Function commutative monoids

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2023-03-13.
Last modified on 2023-05-01.

module group-theory.function-commutative-monoids where
Imports 
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-commutative-monoids
open import group-theory.monoids

Idea

Given a commutative monoid M and a type X, the function commuative monoid M^X consists of functions from X to the underlying type of M. The multiplicative operation and the unit are given pointwise.

Definition

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

  function-Commutative-Monoid : Commutative-Monoid (l1 ⊔ l2)
  function-Commutative-Monoid = Π-Commutative-Monoid X (λ _ → M)

  monoid-function-Commutative-Monoid : Monoid (l1 ⊔ l2)
  monoid-function-Commutative-Monoid =
    monoid-Π-Commutative-Monoid X (λ _ → M)

  set-function-Commutative-Monoid : Set (l1 ⊔ l2)
  set-function-Commutative-Monoid =
    set-Π-Commutative-Monoid X (λ _ → M)

  type-function-Commutative-Monoid : UU (l1 ⊔ l2)
  type-function-Commutative-Monoid =
    type-Π-Commutative-Monoid X (λ _ → M)

  unit-function-Commutative-Monoid : type-function-Commutative-Monoid
  unit-function-Commutative-Monoid =
    unit-Π-Commutative-Monoid X (λ _ → M)

  mul-function-Commutative-Monoid :
    (f g : type-function-Commutative-Monoid) →
    type-function-Commutative-Monoid
  mul-function-Commutative-Monoid =
    mul-Π-Commutative-Monoid X (λ _ → M)

  associative-mul-function-Commutative-Monoid :
    (f g h : type-function-Commutative-Monoid) →
    mul-function-Commutative-Monoid (mul-function-Commutative-Monoid f g) h ＝
    mul-function-Commutative-Monoid f (mul-function-Commutative-Monoid g h)
  associative-mul-function-Commutative-Monoid =
    associative-mul-Π-Commutative-Monoid X (λ _ → M)

  left-unit-law-mul-function-Commutative-Monoid :
    (f : type-function-Commutative-Monoid) →
    mul-function-Commutative-Monoid unit-function-Commutative-Monoid f ＝ f
  left-unit-law-mul-function-Commutative-Monoid =
    left-unit-law-mul-Π-Commutative-Monoid X (λ _ → M)

  right-unit-law-mul-function-Commutative-Monoid :
    (f : type-function-Commutative-Monoid) →
    mul-function-Commutative-Monoid f unit-function-Commutative-Monoid ＝ f
  right-unit-law-mul-function-Commutative-Monoid =
    right-unit-law-mul-Π-Commutative-Monoid X (λ _ → M)

  commutative-mul-function-Commutative-Monoid :
    (f g : type-function-Commutative-Monoid) →
    mul-function-Commutative-Monoid f g ＝ mul-function-Commutative-Monoid g f
  commutative-mul-function-Commutative-Monoid =
    commutative-mul-Π-Commutative-Monoid X (λ _ → M)

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