# Mere equivalences of concrete group actions

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

Created on 2022-07-17.

module group-theory.mere-equivalences-concrete-group-actions where

Imports
open import foundation.functoriality-propositional-truncation
open import foundation.mere-equality
open import foundation.propositional-truncations
open import foundation.propositions
open import foundation.universe-levels

open import group-theory.concrete-group-actions
open import group-theory.concrete-groups
open import group-theory.equivalences-concrete-group-actions


## Definition

mere-equiv-prop-action-Concrete-Group :
{l1 l2 l3 : Level} (G : Concrete-Group l1) →
action-Concrete-Group l2 G → action-Concrete-Group l3 G →
Prop (l1 ⊔ l2 ⊔ l3)
mere-equiv-prop-action-Concrete-Group G X Y =
trunc-Prop (equiv-action-Concrete-Group G X Y)

mere-equiv-action-Concrete-Group :
{l1 l2 l3 : Level} (G : Concrete-Group l1) →
action-Concrete-Group l2 G → action-Concrete-Group l3 G → UU (l1 ⊔ l2 ⊔ l3)
mere-equiv-action-Concrete-Group G X Y =
type-Prop (mere-equiv-prop-action-Concrete-Group G X Y)

is-prop-mere-equiv-action-Concrete-Group :
{l1 l2 l3 : Level} (G : Concrete-Group l1)
(X : action-Concrete-Group l2 G) (Y : action-Concrete-Group l3 G) →
is-prop (mere-equiv-action-Concrete-Group G X Y)
is-prop-mere-equiv-action-Concrete-Group G X Y =
is-prop-type-Prop (mere-equiv-prop-action-Concrete-Group G X Y)

refl-mere-equiv-action-Concrete-Group :
{l1 l2 : Level} (G : Concrete-Group l1) (X : action-Concrete-Group l2 G) →
mere-equiv-action-Concrete-Group G X X
refl-mere-equiv-action-Concrete-Group G X =
unit-trunc-Prop (id-equiv-action-Concrete-Group G X)


## Properties

### Mere equivalences of concrete group actions give mere equalities

mere-eq-mere-equiv-action-Concrete-Group :
{l1 l2 : Level} (G : Concrete-Group l1)
(X : action-Concrete-Group l2 G) (Y : action-Concrete-Group l2 G) →
mere-equiv-action-Concrete-Group G X Y →
mere-eq X Y
mere-eq-mere-equiv-action-Concrete-Group G X Y =
map-trunc-Prop (eq-equiv-action-Concrete-Group G X Y)