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