
+I've written this as an appendix because 1) the aim of the thesis changed
+drastically from the planning report/proposal 2) partly I'm not sure how to
+structure my thesis.
+
+My work so far has very much focused on the formalization, i.e. coding. It's
+unclear to me at this point what I should have in the final report. Here I will
+describe what I have managed to formalize so far and what outstanding challenges
+I'm facing.
+
+\subsection{Implementation overview}
+The overall structure of my project is as follows:
+
+\begin{itemize}
+\item Core categorical concepts
+\subitem Categories
+\subitem Functors
+\subitem Products
+\subitem Exponentials
+\subitem Cartesian closed categories
+\subitem Natural transformations
+\subitem Yoneda embedding
+\subitem Monads
+\subsubitem Monoidal monads
+\subsubitem Kleisli monads
+\subsubitem Voevodsky's construction
+\item Category of \ldots
+\subitem Homotopy sets
+\subitem Categories
+\subitem Relations
+\subitem Functors
+\subitem Free category
+\end{itemize}
+
+I also started work on the category with families as well as the cubical
+category as per the original goal of the thesis. However I have not gotten so
+far with this.
+
+In the following I will give an overview of overall results in each of these
+categories (no pun).
+
+As an overall design-guideline I've defined concepts in a such a way that the
+``data'' and the ``laws'' about that data is split up in seperate modules. 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.
+
+This allows me to reason about things in a more mathematical way, where one can
+reason about two categories by simply focusing on the data. This is acheived by
+creating a function embodying the ``equality principle'' for a given record. In
+the case of monads, to prove two categories propositionally equal it enough to
+provide a proof that their data is equal.
+
+\subsubsection{Categories}
+Defines the basic notion of a category. This definition closely follows that of
+[HoTT]: That is, the standard definition of a category (data; objects, arrows,
+composition and identity, laws; preservation of identity and composition) plus
+the extra condition that it is univalent - namely that you can get an equality
+of two objects from an isomorphism.
+
+I make no distinction between a pre-category and a real category (as in the
+[HoTT]-sense). A pre-category in my implementation would be a category sans the
+witness to univalence.
+
+I also prove that being a category is a proposition. This gives rise to an
+equality principle for monads that focuses on the data-part.
+
+I also show that the opposite category is indeed a category. (\WIP{} I have not
+shown that univalence holds for such a construction)
+
+I also show that taking the opposite is an involution.
+
+\subsubsection{Functors}
+Defines the notion of a functor - also split up into data and laws.
+
+Propositionality for being a functor.
+
+Composition of functors and the identity functor.
+
+\subsubsection{Products}
+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.
+
+\subsubsection{Exponentials}
+Definition of what it means to be an exponential object.
+
+Definition of what it means for a category to have all exponential objects.
+
+\subsubsection{Cartesian closed categories}
+Definition of what it means for a category to be cartesian closed; namely that
+it has all products and all exponentials.
+
+\subsubsection{Natural transformations}
+Definition of transformations\footnote{Maybe this is a name I made up for a
+  family of morphisms} and the naturality condition for these.
+
+Proof that naturality is a mere proposition and the accompanying equality
+principle. Proof that natural transformations are homotopic sets.
+
+The identity natural transformation.
+
+\subsubsection{Yoneda embedding}
+
+The yoneda embedding is typically presented in terms of the category of
+categories (cf. Awodey) \emph however this is not stricly needed - all we need
+is what would be the exponential object in that category - this happens to be
+functors and so this is how we define the yoneda embedding.
+
+\subsubsection{Monads}
+
+Defines an equivalence between these two formulations of a monad:
+
+\subsubsubsection{Monoidal monads}
+
+Defines the standard monoidal representation of a monad:
+
+An endofunctor with two natural transformations (called ``pure'' and ``join'')
+and some laws about these natural transformations.
+
+Propositionality proofs and equality principle is provided.
+
+\subsubsubsection{Kleisli monads}
+
+A presentation of monads perhaps more familiar to a functional programer:
+
+A map on objects and two maps on morphisms (called ``pure'' and ``bind'') and
+some laws about these maps.
+
+Propositionality proofs and equality principle is provided.
+
+\subsubsubsection{Voevodsky's construction}
+
+Provides construction 2.3 as presented in an unpublished paper by the late
+Vladimir Voevodsky. This construction is similiar to the equivalence provided
+for the two 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
+settings, we need to restrict the types to be those that are homotopy sets. Thus the objects of this category are:
+%
+$$\Set_\ell \defeq \sum_{A \tp \MCU_\ell} \isSet\ A$$
+%
+\WIP{} I'm still missing a few details for the proof that this category is
+univalent. Indeed this doesn't not follow immediately from
+%
+$$\mathit{univalence} \tp (A \cong B) \simeq (A \simeq B)$$
+%
+since $A$ and $B$ are of type $\MCU \neq \Set$.
+\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.
+
+Furthermore I provide some helpful lemmas about this raw category. For instance
+I have shown what would be the exponential object in such a category.
+
+These lemmas can be used to provide the actual exponential object in a context
+where we have a witness to this being a category. This is useful if this library
+is later extended to talk about higher categories.
+
+\subsubsection{Functors}
+The category of functors and natural transformations. An immediate corrolary is
+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
+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
+univalent.

proposal/macros.tex

@@ -18,3 +18,9 @@
 \newcommand{\fmap}{\mathit{fmap}}
 \newcommand{\idFun}{\mathit{id}}
 \newcommand{\Sets}{\mathit{Sets}}
+\newcommand{\Set}{\mathit{Set}}
+\newcommand{\MCU}{\UU}
+\newcommand{\isSet}{\mathit{isSet}}
+\newcommand{\tp}{\;\mathord{:}\;}
+\newcommand{\subsubsubsection}[1]{\textbf{#1}}
+\newcommand{\WIP}[1]{\textbf{[WIP]}}

proposal/proposal.tex

@@ -13,6 +13,8 @@
 \usepackage{multicol}
 \usepackage{amsmath,amssymb}
 \usepackage[toc,page]{appendix}
+\usepackage{xspace}
+
 % \setlength{\parskip}{10pt}
 
 % \usepackage{tikz}
@@ -282,5 +284,6 @@ The thesis shall conclude with a discussion about the benefits of Cubical Agda.
 \bibliography{refs}
 \begin{appendices}
 \input{planning.tex}
+\input{halftime.tex}
 \end{appendices}
 \end{document}