cat/doc/conclusion.tex

\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
Agda enjoys Uniqueness of Identity Proofs (UIP) though a flag exists
to turn this off, which is 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; categories, functors, products, exponentials, Cartesian
closed categories, natural transformations, the yoneda embedding,
monads and more. Category theory is an interesting case-study for the
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
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
to demonstrate that any two product objects in a category are
isomorphic. Furthermore the thesis showed two formulations of monads
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
also saw how paths can be used to extract functions. A path between
two types induce an isomorphism between the two types. This
e.g. permits developers to write a monad instance for a given type
using the Kleisli formulation. By transporting along the path between
the monoidal- and Kleisli- formulation one can reuse all the
operations and results shown for monoidal- monads in the context of
kleisli monads.
%%
%% problem with inductive type
%% overcome with cubical
%% the path type
%% homotopy levels
%% depdendent paths
%%
%% category theory
%% algebra of functions ~ funExt
%% identify isomorphic types ~ univalence
%% using categories to prove properties
%% computational properties
%% reusability, compositional