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Equations Inline Footnotes

open import Cat.Displayed.Instances.Factorisations
open import Cat.Morphism.Factorisation.Algebraic
open import Cat.Instances.Shape.Interval
open import Cat.Displayed.Section
open import Cat.Diagram.Comonad
open import Cat.Morphism.Lifts
open import Cat.Prelude

import Cat.Monoidal.Instances.Factorisations as Ffm
import Cat.Functor.Bifunctor as Bi
import Cat.Reasoning

open is-comonad hiding (δ)
open Lwfs-on hiding (δ)
open Section
open Functor
open _=>s_
open _=>_

module Cat.Monoidal.Instances.Factorisations.Left
  {o ℓ} (C : Precategory o ℓ) where

private module Arr = Cat.Reasoning (Arr C)
open Cat.Reasoning C
open Ffm C

Monoidal structure on lwfs🔗

The monoidal category structure on functorial factorisations lifts to left weak factorisation structures.

The comultiplication on the tensor unit is trivial.

Lwfs-on-unit : Lwfs-on Ff-unit
Lwfs-on-unit .L-δ .η (X , Y , f) = Arr.id
Lwfs-on-unit .L-δ .is-natural x y f = ext (id-comm-sym ,ₚ id-comm-sym)
Lwfs-on-unit .L-comonad .δ-unitl = ext (idl id ,ₚ idl id)
Lwfs-on-unit .L-comonad .δ-unitr = ext (idl id ,ₚ idl id)
Lwfs-on-unit .L-comonad .δ-assoc = ext refl

The comultiplication on tensors is where most of the work goes.

module _ {F G : Factorisation C} (Fl : Lwfs-on F) (Gl : Lwfs-on G) where

  private
    module FG = Factorisation (F ⊗ᶠᶠ G)
    module F = Lwfs Fl
    module G = Lwfs Gl
    open F renaming (ρ→ to ρᶠ ; λ→ to λᶠ) using ()
    open G renaming (ρ→ to ρᵍ ; λ→ to λᵍ) using ()

    λλ : ∀ {x y} (f : Hom x y) → Hom x (G.Mid₀ (ρᶠ f))
    λλ f = λᵍ (ρᶠ f) ∘ λᶠ f

    λ→λλ : ∀ {x y} (f : Hom x y) → Homᵃ C (λᶠ f) (λλ f)
    λ→λλ f .top = id
    λ→λλ f .bot = λᵍ (ρᶠ f)
    λ→λλ f .com = elimr refl

    δ-sq : ∀ {x y} (f : Hom x y) → Homᵃ C (λᵍ (ρᶠ f)) (ρᶠ (λλ f))
    δ-sq f using (llift , α , β) ← F.lift-λρ (λ→λλ f) = record
      { top = llift
      ; bot = id
      ; com = β ∙ introl refl
      }

The component of the comultiplication $\delta$ at a map $X\stackrel{f}{\to }Y$ is given by the dashed arrow $l_2$ in the diagram below. The auxiliary arrow $l_1$ serves primarily to establish that everything necessary commutes. Note that the ellipse with faces ${\rho }_{{\rho }_{\dots }}$ and $l_2$ is only the identity on $G({\rho }_f)\text{,}$ and this gives one of the counit laws for $\delta \text{.}$

    lδ : FG.L => (FG.L F∘ FG.L)
    lδ .η (X , Y , f) = record where
      (l1  , α , _) = F.lift-λρ (λ→λλ f)
      (bot , β , _) = G.lift-λρ (δ-sq f)

      top = id
      com : (λᵍ (ρᶠ (λλ f)) ∘ λᶠ (λλ f)) ∘ id ≡ bot ∘ λᵍ (ρᶠ f) ∘ λᶠ f
      com = sym (pulll β ∙∙ pullr α ∙∙ pulll refl)

    lδ .is-natural x y f = ext $ id-comm-sym ,ₚ
        pullr (sym (G.δ-nat _))
      ∙ extendl (G.weave (ext
          ( pullr (sym (F.δ-nat _))
          ∙ extendl (F.weave (ext (id-comm-sym ,ₚ G .S₁ _ .sq₀)))
          ,ₚ id-comm-sym)))

As indicated, we can obtain one of the counit laws by pondering the orb contained in the diagram, and the other counit law pops up by calculating until we can apply $F$ and $G\text{’s}$ unit laws in turn. The only lengthy calculation is associativity, but the idea is the same: calculate to expose opportunities to apply the associativity of both factors.

  Lwfs-on-tensor : Lwfs-on (F ⊗ᶠᶠ G)
  Lwfs-on-tensor .L-δ = lδ
  Lwfs-on-tensor .L-comonad .δ-assoc {x , y , f} = ext $ refl ,ₚ sym p1 where
    p0 =
      F.map (λ→λλ (λλ f)) ∘ F.δ (λλ f) ∘ F.map (λ→λλ f) ∘ F.δ f
        ≡⟨ ap₂ _∘_ refl (pulll (sym (F.δ-nat _)) ∙ pullr (sym F.δ-assoc)) ⟩≡
      F.map (λ→λλ (λλ f)) ∘ F.map (F.L₁ (λ→λλ f)) ∘ F.map (F.δˢ f) ∘ F.δ f
        ≡⟨ pulll (F.collapse refl) ⟩≡
      F.map (λ→λλ (λλ f) Arr.∘ F.L₁ (λ→λλ f)) ∘ F.map (F.δˢ f) ∘ F.δ f
        ≡⟨ extendl (F.weave (ext (elimr F.δ-top ,ₚ pullr refl ∙ sym (G.lift-λρ (δ-sq f) .snd .fst)))) ⟩≡
      F.map (lδ .η (_ , _ , f)) ∘ F.map (λ→λλ f) ∘ F.δ f
        ∎

    p1 =
      (G.map (δ-sq (λλ f)) ∘ G.δ (ρᶠ (λλ f))) ∘ G.map (δ-sq f) ∘ G.δ (ρᶠ f)
        ≡⟨ pullr (extendl (sym (G.δ-nat _))) ⟩≡
      G.map (δ-sq (λλ f)) ∘ G.map (G.L₁ (δ-sq f)) ∘ G.δ (λᵍ (ρᶠ f)) ∘ G.δ (ρᶠ f)
        ≡⟨ ap₂ _∘_ refl (ap₂ _∘_ refl (sym G.δ-assoc)) ⟩≡
      G.map (δ-sq (λλ f)) ∘ G.map (G.L₁ (δ-sq f)) ∘ G.map (G.δˢ (ρᶠ f)) ∘ G.δ (ρᶠ f)
        ≡⟨ ap₂ _∘_ refl (pulll (G.collapse refl)) ⟩≡
      G.map (δ-sq (λλ f)) ∘ G.map (G.L₁ (δ-sq f) Arr.∘ G.δˢ (ρᶠ f)) ∘ G.δ (ρᶠ f)
        ≡⟨ extendl (G.weave (ext (pullr (ap₂ _∘_ refl (elimr G.δ-top)) ∙ p0 ,ₚ pulll (eliml refl) ∙ intror refl))) ⟩≡
      G.map _ ∘ G.map (δ-sq f) ∘ G.δ (ρᶠ f)
        ∎

  Lwfs-on-tensor .L-comonad .δ-unitl {x , y , f} = ext $ idl id ,ₚ
    pulll (G.collapse (ext (pulll (F.collapse
      (ext (idl id ,ₚ sym (G.factors (ρᶠ f))))) ∙ F.δ-unitl ,ₚ idr _)))
    ∙ G.δ-unitl

  Lwfs-on-tensor .L-comonad .δ-unitr {x , y , f} = ext $ idl id ,ₚ
    G.lift-λρ (δ-sq f) .snd .snd