260 lines
9.3 KiB
TeX
260 lines
9.3 KiB
TeX
\chapter{Halftime report}
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I've written this as an appendix because 1) the aim of the thesis changed
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drastically from the planning report/proposal 2) partly I'm not sure how to
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structure my thesis.
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My work so far has very much focused on the formalization, i.e. coding. It's
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unclear to me at this point what I should have in the final report. Here I will
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describe what I have managed to formalize so far and what outstanding challenges
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I'm facing.
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\section{Implementation overview}
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The overall structure of my project is as follows:
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\begin{itemize}
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\item Core categorical concepts
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\subitem Categories
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\subitem Functors
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\subitem Products
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\subitem Exponentials
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\subitem Cartesian closed categories
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\subitem Natural transformations
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\subitem Yoneda embedding
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\subitem Monads
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\subsubitem Monoidal monads
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\subsubitem Kleisli monads
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\subsubitem Voevodsky's construction
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\item Category of \ldots
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\subitem Homotopy sets
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\subitem Categories
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\subitem Relations
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\subitem Functors
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\subitem Free category
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\end{itemize}
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I also started work on the category with families as well as the cubical
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category as per the original goal of the thesis. However I have not gotten so
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far with this.
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In the following I will give an overview of overall results in each of these
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categories (no pun).
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As an overall design-guideline I've defined concepts in a such a way that the
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``data'' and the ``laws'' about that data is split up in seperate modules. As an
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example a category is defined to have two members: `raw` which is a collection
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of the data and `isCategory` which asserts some laws about that data.
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This allows me to reason about things in a more mathematical way, where one can
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reason about two categories by simply focusing on the data. This is acheived by
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creating a function embodying the ``equality principle'' for a given record. In
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the case of monads, to prove two categories propositionally equal it enough to
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provide a proof that their data is equal.
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\subsection{Categories}
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Defines the basic notion of a category. This definition closely follows that of
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[HoTT]: That is, the standard definition of a category (data; objects, arrows,
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composition and identity, laws; preservation of identity and composition) plus
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the extra condition that it is univalent - namely that you can get an equality
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of two objects from an isomorphism.
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I make no distinction between a pre category and a real category (as in the
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[HoTT]-sense). A pre category in my implementation would be a category sans the
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witness to univalence.
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I also prove that being a category is a proposition. This gives rise to an
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equality principle for monads that focuses on the data-part.
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I also show that the opposite category is indeed a category. (\WIP{} I have not
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shown that univalence holds for such a construction)
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I also show that taking the opposite is an involution.
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\subsection{Functors}
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Defines the notion of a functor - also split up into data and laws.
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Propositionality for being a functor.
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Composition of functors and the identity functor.
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\subsection{Products}
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Definition of what it means for an object to be a product in a given category.
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Definition of what it means for a category to have all products.
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\WIP{} Prove propositionality for being a product and having products.
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\subsection{Exponentials}
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Definition of what it means to be an exponential object.
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Definition of what it means for a category to have all exponential objects.
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\subsection{Cartesian closed categories}
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Definition of what it means for a category to be cartesian closed; namely that
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it has all products and all exponentials.
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\subsection{Natural transformations}
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Definition of transformations\footnote{Maybe this is a name I made up for a
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family of morphisms} and the naturality condition for these.
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Proof that naturality is a mere proposition and the accompanying equality
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principle. Proof that natural transformations are homotopic sets.
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The identity natural transformation.
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\subsection{Yoneda embedding}
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The yoneda embedding is typically presented in terms of the category of
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categories (cf. Awodey) \emph however this is not stricly needed - all we need
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is what would be the exponential object in that category - this happens to be
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functors and so this is how we define the yoneda embedding.
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\subsection{Monads}
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Defines an equivalence between these two formulations of a monad:
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\subsubsection{Monoidal monads}
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Defines the standard monoidal representation of a monad:
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An endofunctor with two natural transformations (called ``pure'' and ``join'')
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and some laws about these natural transformations.
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Propositionality proofs and equality principle is provided.
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\subsubsection{Kleisli monads}
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A presentation of monads perhaps more familiar to a functional programer:
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A map on objects and two maps on morphisms (called ``pure'' and ``bind'') and
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some laws about these maps.
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Propositionality proofs and equality principle is provided.
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\subsubsection{Voevodsky's construction}
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Provides construction 2.3 as presented in an unpublished paper by Vladimir
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Voevodsky. This construction is similiar to the equivalence provided for the two
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preceding formulations
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\footnote{ TODO: I would like to include in the thesis some motivation for why
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this construction is particularly interesting.}
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\subsection{Homotopy sets}
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The typical category of sets where the objects are modelled by an Agda set
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(henceforth ``$\Type$'') at a given level is not a valid category in this cubical
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settings, we need to restrict the types to be those that are homotopy sets. Thus
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the objects of this category are:
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%
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$$\hSet_\ell \defeq \sum_{A \tp \MCU_\ell} \isSet\ A$$
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%
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The definition of univalence for categories I have defined is:
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%
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$$\isEquiv\ (\hA \equiv \hB)\ (\hA \cong \hB)\ \idToIso$$
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%
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Where $\hA and \hB$ denote objects in the category. Note that this is stronger
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than
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%
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$$(\hA \equiv \hB) \simeq (\hA \cong \hB)$$
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%
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Because we require that the equivalence is constructed from the witness to:
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%
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$$\id \comp f \equiv f \x f \comp \id \equiv f$$
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%
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And indeed univalence does not follow immediately from univalence for types:
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%
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$$(A \equiv B) \simeq (A \simeq B)$$
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%
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Because $A\ B \tp \Type$ whereas $\hA\ \hB \tp \hSet$.
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For this reason I have shown that this category satisfies the following
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equivalent formulation of being univalent:
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%
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$$\prod_{A \tp hSet} \isContr \left( \sum_{X \tp hSet} A \cong X \right)$$
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%
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But I have not shown that it is indeed equivalent to my former definition.
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\subsection{Categories}
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Note that this category does in fact not exist. In stead I provide the
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definition of the ``raw'' category as well as some of the laws.
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Furthermore I provide some helpful lemmas about this raw category. For instance
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I have shown what would be the exponential object in such a category.
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These lemmas can be used to provide the actual exponential object in a context
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where we have a witness to this being a category. This is useful if this library
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is later extended to talk about higher categories.
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\subsection{Functors}
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The category of functors and natural transformations. An immediate corrolary is
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the set of presheaf categories.
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\WIP{} I have not shown that the category of functors is univalent.
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\subsection{Relations}
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The category of relations. \WIP{} I have not shown that this category is
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univalent. Not sure I intend to do so either.
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\subsection{Free category}
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The free category of a category. \WIP{} I have not shown that this category is
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univalent.
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\section{Current Challenges}
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Besides the items marked \WIP{} above I still feel a bit unsure about what to
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include in my report. Most of my work so far has been specifically about
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developing this library. Some ideas:
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%
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\begin{itemize}
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\item
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Modularity properties
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\item
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Compare with setoid-approach to solve similiar problems.
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\item
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How to structure an implementation to best deal with types that have no
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structure (propositions) and those that do (sets and everything above)
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\end{itemize}
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%
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\section{Ideas for future developments}
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\subsection{Higher categories}
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I only have a notion of (1-)categories. Perhaps it would be nice to also
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formalize higher categories.
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\subsection{Hierarchy of concepts related to monads}
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In Haskell the type-class Monad sits in a hierarchy atop the notion of a functor
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and applicative functors. There's probably a similiar notion in the
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category-theoretic approach to developing this.
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As I have already defined monads from these two perspectives, it would be
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interesting to take this idea even further and actually show how monads are
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related to applicative functors and functors. I'm not entirely sure how this
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would look in Agda though.
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\subsection{Use formulation on the standard library}
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I also thought it would be interesting to use this library to show certain
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properties about functors, applicative functors and monads used in the Agda
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Standard library. So I went ahead and tried to show that agda's standard
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library's notion of a functor (along with suitable laws) is equivalent to my
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formulation (in the category of homotopic sets). I ran into two problems here,
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however; the first one is that the standard library's notion of a functor is
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indexed by the object map:
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%
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$$
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\Functor \tp (\Type \to \Type) \to \Type
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$$
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%
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Where $\Functor\ F$ has the member:
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%
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$$
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\fmap \tp (A \to B) \to F A \to F B
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$$
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%
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Whereas the object map in my definition is existentially quantified:
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%
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$$
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\Functor \tp \Type
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$$
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%
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And $\Functor$ has these members:
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\begin{align*}
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F & \tp \Type \to \Type \\
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\fmap & \tp (A \to B) \to F A \to F B\}
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\end{align*}
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%
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