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
ITT and standard Agda enjoys Uniqueness of Identity Proofs (UIP). This
is not 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. 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