Update proposal

cat.md

@@ -1,26 +1,60 @@
 ---
 title: Formalizing category theory in Agda - Project description
 date: May 27th 2017
-author: Frederik Hanghøj Iversen
+author: Frederik Hanghøj Iversen `<hanghj@student.chalmers.se>`
 bibliography: refs.bib
 ---
 
 Background
 ==========
 
+Functional extensionality gives rise to a notion of equality of functions not
+present in intensional dependent type theory. A type-system called cubical
+type-theory is outlined in [@cohen-2016] that recovers the computational interprtation of the univalence theorem.
+
+Keywords: The category of sets, limits, colimits, functors, natural
+transformations, kleisly category, yoneda lemma, closed cartesian categories,
+propositional logic.
+
 Aim
 ===
 
+The aim of the project is two-fold. The first part of the project will be
+concerned with formalizing some concepts from category theory in Agda's
+type-system: functors, applicative functors, monads, etc.. The second part of
+the project could take different directions:
+
+* It might involve using this formalization to prove properties of functional
+  programs.
+* It may be used to prove the Modal used in Cubical Type Theory using Preshiefs.
+
 Timeplan
 ========
 
+The first part of the project will focus on studying and understanding the
+foundations for this project namely; familiarizing myself with basic concepts
+from category theory, understanding how cubical type theory gives rise to
+expressing functional extensionality and the univalence theorem.
+
+After I have understood these fundamental concepts I will use them in the
+formalization of functors, applicative functors, monads, etc.. in Agda. This
+should be done before the end of the first semester of the school-year
+2017/2018.
+
+At this point I will also have settled on a direction for the rest of the
+project and developed a time-plan for the second phase of the project. But
+cerainly it will involve applying the result of phase 1 in some context as
+mentioned in [the project aim][aim].
+
 Resources
 =========
 
 * Cubical demo by Andrea Vezossi: [@cubical-demo]
-* Book on cubical type theory [@cohen-2016]
+* Paper on cubical type theory [@cohen-2016]
 * Book on homotopy type theory: [@hott-2013]
 * Book on category theory: [@awodey-2006]
+* Modal logic - Modal type theory, see
+  [ncatlab](https://ncatlab.org/nlab/show/modal+type+theory).
 
 References
 ==========