open import Cat.Functor.Equivalence.Complete
open import Cat.Instances.Shape.Terminal
open import Cat.Functor.Conservative
open import Cat.Functor.Equivalence
open import Cat.Diagram.Limit.Base
open import Cat.Diagram.Terminal
open import Cat.Functor.Kan.Base
open import Cat.Diagram.Monad
open import Cat.Prelude

import Cat.Reasoning

module Cat.Diagram.Monad.Limits {o ℓ} {C : Precategory o ℓ} {M : Monad C} where

private
  module EM = Cat.Reasoning (Eilenberg-Moore C M)
  module C = Cat.Reasoning C
  module M = Monad M

open Algebra-hom
open Algebra-on

Limits in categories of algebras🔗

Suppose that $\mathcal{C}$ be a category, $M$ be a monad on $\mathcal{C}$, and $F$ be a $\mathcal{J}$-shaped diagram of $M$-algebras (that is, a functor $F : \mathcal{J} \to \mathcal{C}^M$ into the Eilenberg-Moore category of M). Suppose that an evil wizard has given you a limit for the diagram in $\mathcal{C}$ which underlies $F$, but they have not (being evil and all) told you whether $\lim F$ admits an algebra structure at all.

Perhaps we can make this situation slightly more concrete, by working in a category equivalent to an Eilenberg-Moore category: If we have two groups $G$, $H$, considered as a discrete diagram, then the limit our evil wizard would give us is the product $U(G) \times U(H)$ in $\mathbf{Sets}$. But we already know we can equip this product with a “pointwise” group structure! Does this result generalise?

The answer is positive, though this is one of those cases where abstract nonsense can not help us: gotta roll up our sleeves and calculate away. Suppose we have a diagram as in the setup — we’ll show that the functor $U : \mathcal{C}^M \to \mathcal{C}$ both preserves and reflects limits, in that $K$ is a limiting cone if, and only if, $U(K)$ is.

module _ {jo jℓ} {J : Precategory jo jℓ} (F : Functor J (Eilenberg-Moore C M)) where
  private
    module J = Precategory J
    module F = Functor F
    module FAlg j = Algebra-on (F.₀ j .snd)
  open Functor
  open _=>_

We begin with the following key lemma: Write $K : \mathcal{C}$ for the limit of a diagram $\mathcal{J} \xrightarrow{F} \mathcal{C}^M \xrightarrow{U} \mathcal{C}$. If $K$ carries an $M$-algebra structure $\nu$, and the limit projections $\psi : K \to F(j)$ are $M$-algebra morphisms, then $(K, \nu)$ is the limit of $F$ in $\mathcal{C}^M$.

  make-algebra-limit
    : ∀ {K : Functor ⊤Cat C} {eps : K F∘ !F => Forget C M F∘ F}
    → (lim : is-ran !F (Forget C M F∘ F) K eps)
    → (nu : Algebra-on C M (K .F₀ tt))
    → (∀ j → is-limit.ψ lim j C.∘ nu .ν ≡ FAlg.ν j C.∘ M.M₁ (is-limit.ψ lim j))
    → make-is-limit F (K .F₀ tt , nu)
  make-algebra-limit lim apex-alg comm = em-lim where
    module lim = is-limit lim
    open make-is-limit
    module apex = Algebra-on apex-alg
    open _=>_

The assumptions to this lemma are actually rather hefty: they pretty much directly say that $K$ was already the limit of $F$. However, with this more “elementary” rephrasing, we gain a bit of extra control which will be necessary for constructing limits in $\mathcal{C}^M$ out of limits in $\mathcal{C}$ later.

    em-lim : make-is-limit F _
    em-lim .ψ j .morphism = lim.ψ j
    em-lim .ψ j .commutes = comm j
    em-lim .commutes f    = Algebra-hom-path C (lim.commutes f)
    em-lim .universal eta p .morphism =
      lim.universal (λ j → eta j .morphism) (λ f i → p f i .morphism)
    em-lim .factors eta p =
      Algebra-hom-path C (lim.factors _ _)
    em-lim .unique eta p other q =
      Algebra-hom-path C (lim.unique _ _ _ λ j i → q j i .morphism)
    em-lim .universal eta p .commutes = lim.unique₂ _
      (λ f → C.pulll (F.F₁ f .commutes)
           ∙ C.pullr (sym (M.M-∘ _ _) ∙ ap M.M₁ (ap morphism (p f))))
      (λ j → C.pulll (lim.factors _ _)
           ∙ eta j .commutes)
      (λ j → C.pulll (comm j)
           ∙ C.pullr (sym (M.M-∘ _ _) ∙ ap M.M₁ (lim.factors _ _)))

This key lemma also doubles as a proof that the underlying object functor $U$ reflects limits: We already had an algebra structure “upstairs”!

  Forget-reflects-limits : reflects-limit (Forget C M) F
  Forget-reflects-limits {K} {eps} lim = to-is-limitp
    (make-algebra-limit lim (K .F₀ tt .snd) (λ j → eps .η j .commutes))
    (Algebra-hom-path C refl)

Having shown that $U$ reflects the property of being a limit, we now turn to showing it reflects limits in general. Using our key lemma, it will suffice to construct an algebra structure on $\lim_j UF(j)$, then show that the projection maps $\psi_j : (\lim_j UF(j)) \to UF(j)$ extend to algebra homomorphisms.

  Forget-lift-limit : Limit (Forget C M F∘ F) → Limit F
  Forget-lift-limit lim-over = to-limit $ to-is-limit $ make-algebra-limit
    (Limit.has-limit lim-over) apex-algebra (λ j → lim-over.factors _ _)
    where
    module lim-over = Limit lim-over
    module lim = Ran lim-over

An algebra structure consists, centrally, of a map $M(\lim_j UF(j)) \to \lim_j UF(j)$: a map into the limit. Maps into limits are the happy case, since $\lim_j UF(j)$ is essentially defined by a (natural) isomorphism between the sets $\mathbf{Hom}(X, \lim_j UF(j))$ and $\lim_j \mathbf{Hom}(X, UF(j))$, the latter limit being computed in $\mathbf{Sets}$. We understand limits of sets very well: they’re (subsets of) sets of tuples!

Our algebra map is thus defined componentwise: Since we have a bunch of maps $\nu_j : M(UF(j)) \to UF(j)$, coming from the algebra structures on each $F(j)$, we can “tuple” them into a big map $\nu = \langle \nu_j \psi_j \rangle _j$. Abusing notation slightly, we’re defining $\nu(a, b, ...)$ to be $(\nu_a(a), \nu_b(b), ...)$.

    apex-algebra : Algebra-on C M lim-over.apex
    apex-algebra .ν =
      lim-over.universal (λ j → FAlg.ν j C.∘ M.M₁ (lim-over.ψ j)) comm where abstract
      comm : ∀ {x y} (f : J.Hom x y)
            → F.₁ f .morphism C.∘ FAlg.ν x C.∘ M.M₁ (lim-over.ψ x)
            ≡ FAlg.ν y C.∘ M.M₁ (lim-over.ψ y)
      comm {x} {y} f =
        F.₁ f .morphism C.∘ FAlg.ν x C.∘ M.M₁ (lim-over.ψ x)        ≡⟨ C.extendl (F.₁ f .commutes) ⟩≡
        FAlg.ν y C.∘ M.M₁ (F.₁ f .morphism) C.∘ M.M₁ (lim-over.ψ x) ≡˘⟨ C.refl⟩∘⟨ M.M-∘ _ _ ⟩≡˘
        FAlg.ν y C.∘ M.M₁ (F.₁ f .morphism C.∘ lim-over.ψ x)        ≡⟨ C.refl⟩∘⟨ ap M.M₁ (lim-over.commutes f) ⟩≡
        FAlg.ν y C.∘ M.M₁ (lim-over.ψ y)                            ∎

To show that $\nu$ really is an algebra structure, we’ll reason componentwise, too: we must show that $\nu(\eta_a, \eta_b, ...)$ is the identity map: but we can compute

$$\nu(\eta_a, \eta_b, ...) = (\nu_a\eta_a, \nu_b\eta_b, ...) = (\mathrm{id}_{}, \mathrm{id}_{}, ...) = \mathrm{id}_{}\text{!}$$

    apex-algebra .ν-unit = lim-over.unique₂ _ lim-over.commutes
      (λ j → C.pulll (lim-over.factors _ _)
          ·· C.pullr (sym $ M.unit.is-natural _ _ _)
          ·· C.cancell (FAlg.ν-unit j))
      (λ j → C.idr _)
    apex-algebra .ν-mult = lim-over.unique₂ _
      (λ f → C.pulll $ C.pulll (F.₁ f .commutes)
           ∙ C.pullr (sym (M.M-∘ _ _) ∙ ap M.M₁ (lim-over.commutes f)))
      (λ j → C.pulll (lim-over.factors _ _)
          ·· C.pullr (sym (M.M-∘ _ _) ∙ ap M.M₁ (lim-over.factors _ _) ∙ M.M-∘ _ _)
          ·· C.extendl (FAlg.ν-mult j)
          ·· ap (FAlg.ν j C.∘_) (M.mult.is-natural _ _ _)
          ·· C.assoc _ _ _)
      (λ j → C.pulll (lim-over.factors _ _))

Putting our previous results together, we conclude by observing that, if $\mathcal{C}$ is a complete category, then so is $\mathcal{C}^M$, regardless of how ill-behaved the monad $M$ might be.

Eilenberg-Moore-is-complete
  : ∀ {a b} → is-complete a b C → is-complete a b (Eilenberg-Moore _ M)
Eilenberg-Moore-is-complete complete F =
  Forget-lift-limit F (complete _)