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This repository contains a formalisation of relative monads. Monads are relative monads on the identity functor so it is not technically necessary to repeat the same constructions for ordinary monads. Nevertheless I include separate implementations of ordinary monads and related constructions as a warmup.

Most of the code is quite polished. The stuff related to categories of adjunctions is still pretty gory.

Basic category theory

  • Categories
  • Initial object
  • Terminal object
  • CoProducts
  • Functors
  • Full and Faithfulness
  • Natural Transformations
  • Day Convolution
  • Yoneda Lemma

Relative monad theory

  • (Rel.) Monads
  • (Rel.) Adjunctions
  • (Rel.) Kleisli Category
  • (Rel.) Kleisli Adjunction
  • (Rel.) EM Category
  • (Rel.) EM Adjunction
  • (Rel.) Modules
  • Category of (Rel.) Adjunctions for a (Relative) Monad
  • Proof that (Rel.) Kleisli is the Initial Object in the Cat. of (Rel.) Adjs.
  • Proof that (Rel.) EM is the Terminal Object in the Cat. of (Rel.) Adjs.

Examples

  • Proof that Well-Scoped Lambda Terms are a Rel. Monad.
  • Proof that a Lambda-Model is a Rel. EM Algebra.
  • Proof that Well-Typed Lambda Terms are a Rel. Monad.
  • Proof that an evaluator for Well-Typed Lambda Terms is a Rel. EM Algebra.
  • Proof that weak arrows are relative monads on Yoneda
  • Proof that Rel. Monads on Fin are Lawvere theories
  • Proof that Algebras for Rel. Monads on Fin are models of Lawvere theories.
  • Proof that vector spaces are Rel. Monads
  • Proof that left-modules are algebras of Rel. Monads

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Relative Monad Library for Agda

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