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Monad.lagda.md
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<!--
```agda
open import Cat.Monoidal.Instances.Cartesian
open import Cat.Functor.Coherence
open import Cat.Instances.Product
open import Cat.Monoidal.Strength
open import Cat.Monoidal.Braided
open import Cat.Monoidal.Functor
open import Cat.Monoidal.Reverse
open import Cat.Diagram.Monad
open import Cat.Monoidal.Base
open import Cat.Functor.Base
open import Cat.Prelude
import Cat.Functor.Reasoning
import Cat.Reasoning
```
-->
```agda
module Cat.Monoidal.Strength.Monad where
```
# Strong monads {defines="strong-monad monad-strength left-monad-strength right-monad-strength"}
Recall that a (left) [[strength]] for an endofunctor $M : \cC \to
\cC$ on a [[monoidal category]] consists of a natural transformation
$A \otimes MB \to M (A \otimes B)$ allowing us to "slide" things from
the ambient context into the functor. If $M$ is equipped with the
structure of a [[monad]], then it is natural to refine this notion to be
compatible with the monad, yielding the notion of a (left/right-)
**strong monad**.
<!--
```agda
module _ {o ℓ} {C : Precategory o ℓ} (Cᵐ : Monoidal-category C) (monad : Monad C) where
open Cat.Reasoning C
open Monoidal-category Cᵐ
open Monad monad
```
-->
```agda
record Left-monad-strength : Type (o ⊔ ℓ) where
field
functor-strength : Left-strength Cᵐ M
open Left-strength functor-strength public
field
left-strength-η : ∀ {A B} → σ ∘ (id ⊗₁ η B) ≡ η (A ⊗ B)
left-strength-μ : ∀ {A B} → σ ∘ (id ⊗₁ μ B) ≡ μ (A ⊗ B) ∘ M₁ σ ∘ σ
record Right-monad-strength : Type (o ⊔ ℓ) where
field
functor-strength : Right-strength Cᵐ M
open Right-strength functor-strength public
field
right-strength-η : ∀ {A B} → τ ∘ (η A ⊗₁ id) ≡ η (A ⊗ B)
right-strength-μ : ∀ {A B} → τ ∘ (μ A ⊗₁ id) ≡ μ (A ⊗ B) ∘ M₁ τ ∘ τ
record Monad-strength : Type (o ⊔ ℓ) where
field
strength-left : Left-monad-strength
strength-right : Right-monad-strength
open Left-monad-strength strength-left hiding (functor-strength) public
open Right-monad-strength strength-right hiding (functor-strength) public
field
strength-α→ : ∀ {A B C}
→ M₁ (α→ A B C) ∘ τ ∘ (σ ⊗₁ id) ≡ σ ∘ (id ⊗₁ τ) ∘ α→ A (M₀ B) C
```
Strong monads are of particular importance in the semantics of effectful
programming languages: while monads are used to model effects, they do
not capture the fact that monadic computations can make use of
information from the *context*; for example, consider the following
pseudo-Haskell program (in `do` notation, then in two possible
desugared forms):
<div class="mathpar">
```haskell
do
a ← ma ∷ M A
b ← mb ∷ M B
pure (a , b)
```
$=$
```haskell
ma >>= λ a →
mb >>= λ b →
pure (a , b)
```
$=$
```haskell
join (fmap (λ a →
fmap (λ b →
(a , b)) mb) ma)
```
</div>
Notice that `mb`, and then `a`, are available *under*
$\lambda$-*abstractions*: this is no problem in a functional programming
language like Haskell, because monads are automatically *enriched* in
the sense that the functorial action `fmap` is an *internal* morphism;
in other words, [[$\Sets$-monads are strong]]. But the
mathematical denotation of the above program in a general monoidal
category makes crucial use of the strengths, as we will see below.
With this perspective in mind, the additional coherences imposed on a
*monad* strength are quite natural: using the strength to slide into a
"pure" computation (that is, one in the image of the unit) should yield
a pure computation, and using the strength twice before multiplying
should be the same as using it once after multiplying: they express a
sort of "internal naturality" condition for the unit and multiplication
with respect to the enrichment induced by the strength.
<!--
```agda
functor-strength : Strength Cᵐ M
functor-strength .Strength.strength-left = strength-left .Left-monad-strength.functor-strength
functor-strength .Strength.strength-right = strength-right .Right-monad-strength.functor-strength
functor-strength .Strength.strength-α→ = strength-α→
private unquoteDecl left-eqv = declare-record-iso left-eqv (quote Left-monad-strength)
Left-monad-strength-path
: ∀ {a b}
→ a .Left-monad-strength.functor-strength ≡ b .Left-monad-strength.functor-strength
→ a ≡ b
Left-monad-strength-path p = Iso.injective left-eqv (Σ-prop-path (λ _ → hlevel 1) p)
private unquoteDecl right-eqv = declare-record-iso right-eqv (quote Right-monad-strength)
Right-monad-strength-path
: ∀ {a b}
→ a .Right-monad-strength.functor-strength ≡ b .Right-monad-strength.functor-strength
→ a ≡ b
Right-monad-strength-path p = Iso.injective right-eqv (Σ-prop-path (λ _ → hlevel 1) p)
private unquoteDecl strength-eqv = declare-record-iso strength-eqv (quote Monad-strength)
Monad-strength-path
: ∀ {a b}
→ a .Monad-strength.strength-left ≡ b .Monad-strength.strength-left
→ a .Monad-strength.strength-right ≡ b .Monad-strength.strength-right
→ a ≡ b
Monad-strength-path p q = Iso.injective strength-eqv (Σ-pathp p (Σ-prop-pathp (λ _ _ → hlevel 1) q))
```
-->
## Monoidal functors from strong monads {defines="monoidal-functors-from-strong-monads"}
<!--
```agda
module _ {o ℓ}
{C : Precategory o ℓ} {Cᵐ : Monoidal-category C}
{monad : Monad C}
where
open Cat.Reasoning C
open Monoidal-category Cᵐ
open Monad monad
private
module M = Cat.Functor.Reasoning M
open is-iso
module _ (s : Monad-strength Cᵐ monad) where
open Monad-strength s
open Lax-monoidal-functor-on
```
-->
The above program wasn't picked at random -- it witnesses the common
functional programming wisdom that "every monad is an applicative
functor[^applicative]", whose theoretical underpinning is that, given a
*strong* monad $M$, we can equip $M$ with the structure of a [[lax monoidal
functor]].
[^applicative]: Applicative functors, or *idioms*, are usually defined
as [[lax monoidal functors]] equipped with a compatible strength (not to
be confused with [[strong monoidal functors]]).
In fact, we can do so in *two* different ways, corresponding to
sequencing the effects from left to right or from right to left:
~~~{.quiver}
\[\begin{tikzcd}
& {M (A \otimes MB)} & {M^2 (A \otimes B)} \\
{MA \otimes MB} &&& {M (A \otimes B)} \\
& {M (MA \otimes B)} & {M^2 (A \otimes B)}
\arrow["\tau", from=2-1, to=1-2]
\arrow["M\sigma", from=1-2, to=1-3]
\arrow["\mu", from=1-3, to=2-4]
\arrow["\sigma"', from=2-1, to=3-2]
\arrow["M\tau"', from=3-2, to=3-3]
\arrow["\mu"', from=3-3, to=2-4]
\end{tikzcd}\]
~~~
```agda
left-φ right-φ : -⊗- F∘ (M F× M) => M F∘ -⊗-
left-φ = (mult ◂ -⊗-) ∘nt nat-assoc-to (M ▸ left-strength) ∘nt right-strength'
where
unquoteDecl right-strength' =
cohere-into right-strength' (-⊗- F∘ (M F× M) => M F∘ -⊗- F∘ (Id F× M))
(right-strength ◂ (Id F× M))
right-φ = (mult ◂ -⊗-) ∘nt nat-assoc-to (M ▸ right-strength) ∘nt left-strength'
where
unquoteDecl left-strength' =
cohere-into left-strength' (-⊗- F∘ (M F× M) => M F∘ -⊗- F∘ (M F× Id))
(left-strength ◂ (M F× Id))
```
::: {.definition #commutative-monad alias="commutative-strength"}
If the two ways are the same (thus if the above diagram commutes), we say
that the monad (or the strength) is **commutative**.
:::
```agda
is-commutative-strength : Type (o ⊔ ℓ)
is-commutative-strength = right-φ ≡ left-φ
```
<details>
<summary>
We now complete the definition of the *left-to-right* monoidal structure,
which requires a bit of work. For the unit, we pick $\eta_1$, the unit
of the monad.
```agda
strength→monoidal : Lax-monoidal-functor-on Cᵐ Cᵐ M
strength→monoidal .ε = η Unit
strength→monoidal .F-mult = left-φ
```
</summary>
The associator coherence is witnessed by the following ~~monstrosity~~
commutative diagram.
~~~{.quiver}
\[\begin{tikzcd}[column sep=0.4em]
{(MA\otimes MB)\otimes MC} &&&& {MA\otimes(MB\otimes MC)} \\
{M(A\otimes MB)\otimes MC} & {M((A\otimes MB)\otimes MC)} & {M(A\otimes (MB\otimes MC))} && {MA\otimes M(B\otimes MC)} \\
{M^2(A\otimes B)\otimes MC} & {M(M(A\otimes B)\otimes MC)} & {M(A\otimes M(B\otimes MC))} & {M(A\otimes M^2(B\otimes C))} & {MA\otimes M^2(B\otimes C)} \\
{M(A\otimes B)\otimes MC} & {M^2((A\otimes B)\otimes MC)} & {M^2(A\otimes(B\otimes MC))} & {M^2(A\otimes M(B\otimes C))} & {MA\otimes M(B\otimes C)} \\
{M((A\otimes B)\otimes MC)} && {M(A\otimes (B\otimes MC))} & {M^3(A\otimes (B\otimes C))} & {M(A\otimes M(B\otimes C))} \\
&& {M(A\otimes M(B\otimes C))} \\
{M^2((A\otimes B)\otimes C)} &&&& {M^2(A\otimes (B\otimes C))} \\
{M((A\otimes B)\otimes C)} &&&& {M(A\otimes (B\otimes C))}
\arrow[from=1-1, to=1-5]
\arrow[from=1-1, to=2-1]
\arrow[from=2-1, to=3-1]
\arrow[from=3-1, to=4-1]
\arrow[from=4-1, to=5-1]
\arrow[from=5-1, to=7-1]
\arrow[from=7-1, to=8-1]
\arrow[from=8-1, to=8-5]
\arrow[from=1-5, to=2-5]
\arrow[from=2-5, to=3-5]
\arrow[from=3-5, to=4-5]
\arrow[from=4-5, to=5-5]
\arrow[from=5-5, to=7-5]
\arrow[from=7-5, to=8-5]
\arrow[from=2-1, to=2-2]
\arrow[from=1-5, to=2-3]
\arrow[from=2-2, to=2-3]
\arrow[from=2-5, to=3-3]
\arrow[from=2-3, to=3-3]
\arrow[from=3-5, to=3-4]
\arrow[from=3-3, to=3-4]
\arrow[from=3-4, to=5-5]
\arrow[from=2-2, to=3-2]
\arrow[from=3-1, to=3-2]
\arrow[from=3-3, to=4-3]
\arrow[from=3-2, to=4-2]
\arrow[from=4-2, to=4-3]
\arrow[from=4-2, to=5-1]
\arrow[from=5-1, to=5-3]
\arrow[from=4-3, to=5-3]
\arrow[from=7-1, to=7-5]
\arrow[from=4-3, to=4-4]
\arrow[from=3-4, to=4-4]
\arrow[from=5-3, to=6-3]
\arrow[from=6-3, to=7-5]
\arrow[from=4-4, to=5-4]
\arrow["\mu"', curve={height=6pt}, from=5-4, to=7-5]
\arrow[from=4-4, to=6-3]
\arrow["M\mu", curve={height=-6pt}, from=5-4, to=7-5]
\end{tikzcd}\]
~~~
```agda
strength→monoidal .F-α→ =
M₁ (α→ _ _ _) ∘ (μ _ ∘ M₁ σ ∘ τ) ∘ ((μ _ ∘ M₁ σ ∘ τ) ⊗₁ id) ≡⟨ pulll (extendl (sym (mult.is-natural _ _ _))) ⟩
(μ _ ∘ M₁ (M₁ (α→ _ _ _)) ∘ M₁ σ ∘ τ) ∘ ((μ _ ∘ M₁ σ ∘ τ) ⊗₁ id) ≡⟨ pullr (pullr (pullr refl)) ⟩
μ _ ∘ M₁ (M₁ (α→ _ _ _)) ∘ M₁ σ ∘ τ ∘ ((μ _ ∘ M₁ σ ∘ τ) ⊗₁ id) ≡⟨ refl⟩∘⟨ M.pulll left-strength-α→ ⟩
μ _ ∘ M₁ (σ ∘ (id ⊗₁ σ) ∘ α→ _ _ _) ∘ τ ∘ ((μ _ ∘ M₁ σ ∘ τ) ⊗₁ id) ≡⟨ refl⟩∘⟨ refl⟩∘⟨ ◀.popl right-strength-μ ⟩
μ _ ∘ M₁ (σ ∘ (id ⊗₁ σ) ∘ α→ _ _ _) ∘ (μ _ ∘ M₁ τ ∘ τ) ∘ ((M₁ σ ∘ τ) ⊗₁ id) ≡⟨ refl⟩∘⟨ refl⟩∘⟨ pullr (pullr (◀.popl (τ.is-natural _ _ _))) ⟩
μ _ ∘ M₁ (σ ∘ (id ⊗₁ σ) ∘ α→ _ _ _) ∘ μ _ ∘ M₁ τ ∘ (M₁ (σ ⊗₁ id) ∘ τ) ∘ (τ ⊗₁ id) ≡⟨ refl⟩∘⟨ M.popr (M.popr (pulll (sym (mult.is-natural _ _ _)))) ⟩
μ _ ∘ M₁ σ ∘ M₁ (id ⊗₁ σ) ∘ (μ _ ∘ M₁ (M₁ (α→ _ _ _))) ∘ M₁ τ ∘ (M₁ (σ ⊗₁ id) ∘ τ) ∘ (τ ⊗₁ id) ≡⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ pullr (refl⟩∘⟨ refl⟩∘⟨ pullr refl) ⟩
μ _ ∘ M₁ σ ∘ M₁ (id ⊗₁ σ) ∘ μ _ ∘ M₁ (M₁ (α→ _ _ _)) ∘ M₁ τ ∘ M₁ (σ ⊗₁ id) ∘ τ ∘ (τ ⊗₁ id) ≡⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ M.pulll3 strength-α→ ⟩
μ _ ∘ M₁ σ ∘ M₁ (id ⊗₁ σ) ∘ μ _ ∘ M₁ (σ ∘ (id ⊗₁ τ) ∘ α→ _ _ _) ∘ τ ∘ (τ ⊗₁ id) ≡⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ M.popr (M.popr (sym right-strength-α→)) ⟩
μ _ ∘ M₁ σ ∘ M₁ (id ⊗₁ σ) ∘ μ _ ∘ M₁ σ ∘ M₁ (id ⊗₁ τ) ∘ τ ∘ α→ _ _ _ ≡˘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ extendl (τ.is-natural _ _ _) ⟩
μ _ ∘ M₁ σ ∘ M₁ (id ⊗₁ σ) ∘ μ _ ∘ M₁ σ ∘ τ ∘ (M₁ id ⊗₁ τ) ∘ α→ _ _ _ ≡˘⟨ refl⟩∘⟨ refl⟩∘⟨ extendl (mult.is-natural _ _ _) ⟩
μ _ ∘ M₁ σ ∘ μ _ ∘ M₁ (M₁ (id ⊗₁ σ)) ∘ M₁ σ ∘ τ ∘ (M₁ id ⊗₁ τ) ∘ α→ _ _ _ ≡˘⟨ refl⟩∘⟨ extendl (mult.is-natural _ _ _) ⟩
μ _ ∘ μ _ ∘ M₁ (M₁ σ) ∘ M₁ (M₁ (id ⊗₁ σ)) ∘ M₁ σ ∘ τ ∘ (M₁ id ⊗₁ τ) ∘ α→ _ _ _ ≡˘⟨ extendl mult-assoc ⟩
μ _ ∘ M₁ (μ _) ∘ M₁ (M₁ σ) ∘ M₁ (M₁ (id ⊗₁ σ)) ∘ M₁ σ ∘ τ ∘ (M₁ id ⊗₁ τ) ∘ α→ _ _ _ ≡˘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ M.extendl (σ.is-natural _ _ _) ⟩
μ _ ∘ M₁ (μ _) ∘ M₁ (M₁ σ) ∘ M₁ σ ∘ M₁ (id ⊗₁ M₁ σ) ∘ τ ∘ (M₁ id ⊗₁ τ) ∘ α→ _ _ _ ≡⟨ refl⟩∘⟨ M.pulll3 (sym left-strength-μ) ⟩
μ _ ∘ M₁ (σ ∘ (id ⊗₁ μ _)) ∘ M₁ (id ⊗₁ M₁ σ) ∘ τ ∘ (M₁ id ⊗₁ τ) ∘ α→ _ _ _ ≡˘⟨ refl⟩∘⟨ refl⟩∘⟨ extendl (τ.is-natural _ _ _) ⟩
μ _ ∘ M₁ (σ ∘ (id ⊗₁ μ _)) ∘ τ ∘ (M₁ id ⊗₁ M₁ σ) ∘ (M₁ id ⊗₁ τ) ∘ α→ _ _ _ ≡⟨ refl⟩∘⟨ M.popr (extendl (sym (τ.is-natural _ _ _))) ⟩
μ _ ∘ M₁ σ ∘ τ ∘ (M₁ id ⊗₁ μ _) ∘ (M₁ id ⊗₁ M₁ σ) ∘ (M₁ id ⊗₁ τ) ∘ α→ _ _ _ ≡⟨ pushr (pushr (refl⟩∘⟨ ⊗.pulll3 ((refl⟩∘⟨ M.annihilate (idl _)) ∙ M.eliml refl ,ₚ refl))) ⟩
(μ _ ∘ M₁ σ ∘ τ) ∘ (id ⊗₁ (μ _ ∘ M₁ σ ∘ τ)) ∘ α→ _ _ _ ∎
```
The unitor coherences are relatively easy to prove.
```agda
strength→monoidal .F-λ← =
M₁ λ← ∘ (μ _ ∘ M₁ σ ∘ τ) ∘ (η _ ⊗₁ id) ≡⟨ refl⟩∘⟨ pullr (pullr right-strength-η) ⟩
M₁ λ← ∘ μ _ ∘ M₁ σ ∘ η _ ≡˘⟨ refl⟩∘⟨ refl⟩∘⟨ unit.is-natural _ _ _ ⟩
M₁ λ← ∘ μ _ ∘ η _ ∘ σ ≡⟨ refl⟩∘⟨ cancell right-ident ⟩
M₁ λ← ∘ σ ≡⟨ left-strength-λ← ⟩
λ← ∎
strength→monoidal .F-ρ← =
M₁ ρ← ∘ (μ _ ∘ M₁ σ ∘ τ) ∘ (⌜ id ⌝ ⊗₁ η _) ≡˘⟨ ap¡ M-id ⟩
M₁ ρ← ∘ (μ _ ∘ M₁ σ ∘ τ) ∘ (M₁ id ⊗₁ η _) ≡⟨ refl⟩∘⟨ pullr (pullr (τ.is-natural _ _ _)) ⟩
M₁ ρ← ∘ μ _ ∘ M₁ σ ∘ M₁ (id ⊗₁ η _) ∘ τ ≡⟨ refl⟩∘⟨ refl⟩∘⟨ M.pulll left-strength-η ⟩
M₁ ρ← ∘ μ _ ∘ M₁ (η _) ∘ τ ≡⟨ refl⟩∘⟨ cancell left-ident ⟩
M₁ ρ← ∘ τ ≡⟨ right-strength-ρ← ⟩
ρ← ∎
```
</details>
## Symmetry
In a [[braided monoidal category]], we unsurprisingly say that a monad
strength is *symmetric* if the underlying functor [[strength]] is: a
strength with this property is equivalent to the data of a left (or
right) strength, with the other one obtained by the braiding.
```agda
module _ (Cᵇ : Braided-monoidal Cᵐ) where
is-symmetric-monad-strength : Monad-strength Cᵐ monad → Type (o ⊔ ℓ)
is-symmetric-monad-strength s =
is-symmetric-strength Cᵐ M Cᵇ functor-strength
where open Monad-strength s
```
## Duality
Just as with functor strengths, the definitions of left and right monad
strengths are completely dual up to [[reversing|reverse monoidal
category]] the tensor product.
```agda
monad-strength^rev
: Left-monad-strength (Cᵐ ^rev) monad ≃ Right-monad-strength Cᵐ monad
monad-strength^rev = Iso→Equiv is where
is : Iso _ _
is .fst l = record
{ functor-strength = strength^rev Cᵐ M .fst functor-strength
; right-strength-η = left-strength-η
; right-strength-μ = left-strength-μ
} where open Left-monad-strength l
is .snd .inv r = record
{ functor-strength = Equiv.from (strength^rev Cᵐ M) functor-strength
; left-strength-η = right-strength-η
; left-strength-μ = right-strength-μ
} where open Right-monad-strength r
is .snd .rinv _ = Right-monad-strength-path Cᵐ monad
(Equiv.ε (strength^rev Cᵐ M) _)
is .snd .linv _ = Left-monad-strength-path (Cᵐ ^rev) monad
(Equiv.η (strength^rev Cᵐ M) _)
```
## Sets-monads are strong {defines="sets-monads-are-strong"}
<!--
```agda
module _ {ℓ} (monad : Monad (Sets ℓ)) where
open Monad monad
open Left-monad-strength
```
-->
The fact that [[$\Sets$-endofunctors are strong]] straightforwardly
extends to the fact that $\Sets$-*monads* are strong, by naturality of
the unit and multiplication.
```agda
Sets-monad-strength : Left-monad-strength Setsₓ monad
Sets-monad-strength .functor-strength = Sets-strength M
Sets-monad-strength .left-strength-η = ext λ a b →
sym (unit.is-natural _ _ (a ,_) $ₚ _)
Sets-monad-strength .left-strength-μ = ext λ a mmb →
sym (mult.is-natural _ _ (a ,_) $ₚ _) ∙ ap (μ _) (M-∘ _ _ $ₚ _)
```