2017-05-27 14:09:52 +00:00
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title: Formalizing category theory in Agda - Project description
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date: May 27th 2017
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2017-05-29 09:08:20 +00:00
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author: Frederik Hanghøj Iversen `<hanghj@student.chalmers.se>`
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2017-05-27 14:09:52 +00:00
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bibliography: refs.bib
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---
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Background
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==========
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2017-05-29 09:08:20 +00:00
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Functional extensionality gives rise to a notion of equality of functions not
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present in intensional dependent type theory. A type-system called cubical
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type-theory is outlined in [@cohen-2016] that recovers the computational interprtation of the univalence theorem.
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Keywords: The category of sets, limits, colimits, functors, natural
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transformations, kleisly category, yoneda lemma, closed cartesian categories,
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propositional logic.
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2017-05-27 14:09:52 +00:00
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Aim
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===
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2017-05-29 09:08:20 +00:00
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The aim of the project is two-fold. The first part of the project will be
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concerned with formalizing some concepts from category theory in Agda's
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type-system: functors, applicative functors, monads, etc.. The second part of
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the project could take different directions:
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* It might involve using this formalization to prove properties of functional
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programs.
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* It may be used to prove the Modal used in Cubical Type Theory using Preshiefs.
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2017-05-27 14:09:52 +00:00
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Timeplan
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========
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2017-05-29 09:08:20 +00:00
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The first part of the project will focus on studying and understanding the
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foundations for this project namely; familiarizing myself with basic concepts
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from category theory, understanding how cubical type theory gives rise to
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expressing functional extensionality and the univalence theorem.
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After I have understood these fundamental concepts I will use them in the
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formalization of functors, applicative functors, monads, etc.. in Agda. This
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should be done before the end of the first semester of the school-year
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2017/2018.
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At this point I will also have settled on a direction for the rest of the
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project and developed a time-plan for the second phase of the project. But
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cerainly it will involve applying the result of phase 1 in some context as
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mentioned in [the project aim][aim].
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2017-05-27 14:09:52 +00:00
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Resources
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=========
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* Cubical demo by Andrea Vezossi: [@cubical-demo]
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* Paper on cubical type theory [@cohen-2016]
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* Book on homotopy type theory: [@hott-2013]
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* Book on category theory: [@awodey-2006]
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2017-05-29 09:08:20 +00:00
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* Modal logic - Modal type theory, see
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[ncatlab](https://ncatlab.org/nlab/show/modal+type+theory).
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2017-05-27 14:09:52 +00:00
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References
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==========
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