{-# 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.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.Reasoning

module Cat.Monoidal.Diagram.Monoid where


# Monoids in a monoidal category🔗

Let $(\mathcal{C}, \otimes, 1)$ 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 $\mathcal{C}$, generally just called a “monoid in $\mathcal{C}$”, is really a collection of diagrams in $\mathcal{C}$ centered around an object $M$, the monoid itself.

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:

  record Monoid-on (M : C.Ob) : Type ℓ where
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 $\mathcal{C}$ as a bicategory with a single object $*$ — that is, we deloop it —, then a monoid in $\mathcal{C}$ is given by precisely the same data as a monad in ${\mathbf{B}}\mathcal{C}$, on the object $*$.

  monad→monoid : (M : Monad BC tt) → Monoid-on (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

module M = Monoid-on M


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 $\mathcal{C}$ can be made into a category, much like the monoids in the category of sets. In fact, we shall see later that when ${{\mathbf{Sets}}}$ is equipped with its Cartesian monoidal structure, ${\mathrm{Mon}}({{\mathbf{Sets}}}) \cong {\mathrm{Mon}}$. Rather than defining ${\mathrm{Mon}}(\mathcal{C})$ directly as a category, we instead define it as a category ${\mathrm{Mon}}(\mathcal{C}) {\mathrel{\htmlClass{liesover}{\hspace{1.366em}}}}\mathcal{C}$ displayed over $\mathcal{C}$, which fits naturally with the way we have defined Monoid-object-on.

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:

  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.

  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 $\mathcal{C}$, the reasoning isn’t that painful:

  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) _ _


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 identical1 total categories.

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) ${\mathrm{Mon}}(\mathcal{C}) \to {\mathbf{Mon}}$; 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.

  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-⋆ _ _


1. thus isomorphic, thus equivalent↩︎