Function magmas

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2023-03-13.
Last modified on 2023-03-19.

module structured-types.function-magmas where
Imports 
open import foundation.dependent-pair-types
open import foundation.universe-levels

open import structured-types.magmas

Idea

Given a magma M and a type X, the function magma M^X consists of functions from X into the underlying type of M. The operation on M^X is defined pointwise.

Definition

module _
  {l1 l2 : Level} (M : Magma l1) (X : UU l2)
  where

  type-function-Magma : UU (l1 ⊔ l2)
  type-function-Magma = X → type-Magma M

  mul-function-Magma :
    type-function-Magma → type-function-Magma → type-function-Magma
  mul-function-Magma f g x = mul-Magma M (f x) (g x)

  function-Magma : Magma (l1 ⊔ l2)
  pr1 function-Magma = type-function-Magma
  pr2 function-Magma = mul-function-Magma

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