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Monoid.lagda.md
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Monoid.lagda.md
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<!--
```agda
{-# OPTIONS --lossy-unification #-}
open import Cat.Monoidal.Instances.Cartesian
open import Cat.Displayed.Univalence.Thin
open import Cat.Instances.Sets.Complete
open import Cat.Displayed.Functor
open import Cat.Bi.Diagram.Monad
open import Cat.Displayed.Base
open import Cat.Displayed.Path
open import Cat.Monoidal.Base
open import Cat.Bi.Base
open import Cat.Prelude
import Algebra.Monoid.Category as Mon
import Algebra.Monoid as Mon
import Cat.Diagram.Monad as Mo
import Cat.Reasoning
```
-->
```agda
module Cat.Monoidal.Diagram.Monoid where
```
<!--
```agda
module _ {o ℓ} {C : Precategory o ℓ} (M : Monoidal-category C) where
private module C where
open Cat.Reasoning C public
open Monoidal-category M public
```
-->
# Monoids in a monoidal category
Let $(\cC, \otimes, 1)$ be a [monoidal category] you want to study.
It can be, for instance, one of the [endomorphism categories] in a
[bicategory] that you like. A **monoid object in $\cC$**, generally
just called a "monoid in $\cC$", is really a collection of diagrams
in $\cC$ centered around an object $M$, the monoid itself.
[monoidal category]: Cat.Monoidal.Base.html#monoidal-categories
[endomorphism categories]: Cat.Monoidal.Base.html#endomorphism-categories
[bicategory]: Cat.Bi.Base.html
In addition to the object, we also require a "unit" map $\eta : 1 \to M$
and "multiplication" map $\mu : M \otimes M \to M$. Moreover, these maps
should be compatible with the unitors and associator of the underlying
monoidal category:
```agda
record Monoid-on (M : C.Ob) : Type ℓ where
no-eta-equality
field
η : C.Hom C.Unit M
μ : C.Hom (M C.⊗ M) M
μ-unitl : μ C.∘ (η C.⊗₁ C.id) ≡ C.λ←
μ-unitr : μ C.∘ (C.id C.⊗₁ η) ≡ C.ρ←
μ-assoc : μ C.∘ (C.id C.⊗₁ μ) ≡ μ C.∘ (μ C.⊗₁ C.id) C.∘ C.α← _ _ _
```
If we think of $\cC$ as a bicategory with a single object $*$ ---
that is, we _deloop_ it ---, then a monoid in $\cC$ is given by
precisely the same data as a monad in $\bf{B}\cC$, on the object $*$.
<!--
```agda
private
BC = Deloop M
module BC = Prebicategory BC
open Monoid-on
Monoid-pathp
: ∀ {P : I → C.Ob} {x : Monoid-on (P i0)} {y : Monoid-on (P i1)}
→ PathP (λ i → C.Hom C.Unit (P i)) (x .η) (y .η)
→ PathP (λ i → C.Hom (P i C.⊗ P i) (P i)) (x .μ) (y .μ)
→ PathP (λ i → Monoid-on (P i)) x y
Monoid-pathp {x = x} {y} p q i .η = p i
Monoid-pathp {x = x} {y} p q i .μ = q i
Monoid-pathp {P = P} {x} {y} p q i .μ-unitl =
is-prop→pathp
(λ i → C.Hom-set _ (P i) (q i C.∘ (p i C.⊗₁ C.id)) C.λ←)
(x .μ-unitl)
(y .μ-unitl)
i
Monoid-pathp {P = P} {x} {y} p q i .μ-unitr =
is-prop→pathp
(λ i → C.Hom-set _ (P i) (q i C.∘ (C.id C.⊗₁ p i)) C.ρ←)
(x .μ-unitr)
(y .μ-unitr)
i
Monoid-pathp {P = P} {x} {y} p q i .μ-assoc =
is-prop→pathp
(λ i → C.Hom-set _ (P i)
(q i C.∘ (C.id C.⊗₁ q i))
(q i C.∘ (q i C.⊗₁ C.id) C.∘ C.α← _ _ _))
(x .μ-assoc)
(y .μ-assoc)
i
```
-->
```agda
monad→monoid : (M : Monad BC tt) → Monoid-on (M .Monad.M)
monad→monoid M = go where
module M = Monad M
go : Monoid-on M.M
go .η = M.η
go .μ = M.μ
go .μ-unitl = M.μ-unitl
go .μ-unitr = M.μ-unitr
go .μ-assoc = M.μ-assoc
monoid→monad : ∀ {M} → Monoid-on M → Monad BC tt
monoid→monad M = go where
module M = Monoid-on M
go : Monad BC tt
go .Monad.M = _
go .Monad.μ = M.μ
go .Monad.η = M.η
go .Monad.μ-assoc = M.μ-assoc
go .Monad.μ-unitr = M.μ-unitr
go .Monad.μ-unitl = M.μ-unitl
```
Put another way, a monad is just a monoid in the category of
~~endofunctors~~ endo-1-cells, what's the problem?
## The category Mon(C)
The monoid objects in $\cC$ can be made into a category, much like
the [monoids in the category of sets]. In fact, we shall see later that
when $\Sets$ is equipped with its [Cartesian monoidal structure],
$\rm{Mon}(\Sets) \cong \rm{Mon}$. Rather than defining
$\rm{Mon}(\cC)$ directly as a category, we instead define it as a
category $\rm{Mon}(\cC) \liesover \cC$ \r{displayed over}
$\cC$, which fits naturally with the way we have defined
`Monoid-object-on`{.Agda}.
[Cartesian monoidal structure]: Cat.Monoidal.Instances.Cartesian.html
[monoids in the category of sets]: Algebra.Monoid.html
<!--
```agda
module _ {o ℓ} {C : Precategory o ℓ} (M : Monoidal-category C) where
private module C where
open Cat.Reasoning C public
open Monoidal-category M public
```
-->
Therefore, rather than defining a type of monoid homomorphisms, we
define a predicate on maps $f : m \to n$ expressing the condition of
being a monoid homomorphism. This is the familiar condition from
algebra, but expressed in a point-free way:
```agda
record
is-monoid-hom {m n} (f : C.Hom m n)
(mo : Monoid-on M m) (no : Monoid-on M n) : Type ℓ where
private
module m = Monoid-on mo
module n = Monoid-on no
field
pres-η : f C.∘ m.η ≡ n.η
pres-μ : f C.∘ m.μ ≡ n.μ C.∘ (f C.⊗₁ f)
```
Since being a monoid homomorphism is a pair of propositions, the overall
condition is a proposition as well. This means that we will not need to
concern ourselves with proving displayed identity and associativity
laws, a great simplification.
<!--
```agda
private unquoteDecl eqv = declare-record-iso eqv (quote is-monoid-hom)
is-monoid-hom-is-prop : ∀ {m n} {f : C.Hom m n} {mo no} → is-prop (is-monoid-hom f mo no)
is-monoid-hom-is-prop = is-hlevel≃ 1 (Iso→Equiv eqv) hlevel!
open Displayed
open Functor
open is-monoid-hom
```
-->
```agda
Mon[_] : Displayed C ℓ ℓ
Mon[_] .Ob[_] = Monoid-on M
Mon[_] .Hom[_] = is-monoid-hom
Mon[_] .Hom[_]-set f x y = is-prop→is-set is-monoid-hom-is-prop
```
The most complicated step of putting together the displayed category of
monoid objects is proving that monoid homomorphisms are closed under
composition. However, even in the point-free setting of an arbitrary
category $\cC$, the reasoning isn't _that_ painful:
```agda
Mon[ .id′ ] .pres-η = C.idl _
Mon[ .id′ ] .pres-μ = C.idl _ ∙ C.intror (C.-⊗- .F-id)
Mon[_] ._∘′_ fh gh .pres-η = C.pullr (gh .pres-η) ∙ fh .pres-η
Mon[_] ._∘′_ {x = x} {y} {z} {f} {g} fh gh .pres-μ =
(f C.∘ g) C.∘ x .Monoid-on.μ ≡⟨ C.pullr (gh .pres-μ) ⟩
f C.∘ y .Monoid-on.μ C.∘ (g C.⊗₁ g) ≡⟨ C.extendl (fh .pres-μ) ⟩
Monoid-on.μ z C.∘ (f C.⊗₁ f) C.∘ (g C.⊗₁ g) ≡˘⟨ C.refl⟩∘⟨ C.-⊗- .F-∘ _ _ ⟩
Monoid-on.μ z C.∘ (f C.∘ g C.⊗₁ f C.∘ g) ∎
Mon[_] .idr′ f = is-prop→pathp (λ i → is-monoid-hom-is-prop) _ _
Mon[_] .idl′ f = is-prop→pathp (λ i → is-monoid-hom-is-prop) _ _
Mon[_] .assoc′ f g h = is-prop→pathp (λ i → is-monoid-hom-is-prop) _ _
```
<!--
```agda
private
Setsₓ : ∀ {ℓ} → Monoidal-category (Sets ℓ)
Setsₓ = Cartesian-monoidal Sets-products Sets-terminal
Mon : ∀ {ℓ} → Displayed (Sets ℓ) _ _
Mon = Thin-structure-over (Mon.Monoid-structure _)
```
-->
Constructing this displayed category for the Cartesian monoidal
structure on the category of sets, we find that it is but a few
renamings away from the ordinary category of monoids-on-sets. The only
thing out of the ordinary about the proof below is that we can establish
the _displayed categories_ themselves are identical, so it is a trivial
step to show they induce identical^[thus isomorphic, thus equivalent]
total categories.
```agda
Mon[Sets]≡Mon : ∀ {ℓ} → Mon[ Setsₓ ] ≡ Mon {ℓ}
Mon[Sets]≡Mon {ℓ} = Displayed-path F (λ a → is-iso→is-equiv (fiso a)) ff where
open Displayed-functor
open Monoid-on
open is-monoid-hom
open Mon.Monoid-hom
open Mon.Monoid-on
```
The construction proceeds in three steps: First, put together a functor
(displayed over the identity) $\rm{Mon}(\cC) \to \thecat{Mon}$; Then,
prove that its action on objects ("step 2") and action on morphisms
("step 3") are independently equivalences of types. The characterisation
of paths of displayed categories will take care of turning this data
into an identification.
```agda
F : Displayed-functor Mon[ Setsₓ ] Mon Id
F .F₀′ o .identity = o .η (lift tt)
F .F₀′ o ._⋆_ x y = o .μ (x , y)
F .F₀′ o .has-is-monoid .Mon.has-is-semigroup =
record { has-is-magma = record { has-is-set = hlevel! }
; associative = o .μ-assoc $ₚ _
}
F .F₀′ o .has-is-monoid .Mon.idl = o .μ-unitl $ₚ _
F .F₀′ o .has-is-monoid .Mon.idr = o .μ-unitr $ₚ _
F .F₁′ wit .pres-id = wit .pres-η $ₚ _
F .F₁′ wit .pres-⋆ x y = wit .pres-μ $ₚ _
F .F-id′ = prop!
F .F-∘′ = prop!
open is-iso
fiso : ∀ a → is-iso (F .F₀′ {a})
fiso T .inv m .η _ = m .identity
fiso T .inv m .μ (a , b) = m ._⋆_ a b
fiso T .inv m .μ-unitl = funext λ _ → m .idl
fiso T .inv m .μ-unitr = funext λ _ → m .idr
fiso T .inv m .μ-assoc = funext λ _ → m .associative
fiso T .rinv x = Mon.Monoid-structure _ .id-hom-unique
(record { pres-id = refl ; pres-⋆ = λ _ _ → refl })
(record { pres-id = refl ; pres-⋆ = λ _ _ → refl })
fiso T .linv m = Monoid-pathp Setsₓ refl refl
ff : ∀ {a b : Set _} {f : ∣ a ∣ → ∣ b ∣} {a′ b′}
→ is-equiv (F₁′ F {a} {b} {f} {a′} {b′})
ff {a} {b} {f} {a′} {b′} =
prop-ext (is-monoid-hom-is-prop Setsₓ) (hlevel 1)
(λ z → F₁′ F z) invs .snd
where
invs : Mon.Monoid-hom (F .F₀′ a′) (F .F₀′ b′) f
→ is-monoid-hom Setsₓ f a′ b′
invs m .pres-η = funext λ _ → m .pres-id
invs m .pres-μ = funext λ _ → m .pres-⋆ _ _
```