module 1Lab.Path.IdentitySystem where

Identity systems🔗

An identity system is a way of characterising the path spaces of a particular type, without necessarily having to construct a full encode-decode equivalence. Essentially, the data of an identity system is precisely the data required to implement path induction, a.k.a. the J eliminator. Any type with the data of an identity system satisfies its own J, and conversely, if the type satisfies J, it is an identity system.

We unravel the definition of being an identity system into the following data, using a translation that takes advantage of cubical type theory’s native support for paths-over-paths:

record
  is-identity-system {ℓ ℓ'} {A : Type ℓ}
    (R : A → A → Type ℓ')
    (refl : ∀ a → R a a)
    : Type (ℓ ⊔ ℓ')
  where
  no-eta-equality
  field
    to-path      : ∀ {a b} → R a b → a ≡ b
    to-path-over
      : ∀ {a b} (p : R a b)
      → PathP (λ i → R a (to-path p i)) (refl a) p

  is-contr-ΣR : ∀ {a} → is-contr (Σ A (R a))
  is-contr-ΣR .centre    = _ , refl _
  is-contr-ΣR .paths x i = to-path (x .snd) i , to-path-over (x .snd) i

open is-identity-system public

As mentioned before, the data of an identity system gives us exactly what is required to prove J for the relation RR. This is essentially the decomposition of J into contractibility of singletons, but with singletons replaced by RR-singletons.

IdsJ
  : ∀ {ℓ ℓ' ℓ''} {A : Type ℓ} {R : A → A → Type ℓ'} {r : ∀ a → R a a} {a : A}
  → is-identity-system R r
  → (P : ∀ b → R a b → Type ℓ'')
  → P a (r a)
  → ∀ {b} s → P b s
IdsJ ids P pr s =
  transport (λ i → P (ids .to-path s i) (ids .to-path-over s i)) pr
IdsJ-refl
  : ∀ {ℓ ℓ' ℓ''} {A : Type ℓ} {R : A → A → Type ℓ'} {r : ∀ a → R a a} {a : A}
  → (ids : is-identity-system R r)
  → (P : ∀ b → R a b → Type ℓ'')
  → (x : P a (r a))
  → IdsJ ids P x (r a) ≡ x
IdsJ-refl {R = R} {r = r} {a = a} ids P x =
  transport (λ i → P (ids .to-path (r a) i) (ids .to-path-over (r a) i)) x ≡⟨⟩
  subst P' (λ i → ids .to-path (r a) i , ids .to-path-over (r a) i) x      ≡⟨ ap (λ e → subst P' e x) lemma ⟩≡
  subst P' refl x                                                          ≡⟨ transport-refl x ⟩≡
  x ∎
  where
    P' : Σ _ (R a) → Type _
    P' (b , r) = P b r

    lemma : Σ-pathp (ids .to-path (r a)) (ids .to-path-over (r a)) ≡ refl
    lemma = is-contr→is-set (is-contr-ΣR ids) _ _ _ _

to-path-refl-coh
  : ∀ {ℓ ℓ'} {A : Type ℓ} {R : A → A → Type ℓ'} {r : ∀ a → R a a}
  → (ids : is-identity-system R r)
  → ∀ a
  → (Σ-pathp (ids .to-path (r a)) (ids .to-path-over (r a))) ≡ refl
to-path-refl-coh {r = r} ids a =
  is-contr→is-set (is-contr-ΣR ids) _ _
    (Σ-pathp (ids .to-path (r a)) (ids .to-path-over (r a)))
    refl

to-path-refl
  : ∀ {ℓ ℓ'} {A : Type ℓ} {R : A → A → Type ℓ'} {r : ∀ a → R a a} {a : A}
  → (ids : is-identity-system R r)
  → ids .to-path (r a) ≡ refl
to-path-refl {r = r} {a = a} ids = ap (ap fst) $ to-path-refl-coh ids a

to-path-over-refl
  : ∀ {ℓ ℓ'} {A : Type ℓ} {R : A → A → Type ℓ'} {r : ∀ a → R a a} {a : A}
  → (ids : is-identity-system R r)
  → PathP (λ i → PathP (λ j → R a (to-path-refl {a = a} ids i j)) (r a) (r a))
      (ids .to-path-over (r a))
      refl
to-path-over-refl {a = a} ids = ap (ap snd) $ to-path-refl-coh ids a

If we have a relation RR together with reflexivity witness rr, then any equivalence f:R(a,b)≃(a≡b)f : R(a, b) \simeq (a \equiv b) which maps f(r)=reflf(r) = \mathrm{refl} equips (R,r)(R, r) with the structure of an identity system. Of course if we do not particularly care about the specific reflexivity witness, we can simply define rr as f−1(refl)f^{-1}(\mathrm{refl}).

equiv-path→identity-system
  : ∀ {ℓ ℓ'} {A : Type ℓ} {R : A → A → Type ℓ'} {r : ∀ a → R a a}
  → (eqv : ∀ {a b} → R a b ≃ (a ≡ b))
  → (∀ a → Equiv.from eqv refl ≡ r a)
  → is-identity-system R r
equiv-path→identity-system {R = R} {r = r} eqv pres' = ids where
  contract : ∀ {a} → is-contr (Σ _ (R a))
  contract = is-hlevel≃ 0 ((total (λ _ → eqv .fst) , equiv→total (eqv .snd)))
    (contr _ Singleton-is-contr)

  pres : ∀ {a} → eqv .fst (r a) ≡ refl
  pres {a = a} = Equiv.injective₂ (eqv e⁻¹) (Equiv.η eqv _) (pres' _)

  ids : is-identity-system R r
  ids .to-path = eqv .fst
  ids .to-path-over {a = a} {b = b} p i =
    is-prop→pathp
    (λ i → is-contr→is-prop (eqv .snd .is-eqv λ j → eqv .fst p (i ∧ j)))
    (r a , pres)
    (p , refl)
    i .fst

Note that for any (R,r)(R, r), the type of identity system data on (R,r)(R, r) is a proposition. This is because it is exactly equivalent to the type ∑a(Ra)\sum_a (R a) being contractible for every aa, which is a proposition by standard results.

identity-system-gives-path
  : ∀ {ℓ ℓ'} {A : Type ℓ} {R : A → A → Type ℓ'} {r : ∀ a → R a a}
  → is-identity-system R r
  → ∀ {a b} → R a b ≃ (a ≡ b)
identity-system-gives-path {R = R} {r = r} ids =
  Iso→Equiv (ids .to-path , iso from ri li) where
    from : ∀ {a b} → a ≡ b → R a b
    from {a = a} p = transport (λ i → R a (p i)) (r a)

    ri : ∀ {a b} → is-right-inverse (from {a} {b}) (ids .to-path)
    ri = J (λ y p → ids .to-path (from p) ≡ p)
           ( ap (ids .to-path) (transport-refl _)
           ∙ to-path-refl ids)

    li : ∀ {a b} → is-left-inverse (from {a} {b}) (ids .to-path)
    li = IdsJ ids (λ y p → from (ids .to-path p) ≡ p)
          ( ap from (to-path-refl ids)
          ∙ transport-refl _ )

In subtypes🔗

Let f:A↪Bf : A \hookrightarrow B be an embedding. If (R,r)(R, r) is an identity system on BB, then it can be pulled back along ff to an identity system on AA.

module
  _ {ℓ ℓ' ℓ''} {A : Type ℓ} {B : Type ℓ'}
    {R : B → B → Type ℓ''} {r : ∀ b → R b b}
    (ids : is-identity-system R r)
    (f : A ↪ B)
  where

  pullback-identity-system
    : is-identity-system (λ x y → R (f .fst x) (f .fst y)) (λ _ → r _)
  pullback-identity-system .to-path {a} {b} x = ap fst $
    f .snd (f .fst b) (a , ids .to-path x) (b , refl)
  pullback-identity-system .to-path-over {a} {b} p i =
    comp
      (λ j → R (f .fst a) (f .snd (f .fst b) (a , ids .to-path p) (b , refl) i .snd (~ j)))
      (∂ i) λ where
      k (k = i0) → ids .to-path-over p (~ k)
      k (i = i0) → ids .to-path-over p (~ k ∨ i)
      k (i = i1) → p

This is actually part of an equivalence: if the equality identity system on BB (thus any identity system) can be pulled back along ff, then ff is an embedding.

identity-system→embedding
  : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'}
  → (f : A → B)
  → is-identity-system (λ x y → f x ≡ f y) (λ _ → refl)
  → is-embedding f
identity-system→embedding f ids = cancellable→embedding
  (identity-system-gives-path ids)

Univalence🔗

Note that univalence is precisely the statement that equivalences are an identity system on the universe:

univalence-identity-system
  : ∀ {ℓ} → is-identity-system {A = Type ℓ} _≃_ λ _ → id , id-equiv
univalence-identity-system .to-path = ua
univalence-identity-system .to-path-over p =
  Σ-prop-pathp (λ _ → is-equiv-is-prop) $ funextP $ λ a → path→ua-pathp p refl
Path-identity-system
  : ∀ {ℓ} {A : Type ℓ} → is-identity-system (Path A) (λ _ → refl)
Path-identity-system .to-path p = p
Path-identity-system .to-path-over p i j = p (i ∧ j)

is-identity-system-is-prop
  : ∀ {ℓ ℓ'} {A : Type ℓ} {R : A → A → Type ℓ'} {r : ∀ a → R a a}
  → is-prop (is-identity-system R r)
is-identity-system-is-prop {A = A} {R} {r} =
  retract→is-hlevel 1 from to cancel λ x y i a → is-contr-is-prop (x a) (y a) i
  where
    to : is-identity-system R r → ∀ x → is-contr (Σ A (R x))
    to ids x = is-contr-ΣR ids

    sys : ∀ (l : ∀ x → is-contr (Σ A (R x))) a b (s : R a b) (i j : I)
        → Partial (∂ i ∨ ~ j) (Σ A (R a))
    sys l a b s i j (j = i0) = l a .centre
    sys l a b s i j (i = i0) = l a .paths (a , r a) j
    sys l a b s i j (i = i1) = l a .paths (b , s) j

    from : (∀ x → is-contr (Σ A (R x))) → is-identity-system R r
    from x .to-path      {a} {b} s i = hcomp (∂ i) (sys x a b s i) .fst
    from x .to-path-over {a} {b} s i = hcomp (∂ i) (sys x a b s i) .snd

    square : ∀ (x : is-identity-system R r) a b (s : R a b)
           → Square {A = Σ A (R a)}
             (λ i → x .to-path (r a) i , x .to-path-over (r a) i)
             (λ i → x .to-path s i , x .to-path-over s i)
             (λ i → x .to-path s i , x .to-path-over s i)
             refl
    square x a b s i j = hcomp (∂ i ∨ ∂ j) λ where
      k (k = i0) → x .to-path s j , x .to-path-over s j
      k (i = i0) → x .to-path s j , x .to-path-over s j
      k (i = i1) → x .to-path s j , x .to-path-over s j
      k (j = i0) → to-path-refl-coh {R = R} {r = r} x a (~ k) i
      k (j = i1) → b , s

    sys' : ∀ (x : is-identity-system R r) a b (s : R a b) i j k
         → Partial (∂ i ∨ ∂ j ∨ ~ k) (Σ A (R a))
    sys' x a b s i j k (k = i0) = x .to-path (r a) i , x .to-path-over (r a) i
    sys' x a b s i j k (i = i0) = hfill (∂ j) k (sys (to x) a b s j)
    sys' x a b s i j k (i = i1) =
        x .to-path (x .to-path-over s (k ∨ j)) (k ∧ j)
      , x .to-path-over (x .to-path-over s (k ∨ j)) (k ∧ j)
    sys' x a b s i j k (j = i0) =
        x .to-path (r a) (k ∨ i) , x .to-path-over (r a) (k ∨ i)
    sys' x a b s i j k (j = i1) = square x a b s i k

    cancel : is-left-inverse from to
    cancel x i .to-path {a} {b} s j      = hcomp (∂ i ∨ ∂ j) (sys' x a b s i j) .fst
    cancel x i .to-path-over {a} {b} s j = hcomp (∂ i ∨ ∂ j) (sys' x a b s i j) .snd

instance
  H-Level-is-identity-system
    : ∀ {ℓ ℓ'} {A : Type ℓ} {R : A → A → Type ℓ'} {r : ∀ a → R a a} {n}
    → H-Level (is-identity-system R r) (suc n)
  H-Level-is-identity-system = prop-instance is-identity-system-is-prop

identity-system→hlevel
  : ∀ {ℓ ℓ'} {A : Type ℓ} n {R : A → A → Type ℓ'} {r : ∀ x → R x x}
  → is-identity-system R r
  → (∀ x y → is-hlevel (R x y) n)
  → is-hlevel A (suc n)
identity-system→hlevel zero ids hl x y = ids .to-path (hl _ _ .centre)
identity-system→hlevel (suc n) ids hl x y =
  is-hlevel≃ (suc n) (identity-system-gives-path ids e⁻¹) (hl x y)

Sets and Hedberg’s theorem🔗

We now apply the general theory of identity systems to something a lot more mundane: recognising sets. An immediate consequence of having an identity system (R,r)(R, r) on a type AA is that, if RR is pointwise an nn-type, then AA is an (n+1)(n+1)-type. Now, if RR is a reflexive family of propositions, then all we need for (R,r)(R, r) to be an identity system is that R(x,y)→x=yR(x, y) \to x = y, by the previous observation, this implies AA is a set.

set-identity-system
  : ∀ {ℓ ℓ'} {A : Type ℓ} {R : A → A → Type ℓ'} {r : ∀ x → R x x}
  → (∀ x y → is-prop (R x y))
  → (∀ {x y} → R x y → x ≡ y)
  → is-identity-system R r
set-identity-system rprop rpath .to-path = rpath
set-identity-system rprop rpath .to-path-over p =
  is-prop→pathp (λ i → rprop _ _) _ p

If AA is a type with ¬¬-stable equality, then by the theorem above, the pointwise double negation of its identity types is an identity system: and so, if a type has decidable (thus ¬¬-stable) equality, it is a set.

¬¬-stable-identity-system
  : ∀ {ℓ} {A : Type ℓ}
  → (∀ {x y} → ¬ ¬ Path A x y → x ≡ y)
  → is-identity-system (λ x y → ¬ ¬ Path A x y) λ a k → k refl
¬¬-stable-identity-system = set-identity-system λ x y f g →
  funext λ h → absurd (g h)

Discrete→is-set : ∀ {ℓ} {A : Type ℓ} → Discrete A → is-set A
Discrete→is-set {A = A} dec =
  identity-system→hlevel 1 (¬¬-stable-identity-system stable) λ x y f g →
    funext λ h → absurd (g h)
  where
    stable : {x y : A} → ¬ ¬ x ≡ y → x ≡ y
    stable {x = x} {y = y} ¬¬p with dec {x} {y}
    ... | yes p = p
    ... | no ¬p = absurd (¬¬p ¬p)