⚠️ This module is not part of this project and is only included for reference.
It is either part of the 1lab, the cubical library, or a built-in Agda module.
<!-- ```agda open import 1Lab.Function.Embedding open import 1Lab.Equiv.Fibrewise open import 1Lab.HLevel.Retracts 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 ``` --> ```agda module 1Lab.Path.IdentitySystem where ``` # Identity systems {defines=identity-system} 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: ```agda 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. ```agda 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 ``` <!-- ```agda 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 $R$ together with reflexivity witness $r$, then any equivalence $f : R(a, b) \simeq (a \equiv b)$ which maps $f(r) = \refl$ equips $(R, r)$ with the structure of an identity system. Of course if we do not particularly care about the specific reflexivity witness, we can simply define $r$ as $f^{-1}(\refl)$. ```agda 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 $(R, r)$, 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. ```agda 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 $f : A \mono B$ 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 $A$. ```agda 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. ```agda 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) ``` <!-- ```agda 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: ```agda 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 ``` <!-- ```agda 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 {defines="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. ```agda 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. ```agda ¬¬-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 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) ```