open import Cat.Functor.Adjoint.Monad
open import Cat.Functor.Equivalence
open import Cat.Displayed.Total
open import Cat.Functor.Adjoint
open import Cat.Diagram.Monad
open import Cat.Prelude

import Cat.Functor.Reasoning
import Cat.Reasoning

open Total-hom
open Functor
open _=>_

Monadic adjunctions🔗

An adjunction $F ⊣ G$ between functors $F : C \to D$ and $G : D \to C$ is monadic if the induced comparison functor $D \to C^{R \circ L}$ (where the right-hand side is the Eilenberg-Moore category of the monad of the adjunction) is an equivalence of categories.

module
  Cat.Functor.Adjoint.Monadic
  {o₁ h₁ o₂ h₂ : _}
  {C : Precategory o₁ h₁}
  {D : Precategory o₂ h₂}
  {L : Functor C D} {R : Functor D C}
  (L⊣R : L ⊣ R)
  where

private
  module C = Cat.Reasoning C
  module D = Cat.Reasoning D
  module L = Cat.Functor.Reasoning L
  module R = Cat.Functor.Reasoning R
  module adj = _⊣_ L⊣R

L∘R : Monad C
L∘R = Adjunction→Monad L⊣R

open Monad L∘R

private
  module Kleisli = Cat.Reasoning (Kleisli L∘R)
  module EM = Cat.Reasoning (Eilenberg-Moore L∘R)

The composition of R.₁ with the adjunction counit natural transformation gives R an Algebra structure, thus extending R to a functor $D \to C^{L \circ R} .$

Comparison-EM : Functor D (Eilenberg-Moore L∘R)
Comparison-EM .F₀ x = R.₀ x , alg where
  alg : Algebra-on C L∘R (R.₀ x)
  alg .Algebra-on.ν = R.₁ (adj.counit.ε _)
  alg .Algebra-on.ν-unit = adj.zag
  alg .Algebra-on.ν-mult = R.weave (adj.counit.is-natural _ _ _)

Construction of the functorial action of Comparison-EM

Comparison-EM .F₁ x .hom = R.₁ x
Comparison-EM .F₁ x .preserves = R.weave (sym (adj.counit.is-natural _ _ _))
Comparison-EM .F-id    = ext R.F-id
Comparison-EM .F-∘ f g = ext (R.F-∘ _ _)

An adjunction is monadic if Comparison-EM is an equivalence of categories, thus exhibiting $C$ as the category of $R \circ L -algebras:$

is-monadic : Type _
is-monadic = is-equivalence Comparison-EM