Dependent products of commutative monoids

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2023-03-13.
Last modified on 2023-03-21.

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


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.


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

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

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

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

  unit-Π-Commutative-Monoid : type-Π-Commutative-Monoid
  unit-Π-Commutative-Monoid =
    unit-Π-Monoid I  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  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  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  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  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  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

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