bsc-leon-vatthauer/README.md

BSc Leon Vatthauer

Here I am formalizing some notions of this paper https://arxiv.org/pdf/2102.11828.pdf in agda.

Running the project

TODO

Contributions to agda-categories

This project uses the awesome category theory library for agda (agda-categories), it is already very extensive, but some notions needed here are missing, so I contribute them to the library. So far the contributions are:

1. Kleisli triples [merged]
   • Categories.Monad.Construction.Kleisli
2. Distributive categories (and the relation to extensivity) [merged]
   • Categories.Category.Distributive
   • Categories.Category.Extensive.Bundle
   • Categories.Category.Extensive.Properties.Distributive
3. Commutative categories [TODO]

Goals

• Monad.Instance.Delay
  • Formalize the delay monad (as kleisli triple)
  • Show that a strong delay monad is commutative (also needs formalization of strong delay monad)
• ElgotAlgebra.agda
  • Formalize (un-)guarded elgot-algebra.
  • Show the equivalence of #-Folding and #-Compositionality in the unguarded case. (Proposition 10)
• ElgotAlgebras.agda
  • Formalize the category of elgot algebras for a given carrier.
  • Show existence of products in this category
  • Show existence of exponentials (if carrier has exponentials)
• ElgotMonad.agda [TODO]
  • Formalize (strong) (pre) elgot monad
  • Show ElgotMonad->PreElgotMonat
• Monad K [TODO]
  • Definitions using free uniform iteration algebras
  • strength
  • stable free uniform iteration algebras (and the relation to CCC carriers)
• ...
• Theorem 37 [TODO] (final goal)

Roadmap

TODO

TODOs

• Create Roadmap (find what theorem 37 depends on and then create a game plan)
• Refactor ElgotAlgebras.agda using Categories.Morphism.Reasoning (nicer proofs)