module Cat.Monoidal.Strength where

Strong functors🔗

A left strength for a functor on a monoidal category is a natural transformation

interacting nicely with the left unitor and associator.

  record Left-strength : Type (o ⊔ ℓ) where
    field
      left-strength : -⊗- F∘ (Id F× F) => F F∘ -⊗-

    module σ = _=>_ left-strength

    σ : ∀ {A B} → Hom (A ⊗ F₀ B) (F₀ (A ⊗ B))
    σ = σ.η _

    field
      left-strength-λ← : ∀ {A} → F₁ (λ← {A}) ∘ σ ≡ λ←
      left-strength-α→ : ∀ {A B C}
        → F₁ (α→ A B C) ∘ σ ≡ σ ∘ (id ⊗₁ σ) ∘ α→ A B (F₀ C)

Reversely1, a right strength is a natural transformation

interacting nicely with the right unitor and associator.

  record Right-strength : Type (o ⊔ ℓ) where
    field
      right-strength : -⊗- F∘ (F F× Id) => F F∘ -⊗-

    module τ = _=>_ right-strength

    τ : ∀ {A B} → Hom (F₀ A ⊗ B) (F₀ (A ⊗ B))
    τ = τ.η _

    field
      right-strength-ρ← : ∀ {A} → F₁ (ρ← {A}) ∘ τ ≡ ρ←
      right-strength-α← : ∀ {A B C}
        → F₁ (α← A B C) ∘ τ ≡ τ ∘ (τ ⊗₁ id) ∘ α← (F₀ A) B C

A strength for is a pair of a left strength and a right strength inducing a single operation i.e. making the following diagram commute:

  record Strength : Type (o ⊔ ℓ) where
    field
      strength-left : Left-strength
      strength-right : Right-strength

    open Left-strength strength-left public
    open Right-strength strength-right public

    field
      strength-α→ : ∀ {A B C}
        → F₁ (α→ A B C) ∘ τ ∘ (σ ⊗₁ id) ≡ σ ∘ (id ⊗₁ τ) ∘ α→ A (F₀ B) C

A functor equipped with a strength is called a strong functor.

Symmetry🔗

In a symmetric monoidal category (or even just a braided monoidal category, if one is careful about directions), there is an equivalence between the notions of left and right strength: we can obtain one from the other by “conjugating” with the braiding, as illustrated by this diagram.

Therefore, the literature usually speaks of “strength” in a symmetric monoidal category to mean either a left or a right strength, but note that this is not quite the same as a Strength as defined above, which has left and right strengths not necessarily related by the braiding. If they are, we will say that the strength is symmetric; such a strength contains exactly the information of a left (or right) strength.

    is-symmetric-strength : Strength → Type (o ⊔ ℓ)
    is-symmetric-strength s = ∀ {A B} → τ {A} {B} ∘ β→ ≡ F₁ β→ ∘ σ
      where open Strength s

The construction of the equivalence between left and right strengths is extremely tedious, so we leave the details to the curious reader.

    left≃right : Iso Left-strength Right-strength
    left≃right .fst l = r where
      open Left-strength l
      open Right-strength
      r : Right-strength
      r .right-strength .η _ = F₁ β→ ∘ σ ∘ β←
      r .right-strength .is-natural _ _ (f , g) =
        (F₁ β→ ∘ σ ∘ β←) ∘ (F₁ f ⊗₁ g) ≡⟨ pullr (pullr (β←.is-natural _ _ _)) ⟩≡
        F₁ β→ ∘ σ ∘ (g ⊗₁ F₁ f) ∘ β←   ≡⟨ refl⟩∘⟨ extendl (σ.is-natural _ _ _) ⟩≡
        F₁ β→ ∘ F₁ (g ⊗₁ f) ∘ σ ∘ β←   ≡⟨ F.extendl (β→.is-natural _ _ _) ⟩≡
        F₁ (f ⊗₁ g) ∘ F₁ β→ ∘ σ ∘ β←   ∎
      r .right-strength-ρ← =
        F₁ ρ← ∘ F₁ β→ ∘ σ ∘ β← ≡⟨ F.pulll ρ←-β→ ⟩≡
        F₁ λ← ∘ σ ∘ β←         ≡⟨ pulll left-strength-λ← ⟩≡
        λ← ∘ β←                ≡⟨ λ←-β← ⟩≡
        ρ←                     ∎
      r .right-strength-α← =
        F₁ (α← _ _ _) ∘ F₁ β→ ∘ σ ∘ β←                                       ≡⟨ refl⟩∘⟨ refl⟩∘⟨ pushl3 (sym (lswizzle (σ.is-natural _ _ _) (F.annihilate (◀.annihilate (β≅ .invl))))) ⟩≡
        F₁ (α← _ _ _) ∘ F₁ β→ ∘ F₁ (β→ ⊗₁ id) ∘ σ ∘ (β← ⊗₁ ⌜ F₁ id ⌝) ∘ β←   ≡⟨ ap! F-id ⟩≡
        F₁ (α← _ _ _) ∘ F₁ β→ ∘ F₁ (β→ ⊗₁ id) ∘ σ ∘ (β← ⊗₁ id) ∘ β←          ≡⟨ F.extendl3 (sym β→-id⊗β→-α→) ⟩≡
        F₁ β→ ∘ F₁ (id ⊗₁ β→) ∘ F₁ (α→ _ _ _) ∘ σ ∘ (β← ⊗₁ id) ∘ β←          ≡⟨ refl⟩∘⟨ refl⟩∘⟨ pulll left-strength-α→ ⟩≡
        F₁ β→ ∘ F₁ (id ⊗₁ β→) ∘ (σ ∘ (id ⊗₁ σ) ∘ α→ _ _ _) ∘ (β← ⊗₁ id) ∘ β← ≡⟨ refl⟩∘⟨ refl⟩∘⟨ pullr (pullr refl) ⟩≡
        F₁ β→ ∘ F₁ (id ⊗₁ β→) ∘ σ ∘ (id ⊗₁ σ) ∘ α→ _ _ _ ∘ (β← ⊗₁ id) ∘ β←   ≡⟨ refl⟩∘⟨ pushr (pushr (refl⟩∘⟨ sym β←-β←⊗id-α←)) ⟩≡
        F₁ β→ ∘ (F₁ (id ⊗₁ β→) ∘ σ ∘ (id ⊗₁ σ)) ∘ β← ∘ (β← ⊗₁ id) ∘ α← _ _ _ ≡˘⟨ refl⟩∘⟨ pulll (▶.shufflel (σ.is-natural _ _ _)) ⟩≡˘
        F₁ β→ ∘ σ ∘ (id ⊗₁ (F₁ β→ ∘ σ)) ∘ β← ∘ (β← ⊗₁ id) ∘ α← _ _ _         ≡⟨ pushr (pushr (extendl (sym (β←.is-natural _ _ _)))) ⟩≡
        (F₁ β→ ∘ σ ∘ β←) ∘ ((F₁ β→ ∘ σ) ⊗₁ id) ∘ (β← ⊗₁ id) ∘ α← _ _ _       ≡⟨ refl⟩∘⟨ ◀.pulll (sym (assoc _ _ _)) ⟩≡
        (F₁ β→ ∘ σ ∘ β←) ∘ ((F₁ β→ ∘ σ ∘ β←) ⊗₁ id) ∘ α← _ _ _               ∎
    left≃right .snd .inv r = l where
      open Right-strength r
      open Left-strength
      l : Left-strength
      l .left-strength .η _ = F₁ β← ∘ τ ∘ β→
      l .left-strength .is-natural _ _ (f , g) =
        (F₁ β← ∘ τ ∘ β→) ∘ (f ⊗₁ F₁ g) ≡⟨ pullr (pullr (β→.is-natural _ _ _)) ⟩≡
        F₁ β← ∘ τ ∘ (F₁ g ⊗₁ f) ∘ β→   ≡⟨ refl⟩∘⟨ extendl (τ.is-natural _ _ _) ⟩≡
        F₁ β← ∘ F₁ (g ⊗₁ f) ∘ τ ∘ β→   ≡⟨ F.extendl (β←.is-natural _ _ _) ⟩≡
        F₁ (f ⊗₁ g) ∘ F₁ β← ∘ τ ∘ β→   ∎
      l .left-strength-λ← =
        F₁ λ← ∘ F₁ β← ∘ τ ∘ β→ ≡⟨ F.pulll λ←-β← ⟩≡
        F₁ ρ← ∘ τ ∘ β→         ≡⟨ pulll right-strength-ρ← ⟩≡
        ρ← ∘ β→                ≡⟨ ρ←-β→ ⟩≡
        λ←                     ∎
      l .left-strength-α→ =
        F₁ (α→ _ _ _) ∘ F₁ β← ∘ τ ∘ β→                                       ≡⟨ refl⟩∘⟨ refl⟩∘⟨ pushl3 (sym (lswizzle (τ.is-natural _ _ _) (F.annihilate (▶.annihilate (β≅ .invr))))) ⟩≡
        F₁ (α→ _ _ _) ∘ F₁ β← ∘ F₁ (id ⊗₁ β←) ∘ τ ∘ (⌜ F₁ id ⌝ ⊗₁ β→) ∘ β→   ≡⟨ ap! F-id ⟩≡
        F₁ (α→ _ _ _) ∘ F₁ β← ∘ F₁ (id ⊗₁ β←) ∘ τ ∘ (id ⊗₁ β→) ∘ β→          ≡⟨ F.extendl3 ((refl⟩∘⟨ β←.is-natural _ _ _) ∙ sym β←-β←⊗id-α←) ⟩≡
        F₁ β← ∘ F₁ (β← ⊗₁ id) ∘ F₁ (α← _ _ _) ∘ τ ∘ (id ⊗₁ β→) ∘ β→          ≡⟨ refl⟩∘⟨ refl⟩∘⟨ pulll right-strength-α← ⟩≡
        F₁ β← ∘ F₁ (β← ⊗₁ id) ∘ (τ ∘ (τ ⊗₁ id) ∘ α← _ _ _) ∘ (id ⊗₁ β→) ∘ β→ ≡⟨ refl⟩∘⟨ refl⟩∘⟨ pullr (pullr refl) ⟩≡
        F₁ β← ∘ F₁ (β← ⊗₁ id) ∘ τ ∘ (τ ⊗₁ id) ∘ α← _ _ _ ∘ (id ⊗₁ β→) ∘ β→   ≡⟨ refl⟩∘⟨ pushr (pushr (refl⟩∘⟨ ((refl⟩∘⟨ sym (β→.is-natural _ _ _)) ∙ sym β→-id⊗β→-α→))) ⟩≡
        F₁ β← ∘ (F₁ (β← ⊗₁ id) ∘ τ ∘ (τ ⊗₁ id)) ∘ β→ ∘ (id ⊗₁ β→) ∘ α→ _ _ _ ≡˘⟨ refl⟩∘⟨ pulll (◀.shufflel (τ.is-natural _ _ _)) ⟩≡˘
        F₁ β← ∘ τ ∘ ((F₁ β← ∘ τ) ⊗₁ id) ∘ β→ ∘ (id ⊗₁ β→) ∘ α→ _ _ _         ≡⟨ pushr (pushr (extendl (sym (β→.is-natural _ _ _)))) ⟩≡
        (F₁ β← ∘ τ ∘ β→) ∘ (id ⊗₁ (F₁ β← ∘ τ)) ∘ (id ⊗₁ β→) ∘ α→ _ _ _       ≡⟨ refl⟩∘⟨ ▶.pulll (sym (assoc _ _ _)) ⟩≡
        (F₁ β← ∘ τ ∘ β→) ∘ (id ⊗₁ (F₁ β← ∘ τ ∘ β→)) ∘ α→ _ _ _               ∎
    left≃right .snd .rinv r = Right-strength-path $ ext λ (A , B) →
      F₁ β→ ∘ (F₁ β← ∘ τ ∘ β→) ∘ β← ≡⟨ extendl (F.cancell (β≅ .invl)) ⟩≡
      τ ∘ β→ ∘ β←                   ≡⟨ elimr (β≅ .invl) ⟩≡
      τ                             ∎
      where open Right-strength r
    left≃right .snd .linv l = Left-strength-path $ ext λ (A , B) →
      F₁ β← ∘ (F₁ β→ ∘ σ ∘ β←) ∘ β→ ≡⟨ extendl (F.cancell (β≅ .invr)) ⟩≡
      σ ∘ β← ∘ β→                   ≡⟨ elimr (β≅ .invr) ⟩≡
      σ                             ∎
      where open Left-strength l

Duality🔗

As hinted to above, a right strength for on can equivalently be defined as a left strength on the reverse monoidal category It is entirely trivial to show that the two definitions are equivalent:

  strength^rev : Left-strength (M ^rev) F ≃ Right-strength M F
  strength^rev = Iso→Equiv is where
    is : Iso _ _
    is .fst l = record
      { right-strength = NT (λ _ → σ) λ _ _ _ → σ.is-natural _ _ _
      ; right-strength-ρ← = left-strength-λ←
      ; right-strength-α← = left-strength-α→
      } where open Left-strength l
    is .snd .inv r = record
      { left-strength = NT (λ _ → τ) λ _ _ _ → τ.is-natural _ _ _
      ; left-strength-λ← = right-strength-ρ←
      ; left-strength-α→ = right-strength-α←
      } where open Right-strength r
    is .snd .rinv _ = Right-strength-path _ _ trivial!
    is .snd .linv _ = Left-strength-path _ _ trivial!

Sets-endofunctors are strong🔗

Every endofunctor on seen as a cartesian monoidal category, can be equipped with a canonical symmetric strength: the tensor product is the actual product of sets, so, given we can simply apply the functorial action of on the function yielding a function

  Sets-strength : Left-strength Setsₓ F
  Sets-strength .left-strength .η (A , B) (a , Fb) = F₁ (a ,_) Fb
  Sets-strength .left-strength .is-natural (A , B) (C , D) (ac , bd) =
    ext λ a Fb → (sym (F-∘ _ _) ∙ F-∘ _ _) $ₚ Fb
  Sets-strength .left-strength-λ← =
    ext λ _ Fa → (sym (F-∘ _ _) ∙ F-id) $ₚ Fa
  Sets-strength .left-strength-α→ =
    ext λ a b Fc → (sym (F-∘ _ _) ∙ F-∘ _ _) $ₚ Fc

This is an instance of a more general fact: in a closed monoidal category (that is, one with an adjunction for example coming from a cartesian closed category), left strengths for endofunctors are equivalent to enrichments of F: that is, natural transformations

internalising the functorial action Then, what we have shown boils down to the fact that every endofunctor on is trivially


  1. That is, on the other side of the reverse monoidal category duality.↩︎