open import 1Lab.Function.Embedding
open import 1Lab.Equiv.Fibrewise
open import 1Lab.HLevel.Closure
open import 1Lab.Type.Sigma
open import 1Lab.Univalence
open import 1Lab.Type.Pi
open import 1Lab.HLevel
open import 1Lab.Equiv
open import 1Lab.Path
open import 1Lab.Type

open import Data.Dec.Base

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 $R .$ This is essentially the decomposition of J into contractibility of singletons, but with singletons replaced by $R -singletons.$

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

Note that for any $(R, r),$ the type of identity system data on $(R, r)$ is a proposition. This is because it is exactly equivalent to the type $\sum_{a} (R a)$ being contractible for every $a,$ which is a proposition by standard results. One direction is is-contr-ΣR; we prove the converse direction now.

contr→identity-system
  : ∀ {ℓ ℓ'} {A : Type ℓ} {R : A → A → Type ℓ'} {r : ∀ a → R a a}
  → (∀ {a} → is-contr (Σ _ (R a)))
  → is-identity-system R r
contr→identity-system {R = R} {r} c = ids where
  paths' : ∀ {a} (p : Σ _ (R a)) → (a , r a) ≡ p
  paths' _ = is-contr→is-prop c _ _

  ids : is-identity-system R r
  ids .to-path p = ap fst (paths' (_ , p))
  ids .to-path-over p = ap snd (paths' (_ , p))

If we have a relation $R$ together with reflexivity witness $r,$ then any equivalence $f : R (a, b) ≃ (a \equiv b)$ equips $(R, r)$ with the structure of an identity system, by contractibility of singletons. Of course if we do not particularly care about the specific reflexivity witness, we can simply define $r$ as $f^{- 1} (refl) .$

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

Conversely, any identity system $(R, r)$ implies an equivalence $R (a, b) ≃ (a \equiv b) .$

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 _ )

Based identity systems🔗

It is sometimes useful to characterise the based identity type at a point $a : A,$ i.e. the family $a \equiv -,$ instead of the whole binary family of paths $- \equiv - .$ To that end, we introduce a unary variant of identity systems called based identity systems, or unary identity systems.

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

open is-based-identity-system public

is-contr-ΣC
  : ∀ {ℓ ℓ'} {A : Type ℓ} {a : A} {C : A → Type ℓ'} {r : C a}
  → is-based-identity-system a C r
  → is-contr (Σ A C)
is-contr-ΣC {r = r} ids .centre         = _ , r
is-contr-ΣC {r = r} ids .paths x i .fst = ids .to-path (x .snd) i
is-contr-ΣC {r = r} ids .paths x i .snd = ids .to-path-over (x .snd) i

As previously, the data of a based identity system at $a : A$ is precisely what is required to implement based path induction at $a .$

IdsJ-based
  : ∀ {ℓ ℓ' ℓ''} {A : Type ℓ} {a : A} {C : A → Type ℓ'} {r : C a}
  → is-based-identity-system a C r
  → (P : ∀ b → C b → Type ℓ'')
  → P a r
  → ∀ {b} s → P b s
IdsJ-based ids P pr s = transport
  (λ i → P (ids .to-path s i) (ids .to-path-over s i)) pr

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

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

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

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

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

contr→based-identity-system
  : ∀ {ℓ ℓ'} {A : Type ℓ} {a : A} {C : A → Type ℓ'} {r : C a}
  → is-contr (Σ _ C)
  → is-based-identity-system a C r
contr→based-identity-system {a = a} {C = C} {r} c = ids where
  paths' : ∀ (p : Σ _ C) → (a , r) ≡ p
  paths' _ = is-contr→is-prop c _ _

  ids : is-based-identity-system a C r
  ids .to-path p = ap fst (paths' (_ , p))
  ids .to-path-over p = ap snd (paths' (_ , p))

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

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

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

In subtypes🔗

Let $f : A ↪ B$ be an embedding. If $(R, r)$ is an identity system on $B,$ then it can be pulled back along $f$ to an identity system on $A .$

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 $B$ (thus any identity system) can be pulled back along $f,$ then $f$ 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)

module
  _ {ℓ ℓ'} {A : Type ℓ}
    {R S : A → A → Type ℓ'}
    {r : ∀ a → R a a} {s : ∀ a → S a a}
    (ids : is-identity-system R r)
    (eqv : ∀ x y → R x y ≃ S x y)
    (pres : ∀ x → eqv x x .fst (r x) ≡ s x)
  where

  transfer-identity-system : is-identity-system S s
  transfer-identity-system .to-path sab = ids .to-path (Equiv.from (eqv _ _) sab)
  transfer-identity-system .to-path-over {a} {b} p i = hcomp (∂ i) λ where
    j (j = i0) → Equiv.to (eqv _ _) (ids .to-path-over (Equiv.from (eqv _ _) p) i)
    j (i = i0) → pres a j
    j (i = i1) → Equiv.ε (eqv _ _) p j

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

    from : (∀ x → is-contr (Σ A (R x))) → is-identity-system R r
    from x = contr→identity-system (x _)

    cancel'
      : ∀ (x : is-identity-system R r) {a b} (s : R a b)
      → PathP (λ i → (a , r a) ≡ (b , s))
        (is-contr-ΣR (from (to x)) .paths (b , s))
        (is-contr-ΣR x .paths (b , s))
    cancel' x s = is-prop→squarep (λ _ _ → is-contr→is-prop (is-contr-ΣR x)) _ _ _ _

    cancel : is-left-inverse from to
    cancel x i .to-path s = ap fst (cancel' x s i)
    cancel x i .to-path-over s = ap snd (cancel' x s i)

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 =
  Equiv→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 $(R, r)$ on a type $A$ is that, if $R$ is pointwise an $n -type,$ then $A$ is an $(n + 1) -type.$ Now, if $R$ is a reflexive family of propositions, then all we need for $(R, r)$ to be an identity system is that $R (x, y) \to x = y,$ by the previous observation, this implies $A$ 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 $A$ 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. This is known as Hedberg’s theorem.

¬¬-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)

opaque
  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 (dec→dne ⦃ dec ⦄))
    λ x y f g → funext λ h → absurd (g h)