open import 1Lab.Path.Reasoning
open import 1Lab.Prelude

module Homotopy.Conjugation where

Conjugation of paths🔗

private variable
  ℓ : Level
  A : Type ℓ
  x y z : A
  p q r : x ≡ y

open is-iso

In any type $G$ for which we know two points $x,y:G\text{,}$ the existence of any identification $p:x\equiv y$ induces an equivalence between the loop spaces $x\equiv x$ and $y\equiv y\text{,}$ given by transport in the usual way. However, since we know ahead-of-time what transport in a type of paths computes to, we can take a short-cut and define the equivalence directly: it is given by conjugation with $p\text{.}$

opaque
  conj : ∀ {ℓ} {A : Type ℓ} {x y : A} → y ≡ x → y ≡ y → x ≡ x
  conj p q = sym p ∙∙ q ∙∙ p

opaque
  unfolding conj

  conj-defn : (p : y ≡ x) (q : y ≡ y) → conj p q ≡ sym p ∙ q ∙ p
  conj-defn p q = double-composite (sym p) q p

  conj-defn' : (p : y ≡ x) (q : y ≡ y) → conj p q ≡ subst (λ x → x ≡ x) p q
  conj-defn' p q = conj-defn p q ∙ sym (subst-path-both q p)

opaque
  unfolding conj

  conj-refl : (l : x ≡ x) → conj refl l ≡ l
  conj-refl l = ∙-idr l

  conj-∙ : (p : x ≡ y) (q : y ≡ z) (r : x ≡ x)
         → conj q (conj p r) ≡ conj (p ∙ q) r
  conj-∙ p q r = ∙∙-stack

  conj-of-refl : (p : y ≡ x) → conj p refl ≡ refl
  conj-of-refl p i j = hcomp (i ∨ ∂ j) λ where
    k (k = i0) → p k
    k (i = i1) → p k
    k (j = i0) → p k
    k (j = i1) → p k

  conj-of-∙ : (p : y ≡ x) (q r : y ≡ y) → conj p (q ∙ r) ≡ conj p q ∙ conj p r
  conj-of-∙ p q r = sym ∙∙-chain

opaque
  unfolding conj

  ap-conj
    : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} {x y : A}
    → (f : A → B) (p : y ≡ x) (q : y ≡ y)
    → ap f (conj p q) ≡ conj (ap f p) (ap f q)
  ap-conj f p q = ap-∙∙ f _ _ _

opaque
  conj⁻conj : conj (sym p) (conj p q) ≡ q
  conj⁻conj {p = p} {q = q} =
       ap (conj _) (conj-defn' _ _)
    ∙∙ conj-defn' _ _
    ∙∙ transport⁻transport (λ i → p i ≡ p i) q

opaque
  square→conj
    : {p : y ≡ x} {q₁ : y ≡ y} {q₂ : x ≡ x}
    → Square p q₁ q₂ p → conj p q₁ ≡ q₂
  square→conj p = conj-defn' _ _ ∙ from-pathp p

opaque
  conj-commutative : {p q : x ≡ x} → q ∙ p ≡ p ∙ q → conj p q ≡ q
  conj-commutative α = conj-defn _ _ ∙∙ ap₂ _∙_ refl α ∙∙ ∙-cancell _ _

conj-is-iso : (p : y ≡ x) → is-iso (conj p)
conj-is-iso p .from q = conj (sym p) q
conj-is-iso p .rinv q = conj⁻conj
conj-is-iso p .linv q = conj⁻conj

opaque
  unfolding conj conj-refl conj-of-refl

  conj-refl-square
    : ∀ {ℓ} {A : Type ℓ} {a₀ : A}
    → Square (conj-refl refl) (conj-of-refl refl) (λ i j → a₀) refl
  conj-refl-square {a₀ = a₀} i j k = hcomp (∂ k ∨ i ∨ j) λ where
    l (l = i0) → a₀
    l (k = i0) → a₀
    l (k = i1) → a₀
    l (i = i1) → a₀
    l (j = i1) → a₀

opaque
  conj-is-equiv : (p : y ≡ x) → is-equiv (conj p)
  conj-is-equiv p = is-iso→is-equiv (conj-is-iso p)

module conj {ℓ} {A : Type ℓ} {x y : A} (p : y ≡ x) = Equiv (conj p , conj-is-equiv p)