open import 1Lab.HLevel.Retracts
open import 1Lab.Path.Groupoid
open import 1Lab.Equiv.Biinv
open import 1Lab.Type.Sigma
open import 1Lab.Univalence
open import 1Lab.HLevel
open import 1Lab.Equiv
open import 1Lab.Path
open import 1Lab.Type

module 1Lab.Equiv.HalfAdjoint where

Adjoint Equivalences🔗

An adjoint equivalence is an isomorphism (f,g,η,ε)(f, g, \eta, \varepsilon) where the homotopies (η\eta, ε\varepsilon) satisfy the triangle identities, thus witnessing ff and gg as adjoint functors. In Homotopy Type Theory, we can use a half adjoint equivalence - satisfying only one of the triangle identities - as a good notion of equivalence.

is-half-adjoint-equiv :  {ℓ₁ ℓ₂} {A : Type ℓ₁} {B : Type ℓ₂} (f : A  B)  Type _
is-half-adjoint-equiv {A = A} {B = B} f =
  Σ[ g  (B  A) ]
  Σ[ η  ((x : A)  g (f x)  x) ]
  Σ[ ε  ((y : B)  f (g y)  y) ]
  ((x : A)  ap f (η x)  ε (f x))

The argument is an adaptation of a standard result of both category theory and homotopy theory - where we can “improve” an equivalence of categories into an adjoint equivalence, or a homotopy equivalence into a strong homotopy equivalence (Vogt’s lemma). In HoTT, we show this synthetically for equivalences between \infty-groupoids.

  :  {ℓ₁ ℓ₂} {A : Type ℓ₁} {B : Type ℓ₂} {f : A  B}
   is-iso f  is-half-adjoint-equiv f
is-iso→is-half-adjoint-equiv {A = A} {B} {f} iiso =
  g , η , ε' , λ x  sym (zig x)
    open is-iso iiso renaming (inv to g ; linv to η ; rinv to ε)

For gg and η\eta, we can take the values provided by is-iso. However, if we want (η,ε)(\eta, \varepsilon) to satisfy the triangle identities, we can not in general take ε=ε\varepsilon' = \varepsilon. We can, however, alter it like thus:

    ε' : (y : B)  f (g y)  y
    ε' y = sym (ε (f (g y)))  ap f (η (g y))  ε y

Drawn as a diagram, the path above factors like:

There is a great deal of redundancy in this definition, given that εy\varepsilon y and εy\varepsilon' y have the same boundary! The point is that while the definition of ε\varepsilon is entirely opaque to us, ε\varepsilon' is written in such a way that we can use properties of paths to make the sym (ε...)\id{sym}\ (\varepsilon ...) and ε\varepsilon cancel:

    zig : (x : A)  ε' (f x)  ap f (η x)
    zig x =
      ε' (f x)                                                    ≡⟨⟩
      sym (ε (f (g (f x))))   ap f (η (g (f x)))    ε (f x)     ≡⟨ ap₂ _∙_ refl (ap₂ _∙_ (ap (ap f) (homotopy-invert η)) refl) 
      sym (ε (f (g (f x))))   ap (f  g  f) (η x)  ε (f x)     ≡⟨ ap₂ _∙_ refl (sym (homotopy-natural ε _)) 
      sym (ε (f (g (f x))))   ε (f (g (f x)))       ap f (η x)  ≡⟨ ∙-cancel-l (ε (f (g (f x)))) (ap f (η x)) 
      ap f (η x)                                                  

The notion of half-adjoint equivalence is a useful stepping stone in writing a more comprehensible proof that isomorphisms are equivalences. Since this result is fundamental, the proof we actually use is written with efficiency of computation in mind - hence, cubically. The proof here is intended to be more educational.

First, we give an equivalent characterisation of paths in fibres, which will be used in proving that half adjoint equivalences are equivalences.

fibre-paths :  {ℓ₁ ℓ₂} {A : Type ℓ₁} {B : Type ℓ₂} {f : A  B} {y : B}
             {f1 f2 : fibre f y}
             (f1  f2)
             (Σ[ γ  f1 .fst  f2 .fst ] (ap f γ  f2 .snd  f1 .snd))
The proof of this is not very enlightening, but it’s included here (rather than being completely invisible) for completeness:
fibre-paths {f = f} {y} {f1} {f2} =
  Path (fibre f y) f1 f2                                                       ≃⟨ Iso→Equiv Σ-path-iso e⁻¹ 
  (Σ[ γ  f1 .fst  f2 .fst ] (subst  x₁  f x₁  _) γ (f1 .snd)  f2 .snd)) ≃⟨ Σ-ap-snd  x  path→equiv (lemma x)) 
  (Σ[ γ  f1 .fst  f2 .fst ] (ap f γ  f2 .snd  f1 .snd))                    ≃∎
    helper : (p' : f (f1 .fst)  y)
            (subst  x  f x  y) refl (f1 .snd)  p')
            (ap f refl  p'  f1 .snd)
    helper p' =
      subst  x  f x  y) refl (f1 .snd)  p' ≡⟨ ap₂ _≡_ (transport-refl _) refl 
      (f1 .snd)  p'                            ≡⟨ Iso→Path (sym , iso sym  x  refl)  x  refl)) 
      p'  f1 .snd                              ≡⟨ ap₂ _≡_ (sym (∙-id-l _)) refl 
      refl  p'  f1 .snd                       ≡⟨⟩
      ap f refl  p'  f1 .snd                  

    lemma :  {x'} {p'}  (γ : f1 .fst  x')
           (subst  x  f x  _) γ (f1 .snd)  p')
           (ap f γ  p'  f1 .snd)
    lemma {x'} {p'} p =
      J  x' γ   p'  (subst  x  f x  _) γ (f1 .snd)  p')
                        (ap f γ  p'  f1 .snd))
        helper p p'

Then, given an element y:By : B, we can construct a fibre of of ff, and, using the above characterisation of paths, prove that this fibre is a centre of contraction:

  :  {ℓ₁ ℓ₂} {A : Type ℓ₁} {B : Type ℓ₂} {f : A  B}
   is-half-adjoint-equiv f  is-equiv f
is-half-adjoint-equiv→is-equiv {A = A} {B} {f} (g , η , ε , zig) .is-eqv y = contr fib contract where
  fib : fibre f y
  fib = g y , ε y

The fibre is given by (g(y),ε(y))(g(y), ε(y)), which we can prove identical to another (x,p)(x, p) using a very boring calculation:

  contract : (fib₂ : fibre f y)  fib  fib₂
  contract (x , p) = (fibre-paths e⁻¹) .fst (x≡gy , path) where
    x≡gy = ap g (sym p)  η x

    path : ap f (ap g (sym p)  η x)  p  ε y
    path =
      ap f (ap g (sym p)  η x)  p               ≡⟨ ap₂ _∙_ (ap-comp-path f (ap g (sym p)) (η x)) refl  sym (∙-assoc _ _ _) 
      ap (f  g) (sym p)  ap f (η x)  p         ≡⟨ ap₂ _∙_ refl (ap₂ _∙_ (zig _) refl)  -- by the triangle identity
      ap (f  g) (sym p)  ε (f x)     p         ≡⟨ ap₂ _∙_ refl (homotopy-natural ε p)   -- by naturality of ε

The calculation of path factors as a bunch of boring adjustments to paths using the groupoid structure of types, and the two interesting steps above: The triangle identity says that ap(f)(ηx)=ε(fx)\id{ap}(f)(\eta x) = \varepsilon(f x), and naturality of ε\varepsilon lets us “push it past pp” to get something we can cancel:

      ap (f  g) (sym p)  ap (f  g) p  ε y     ≡⟨ ∙-assoc _ _ _ 
      (ap (f  g) (sym p)  ap (f  g) p)  ε y   ≡⟨ ap₂ _∙_ (sym (ap-comp-path (f  g) (sym p) p)) refl 
      ap (f  g) (sym p  p)  ε y                ≡⟨ ap₂ _∙_ (ap (ap (f  g)) (∙-inv-r _)) refl 
      ap (f  g) refl  ε y                       ≡⟨⟩
      refl  ε y                                  ≡⟨ ∙-id-l (ε y) 
      ε y                                         

Putting these together, we get an alternative definition of is-iso→is-equiv:

  :  {ℓ₁ ℓ₂} {A : Type ℓ₁} {B : Type ℓ₂} {f : A  B}
   is-iso f  is-equiv f
is-iso→is-equiv' = is-half-adjoint-equiv→is-equiv  is-iso→is-half-adjoint-equiv