Groups of loops in 1-types

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

Created on 2022-03-18.
Last modified on 2023-10-22.

module synthetic-homotopy-theory.groups-of-loops-in-1-types where
Imports 
open import foundation.1-types
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.sets
open import foundation.universe-levels

open import group-theory.groups
open import group-theory.semigroups

open import structured-types.pointed-types

open import synthetic-homotopy-theory.loop-spaces

Idea

The loop space of any pointed 1-type is a group.

Definitions

module _
  {l : Level} (A : Pointed-Type l)
  where

  loop-space-Set : is-set (type-Ω A) → Set l
  pr1 (loop-space-Set is-set-Ω) = type-Ω A
  pr2 (loop-space-Set is-set-Ω) = is-set-Ω

  loop-space-Semigroup : is-set (type-Ω A) → Semigroup l
  pr1 (loop-space-Semigroup is-set-Ω) = loop-space-Set is-set-Ω
  pr1 (pr2 (loop-space-Semigroup is-set-Ω)) p q = p ∙ q
  pr2 (pr2 (loop-space-Semigroup is-set-Ω)) = assoc

  loop-space-Group : is-set (type-Ω A) → Group l
  pr1 (loop-space-Group is-set-Ω) = loop-space-Semigroup is-set-Ω
  pr1 (pr1 (pr2 (loop-space-Group is-set-Ω))) = refl
  pr1 (pr2 (pr1 (pr2 (loop-space-Group is-set-Ω)))) q = left-unit
  pr2 (pr2 (pr1 (pr2 (loop-space-Group is-set-Ω)))) p = right-unit
  pr1 (pr2 (pr2 (loop-space-Group is-set-Ω))) = inv
  pr1 (pr2 (pr2 (pr2 (loop-space-Group is-set-Ω)))) = left-inv
  pr2 (pr2 (pr2 (pr2 (loop-space-Group is-set-Ω)))) = right-inv

loop-space-1-type-Set :
  {l : Level} (A : 1-Type l) (a : type-1-Type A) → Set l
loop-space-1-type-Set A a =
  loop-space-Set (pair (type-1-Type A) a) (is-1-type-type-1-Type A a a)

loop-space-1-type-Semigroup :
  {l : Level} (A : 1-Type l) (a : type-1-Type A) → Semigroup l
loop-space-1-type-Semigroup A a =
  loop-space-Semigroup (pair (type-1-Type A) a) (is-1-type-type-1-Type A a a)

loop-space-1-type-Group :
  {l : Level} (A : 1-Type l) (a : type-1-Type A) → Group l
loop-space-1-type-Group A a =
  loop-space-Group (pair (type-1-Type A) a) (is-1-type-type-1-Type A a a)

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