# Cartesian products of concrete groups

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

Created on 2023-05-10.

module group-theory.cartesian-products-concrete-groups where

Imports
open import foundation.0-connected-types
open import foundation.1-types
open import foundation.cartesian-product-types
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.identity-types
open import foundation.mere-equality
open import foundation.propositional-truncations
open import foundation.propositions
open import foundation.sets
open import foundation.truncated-types
open import foundation.truncation-levels
open import foundation.universe-levels

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

open import higher-group-theory.cartesian-products-higher-groups
open import higher-group-theory.higher-groups

open import structured-types.pointed-types


## Idea

The cartesian product of two concrete groups is defined as the cartesian product of their underlying ∞-group

## Definition

module _
{l1 l2 : Level} (G : Concrete-Group l1) (R : Concrete-Group l2)
where

product-Concrete-Group : Concrete-Group (l1 ⊔ l2)
pr1 product-Concrete-Group =
product-∞-Group
( ∞-group-Concrete-Group G)
( ∞-group-Concrete-Group R)
pr2 product-Concrete-Group =
is-set-equiv
( type-∞-Group (pr1 G) ×
type-∞-Group (pr1 R))
( equiv-type-∞-Group-product-∞-Group
( ∞-group-Concrete-Group G)
( ∞-group-Concrete-Group R))
( is-set-product
( is-set-type-Concrete-Group G)
( is-set-type-Concrete-Group R))

∞-group-product-Concrete-Group : ∞-Group (l1 ⊔ l2)
∞-group-product-Concrete-Group = pr1 product-Concrete-Group

classifying-pointed-type-product-Concrete-Group : Pointed-Type (l1 ⊔ l2)
classifying-pointed-type-product-Concrete-Group =
classifying-pointed-type-∞-Group ∞-group-product-Concrete-Group

classifying-type-product-Concrete-Group : UU (l1 ⊔ l2)
classifying-type-product-Concrete-Group =
classifying-type-∞-Group ∞-group-product-Concrete-Group

shape-product-Concrete-Group : classifying-type-product-Concrete-Group
shape-product-Concrete-Group =
shape-∞-Group ∞-group-product-Concrete-Group

is-0-connected-classifying-type-product-Concrete-Group :
is-0-connected classifying-type-product-Concrete-Group
is-0-connected-classifying-type-product-Concrete-Group =
is-0-connected-classifying-type-∞-Group ∞-group-product-Concrete-Group

mere-eq-classifying-type-product-Concrete-Group :
(X Y : classifying-type-product-Concrete-Group) → mere-eq X Y
mere-eq-classifying-type-product-Concrete-Group =
mere-eq-classifying-type-∞-Group ∞-group-product-Concrete-Group

elim-prop-classifying-type-product-Concrete-Group :
{l2 : Level} (P : classifying-type-product-Concrete-Group → Prop l2) →
type-Prop (P shape-product-Concrete-Group) →
((X : classifying-type-product-Concrete-Group) → type-Prop (P X))
elim-prop-classifying-type-product-Concrete-Group =
elim-prop-classifying-type-∞-Group ∞-group-product-Concrete-Group

type-product-Concrete-Group : UU (l1 ⊔ l2)
type-product-Concrete-Group = type-∞-Group ∞-group-product-Concrete-Group

is-set-type-product-Concrete-Group : is-set type-product-Concrete-Group
is-set-type-product-Concrete-Group = pr2 product-Concrete-Group

set-product-Concrete-Group : Set (l1 ⊔ l2)
set-product-Concrete-Group =
pair type-product-Concrete-Group is-set-type-product-Concrete-Group

abstract
is-1-type-classifying-type-product-Concrete-Group :
is-trunc one-𝕋 classifying-type-product-Concrete-Group
is-1-type-classifying-type-product-Concrete-Group X Y =
apply-universal-property-trunc-Prop
( mere-eq-classifying-type-product-Concrete-Group
shape-product-Concrete-Group
X)
( is-set-Prop (Id X Y))
( λ where
refl →
apply-universal-property-trunc-Prop
( mere-eq-classifying-type-product-Concrete-Group
shape-product-Concrete-Group
Y)
( is-set-Prop (Id shape-product-Concrete-Group Y))
( λ where refl → is-set-type-product-Concrete-Group))

classifying-1-type-product-Concrete-Group : Truncated-Type (l1 ⊔ l2) one-𝕋
classifying-1-type-product-Concrete-Group =
pair
classifying-type-product-Concrete-Group
is-1-type-classifying-type-product-Concrete-Group

Id-product-BG-Set :
(X Y : classifying-type-product-Concrete-Group) → Set (l1 ⊔ l2)
Id-product-BG-Set X Y = Id-Set classifying-1-type-product-Concrete-Group X Y

unit-product-Concrete-Group : type-product-Concrete-Group
unit-product-Concrete-Group = unit-∞-Group ∞-group-product-Concrete-Group

mul-product-Concrete-Group :
(x y : type-product-Concrete-Group) → type-product-Concrete-Group
mul-product-Concrete-Group = mul-∞-Group ∞-group-product-Concrete-Group

mul-product-Concrete-Group' :
(x y : type-product-Concrete-Group) → type-product-Concrete-Group
mul-product-Concrete-Group' x y = mul-product-Concrete-Group y x

associative-mul-product-Concrete-Group :
(x y z : type-product-Concrete-Group) →
Id
(mul-product-Concrete-Group (mul-product-Concrete-Group x y) z)
(mul-product-Concrete-Group x (mul-product-Concrete-Group y z))
associative-mul-product-Concrete-Group =
associative-mul-∞-Group ∞-group-product-Concrete-Group

left-unit-law-mul-product-Concrete-Group :
(x : type-product-Concrete-Group) →
Id (mul-product-Concrete-Group unit-product-Concrete-Group x) x
left-unit-law-mul-product-Concrete-Group =
left-unit-law-mul-∞-Group ∞-group-product-Concrete-Group

right-unit-law-mul-product-Concrete-Group :
(y : type-product-Concrete-Group) →
Id (mul-product-Concrete-Group y unit-product-Concrete-Group) y
right-unit-law-mul-product-Concrete-Group =
right-unit-law-mul-∞-Group ∞-group-product-Concrete-Group

coherence-unit-laws-mul-product-Concrete-Group :
Id
( left-unit-law-mul-product-Concrete-Group unit-product-Concrete-Group)
( right-unit-law-mul-product-Concrete-Group unit-product-Concrete-Group)
coherence-unit-laws-mul-product-Concrete-Group =
coherence-unit-laws-mul-∞-Group ∞-group-product-Concrete-Group

inv-product-Concrete-Group :
type-product-Concrete-Group → type-product-Concrete-Group
inv-product-Concrete-Group = inv-∞-Group ∞-group-product-Concrete-Group

left-inverse-law-mul-product-Concrete-Group :
(x : type-product-Concrete-Group) →
Id
( mul-product-Concrete-Group (inv-product-Concrete-Group x) x)
( unit-product-Concrete-Group)
left-inverse-law-mul-product-Concrete-Group =
left-inverse-law-mul-∞-Group ∞-group-product-Concrete-Group

right-inverse-law-mul-product-Concrete-Group :
(x : type-product-Concrete-Group) →
Id
( mul-product-Concrete-Group x (inv-product-Concrete-Group x))
( unit-product-Concrete-Group)
right-inverse-law-mul-product-Concrete-Group =
right-inverse-law-mul-∞-Group ∞-group-product-Concrete-Group

group-product-Concrete-Group : Group (l1 ⊔ l2)
pr1 (pr1 group-product-Concrete-Group) =
set-product-Concrete-Group
pr1 (pr2 (pr1 group-product-Concrete-Group)) =
mul-product-Concrete-Group
pr2 (pr2 (pr1 group-product-Concrete-Group)) =
associative-mul-product-Concrete-Group
pr1 (pr1 (pr2 group-product-Concrete-Group)) =
unit-product-Concrete-Group
pr1 (pr2 (pr1 (pr2 group-product-Concrete-Group))) =
left-unit-law-mul-product-Concrete-Group
pr2 (pr2 (pr1 (pr2 group-product-Concrete-Group))) =
right-unit-law-mul-product-Concrete-Group
pr1 (pr2 (pr2 group-product-Concrete-Group)) =
inv-product-Concrete-Group
pr1 (pr2 (pr2 (pr2 group-product-Concrete-Group))) =
left-inverse-law-mul-product-Concrete-Group
pr2 (pr2 (pr2 (pr2 group-product-Concrete-Group))) =
right-inverse-law-mul-product-Concrete-Group

op-group-product-Concrete-Group : Group (l1 ⊔ l2)
pr1 (pr1 op-group-product-Concrete-Group) =
set-product-Concrete-Group
pr1 (pr2 (pr1 op-group-product-Concrete-Group)) =
mul-product-Concrete-Group'
pr2 (pr2 (pr1 op-group-product-Concrete-Group)) x y z =
inv (associative-mul-product-Concrete-Group z y x)
pr1 (pr1 (pr2 op-group-product-Concrete-Group)) =
unit-product-Concrete-Group
pr1 (pr2 (pr1 (pr2 op-group-product-Concrete-Group))) =
right-unit-law-mul-product-Concrete-Group
pr2 (pr2 (pr1 (pr2 op-group-product-Concrete-Group))) =
left-unit-law-mul-product-Concrete-Group
pr1 (pr2 (pr2 op-group-product-Concrete-Group)) =
inv-product-Concrete-Group
pr1 (pr2 (pr2 (pr2 op-group-product-Concrete-Group))) =
right-inverse-law-mul-product-Concrete-Group
pr2 (pr2 (pr2 (pr2 op-group-product-Concrete-Group))) =
left-inverse-law-mul-product-Concrete-Group


## Property

  equiv-type-Concrete-Group-product-Concrete-Group :
type-product-Concrete-Group ≃
( type-Concrete-Group G × type-Concrete-Group R)
equiv-type-Concrete-Group-product-Concrete-Group =
equiv-type-∞-Group-product-∞-Group
( ∞-group-Concrete-Group G)
( ∞-group-Concrete-Group R)