Write conclusion

doc/conclusion.tex

@@ -1,3 +1,52 @@
 \chapter{Conclusion}
+This thesis highlighted some of issues with the standard inductive definition of
+propositional equality used in Agda. Functional extensionality and univalence
+are two examples not admissible in Intensional Type Theory (ITT). 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}. It turns out to be useful to built the formalization with this 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 which in
+practice turns out to be considerably more difficult to work with than
+homogeneous paths. The thesis also demonstrated how to use appropriate
+abstraction techniques for dealing with this, such as based path-induction.
 
-\TODO{\ldots}
+This thesis formalized some of the core concepts from category theory including;
+categories, functors, products, exponentials, Cartesian closed categories,
+natural transformations, the yoneda embedding and monads. 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 identity
+isomorphic structures and univalence allows us to make this notion precise. The
+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 monoidal- and Kleisli-
+monads. 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 this formulation to become a monoidal
+monad one can reuse all results about monoidal monads on the Kleisli
+formulation.
+%%
+%% 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