Morphisms of magmas

Content created by Jonathan Prieto-Cubides, Fredrik Bakke and Egbert Rijke.

Created on 2022-05-07.
Last modified on 2023-03-13.

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

open import structured-types.magmas

Idea

A morphism of magmas from M to N is a map between their underlying type that preserves the binary operation

Definition

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

  preserves-mul-Magma : (type-Magma M → type-Magma N) → UU (l1 ⊔ l2)
  preserves-mul-Magma f =
    (x y : type-Magma M) → Id (f (mul-Magma M x y)) (mul-Magma N (f x) (f y))

  hom-Magma : UU (l1 ⊔ l2)
  hom-Magma = Σ (type-Magma M → type-Magma N) preserves-mul-Magma

  map-hom-Magma : hom-Magma → type-Magma M → type-Magma N
  map-hom-Magma = pr1

  preserves-mul-map-hom-Magma :
    (f : hom-Magma) → preserves-mul-Magma (map-hom-Magma f)
  preserves-mul-map-hom-Magma = pr2

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