Monadic adjunctions🔗
An adjunction
between functors
and
is monadic if the induced comparison functor
(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 R∘L : Monad-on _ R∘L = Adjunction→Monad L⊣R open Monad-on R∘L _ = Algebra
The composition of R.₁
with
the adjunction counit
natural
transformation gives R
an Algebra
structure, thus
extending R
to a functor
Comparison-EM : Functor D (Eilenberg-Moore R∘L) Comparison-EM .F₀ x = R.₀ x , alg where alg : Algebra-on R∘L (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 (sym $ adj.counit.is-natural _ _ _)
Construction of the functorial action of Comparison-EM
Comparison-EM .F₁ x .fst = R.₁ x Comparison-EM .F₁ x .snd = R.weave (sym (adj.counit.is-natural _ _ _)) Comparison-EM .F-id = ext R.F-id Comparison-EM .F-∘ f g = ext (R.F-∘ _ _)
By construction, the composition of the comparison functor with the forgetful functor is equal to
Forget∘Comparison≡R : Forget-EM F∘ Comparison-EM ≡ R Forget∘Comparison≡R = Functor-path (λ _ → refl) (λ _ → refl)
To summarise, we have the following triangle:
An adjunction is monadic if Comparison-EM
is an equivalence of categories, thus
exhibiting
as the category of
is-monadic : Type _ is-monadic = is-equivalence Comparison-EM
We also say that the right adjoint is a monadic functor.
Monadic functors create limits🔗
By the description of limits in categories of
algebras, Forget-EM
creates limits.
Furthermore, if the adjunction
is monadic, then Comparison-EM
is an
equivalence of categories, so it also creates limits. Since this
property is closed under composition, monadic functors creates
limits.
monadic→creates-limits : ∀ {oj ℓj} {J : Precategory oj ℓj} → creates-limits-of J R monadic→creates-limits = subst (creates-limits-of _) Forget∘Comparison≡R $ F∘-creates-limits (equivalence→creates-limits monadic) (Forget-EM-creates-limits R∘L)