Equivalences of higher groups

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2023-04-10.
Last modified on 2024-03-23.

module higher-group-theory.equivalences-higher-groups where
open import foundation.0-connected-types
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.function-types
open import foundation.fundamental-theorem-of-identity-types
open import foundation.identity-types
open import foundation.subtype-identity-principle
open import foundation.torsorial-type-families
open import foundation.universe-levels

open import higher-group-theory.higher-groups
open import higher-group-theory.homomorphisms-higher-groups

open import structured-types.pointed-equivalences
open import structured-types.pointed-isomorphisms
open import structured-types.pointed-types

open import synthetic-homotopy-theory.functoriality-loop-spaces


Equivalences of higher groups

module _
  {l1 l2 : Level} (G : ∞-Group l1) (H : ∞-Group l2)

  equiv-∞-Group : UU (l1  l2)
  equiv-∞-Group =
    classifying-pointed-type-∞-Group G ≃∗ classifying-pointed-type-∞-Group H

  module _
    (e : equiv-∞-Group)

    hom-equiv-∞-Group : hom-∞-Group G H
    hom-equiv-∞-Group = pointed-map-pointed-equiv e

    map-equiv-∞-Group : type-∞-Group G  type-∞-Group H
    map-equiv-∞-Group = map-hom-∞-Group G H hom-equiv-∞-Group

    pointed-equiv-equiv-∞-Group :
      pointed-type-∞-Group G ≃∗ pointed-type-∞-Group H
    pointed-equiv-equiv-∞-Group = pointed-equiv-Ω-pointed-equiv e

    equiv-equiv-∞-Group : type-∞-Group G  type-∞-Group H
    equiv-equiv-∞-Group = equiv-map-Ω-pointed-equiv e

The identity equivalence of higher groups

id-equiv-∞-Group :
  {l1 : Level} (G : ∞-Group l1)  equiv-∞-Group G G
id-equiv-∞-Group G = id-pointed-equiv

Isomorphisms of ∞-groups

module _
  {l1 l2 : Level} (G : ∞-Group l1) (H : ∞-Group l2)

  is-iso-∞-Group : hom-∞-Group G H  UU (l1  l2)
  is-iso-∞-Group = is-pointed-iso


The total space of equivalences of higher groups is contractible

is-torsorial-equiv-∞-Group :
  {l1 : Level} (G : ∞-Group l1) 
  is-torsorial  (H : ∞-Group l1)  equiv-∞-Group G H)
is-torsorial-equiv-∞-Group G =
    ( is-torsorial-pointed-equiv (classifying-pointed-type-∞-Group G))
    ( λ X  is-prop-is-0-connected (type-Pointed-Type X))
    ( classifying-pointed-type-∞-Group G)
    ( id-pointed-equiv)
    ( is-0-connected-classifying-type-∞-Group G)

equiv-eq-∞-Group :
  {l1 : Level} (G H : ∞-Group l1)  (G  H)  equiv-∞-Group G H
equiv-eq-∞-Group G .G refl = id-equiv-∞-Group G

is-equiv-equiv-eq-∞-Group :
  {l1 : Level} (G H : ∞-Group l1)  is-equiv (equiv-eq-∞-Group G H)
is-equiv-equiv-eq-∞-Group G =
    ( is-torsorial-equiv-∞-Group G)
    ( equiv-eq-∞-Group G)

extensionality-∞-Group :
  {l1 : Level} (G H : ∞-Group l1)  (G  H)  equiv-∞-Group G H
pr1 (extensionality-∞-Group G H) = equiv-eq-∞-Group G H
pr2 (extensionality-∞-Group G H) = is-equiv-equiv-eq-∞-Group G H

eq-equiv-∞-Group :
  {l1 : Level} (G H : ∞-Group l1)  equiv-∞-Group G H  G  H
eq-equiv-∞-Group G H = map-inv-equiv (extensionality-∞-Group G H)

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