Dependent products of commutative monoids

module group-theory.dependent-products-commutative-monoids where

Imports

open import foundation.dependent-pair-types
open import foundation.function-extensionality
open import foundation.identity-types
open import foundation.sets
open import foundation.universe-levels

open import group-theory.commutative-monoids
open import group-theory.dependent-products-monoids
open import group-theory.monoids

Idea

Given a family of commutative monoids Mᵢ indexed by i : I, the dependent product Π(i : I), Mᵢ is a commutative monoid consisting of dependent functions taking i : I to an element of the underlying type of Mᵢ. The multiplicative operation and the unit are given pointwise.

Definition

module _
  {l1 l2 : Level} (I : UU l1) (M : I → Commutative-Monoid l2)
  where

  monoid-Π-Commutative-Monoid : Monoid (l1 ⊔ l2)
  monoid-Π-Commutative-Monoid =
    Π-Monoid I (λ i → monoid-Commutative-Monoid (M i))

  set-Π-Commutative-Monoid : Set (l1 ⊔ l2)
  set-Π-Commutative-Monoid =
    set-Π-Monoid I (λ i → monoid-Commutative-Monoid (M i))

  type-Π-Commutative-Monoid : UU (l1 ⊔ l2)
  type-Π-Commutative-Monoid =
    type-Π-Monoid I (λ i → monoid-Commutative-Monoid (M i))

  unit-Π-Commutative-Monoid : type-Π-Commutative-Monoid
  unit-Π-Commutative-Monoid =
    unit-Π-Monoid I (λ i → monoid-Commutative-Monoid (M i))

  mul-Π-Commutative-Monoid :
    (f g : type-Π-Commutative-Monoid) → type-Π-Commutative-Monoid
  mul-Π-Commutative-Monoid =
    mul-Π-Monoid I (λ i → monoid-Commutative-Monoid (M i))

  associative-mul-Π-Commutative-Monoid :
    (f g h : type-Π-Commutative-Monoid) →
    mul-Π-Commutative-Monoid (mul-Π-Commutative-Monoid f g) h ＝
    mul-Π-Commutative-Monoid f (mul-Π-Commutative-Monoid g h)
  associative-mul-Π-Commutative-Monoid =
    associative-mul-Π-Monoid I (λ i → monoid-Commutative-Monoid (M i))

  left-unit-law-mul-Π-Commutative-Monoid :
    (f : type-Π-Commutative-Monoid) →
    mul-Π-Commutative-Monoid unit-Π-Commutative-Monoid f ＝ f
  left-unit-law-mul-Π-Commutative-Monoid =
    left-unit-law-mul-Π-Monoid I (λ i → monoid-Commutative-Monoid (M i))

  right-unit-law-mul-Π-Commutative-Monoid :
    (f : type-Π-Commutative-Monoid) →
    mul-Π-Commutative-Monoid f unit-Π-Commutative-Monoid ＝ f
  right-unit-law-mul-Π-Commutative-Monoid =
    right-unit-law-mul-Π-Monoid I (λ i → monoid-Commutative-Monoid (M i))

  commutative-mul-Π-Commutative-Monoid :
    (f g : type-Π-Commutative-Monoid) →
    mul-Π-Commutative-Monoid f g ＝ mul-Π-Commutative-Monoid g f
  commutative-mul-Π-Commutative-Monoid f g =
    eq-htpy (λ i → commutative-mul-Commutative-Monoid (M i) (f i) (g i))

  Π-Commutative-Monoid : Commutative-Monoid (l1 ⊔ l2)
  pr1 Π-Commutative-Monoid = monoid-Π-Commutative-Monoid
  pr2 Π-Commutative-Monoid = commutative-mul-Π-Commutative-Monoid