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| title | date | author | bibliography |
|---|---|---|---|
| Formalizing category theory in Agda - Project description | May 27th 2017 | Frederik Hanghøj Iversen `<hanghj@student.chalmers.se>` | 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. This formalization should aim to incorporate definitions and theorems that allow us to express the correpondence in the second part: Namely showing the correpondence between well-typed terms in cubical type theory as presented in Huber and Thierry's paper and with that of some concepts from Category Theory.
This latter part is not entirely clear for me yet, I know that all these are relevant keywords:
- The category, C, of names and substitutions
- Cubical Sets, i.e.: Functors from C to Set (the category of sets)
- Presheaves
- Fibers and fibrations
These are all formulated in the language of Category Theory. The purpose it to show what they correspond to in the in Cubical Type Theory. As I understand it at least the last buzzword on this list corresponds to Type Families.
I'm not sure how I'll go about expressing this in Agda. I suspect it might be a matter of demostrating that these two formulations are isomorphic.
So far I have some experience with at least expressing some categorical concepts in Agda using this new notion of equality. That is, equaility is in some sense a continuuous path from a point of some type to another. So at the moment, my understanding of cubical type theory is just that it has another notion of equality but is otherwise pretty much the same.
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]
- 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.