module 1Lab.Type.Sigma where
Properties of Σ types🔗
This module contains properties of types, not necessarily organised in any way.
Groupoid structure🔗
The first thing we prove is that paths in sigmas are sigmas of paths. The type signatures make it clearer:
Σ-pathp-iso : {A : I → Type ℓ} {B : (i : I) → A i → Type ℓ₁} {x : Σ (A i0) (B i0)} {y : Σ (A i1) (B i1)} → Iso (Σ[ p ∈ PathP A (x .fst) (y .fst) ] (PathP (λ i → B i (p i)) (x .snd) (y .snd))) (PathP (λ i → Σ (A i) (B i)) x y) Σ-path-iso : {x y : Σ A B} → Iso (Σ[ p ∈ x .fst ≡ y .fst ] (subst B p (x .snd) ≡ y .snd)) (x ≡ y)
The first of these, using a dependent path, is easy to prove
directly, because paths in cubical type theory automatically
inherit the structure of their domain types. The second is a consequence
of the first, using the fact that PathPs and paths over a transport are the same.
fst Σ-pathp-iso (p , q) i = p i , q i is-iso.inv (snd Σ-pathp-iso) p = (λ i → p i .fst) , (λ i → p i .snd) is-iso.rinv (snd Σ-pathp-iso) x = refl is-iso.linv (snd Σ-pathp-iso) x = refl Σ-path-iso {B = B} {x} {y} = transport (λ i → Iso (Σ[ p ∈ x .fst ≡ y .fst ] (PathP≡Path (λ j → B (p j)) (x .snd) (y .snd) i)) (x ≡ y)) Σ-pathp-iso
Closure under equivalences🔗
Univalence automatically implies that every type former respects
equivalences. However, this theorem is limited to equivalences between
types in the same universe. Thus, we provide Σ-ap-fst, Σ-ap-snd, and Σ-ap, which allows one to
perturb a Σ by equivalences across
levels:
Σ-ap-snd : ((x : A) → P x ≃ Q x) → Σ A P ≃ Σ A Q Σ-ap-fst : (e : A ≃ A') → Σ A (B ∘ e .fst) ≃ Σ A' B Σ-ap : (e : A ≃ A') → ((x : A) → P x ≃ Q (e .fst x)) → Σ A P ≃ Σ A' Q Σ-ap e e' = Σ-ap-snd e' ∙e Σ-ap-fst e
The proofs of these theorems are not very enlightening, but they are included for completeness.
Σ-ap-snd {A = A} {P = P} {Q = Q} pointwise = Iso→Equiv morp where pwise : (x : A) → Iso (P x) (Q x) pwise x = _ , is-equiv→is-iso (pointwise x .snd) morp : Iso (Σ _ P) (Σ _ Q) fst morp (i , x) = i , pointwise i .fst x is-iso.inv (snd morp) (i , x) = i , pwise i .snd .is-iso.inv x is-iso.rinv (snd morp) (i , x) = ap₂ _,_ refl (pwise i .snd .is-iso.rinv _) is-iso.linv (snd morp) (i , x) = ap₂ _,_ refl (pwise i .snd .is-iso.linv _) Σ-ap-fst {A = A} {A' = A'} {B = B} e = intro , isEqIntro where intro : Σ _ (B ∘ e .fst) → Σ _ B intro (a , b) = e .fst a , b isEqIntro : is-equiv intro isEqIntro .is-eqv x = contr ctr isCtr where PB : ∀ {x y} → x ≡ y → B x → B y → Type _ PB p = PathP (λ i → B (p i)) open Σ x renaming (fst to a'; snd to b) open Σ (e .snd .is-eqv a' .is-contr.centre) renaming (fst to ctrA; snd to α) ctrB : B (e .fst ctrA) ctrB = subst B (sym α) b ctrP : PB α ctrB b ctrP i = coe1→i (λ i → B (α i)) i b ctr : fibre intro x ctr = (ctrA , ctrB) , Σ-pathp α ctrP isCtr : ∀ y → ctr ≡ y isCtr ((r , s) , p) = λ i → (a≡r i , b!≡s i) , Σ-pathp (α≡ρ i) (coh i) where open Σ (Σ-pathp-iso .snd .is-iso.inv p) renaming (fst to ρ; snd to σ) open Σ (Σ-pathp-iso .snd .is-iso.inv (e .snd .is-eqv a' .is-contr.paths (r , ρ))) renaming (fst to a≡r; snd to α≡ρ) b!≡s : PB (ap (e .fst) a≡r) ctrB s b!≡s i = comp (λ k → B (α≡ρ i (~ k))) (∂ i) λ where k (i = i0) → ctrP (~ k) k (i = i1) → σ (~ k) k (k = i0) → b coh : PathP (λ i → PB (α≡ρ i) (b!≡s i) b) ctrP σ coh i j = fill (λ k → B (α≡ρ i (~ k))) (∂ i) (~ j) λ where k (i = i0) → ctrP (~ k) k (i = i1) → σ (~ k) k (k = i0) → b Σ-assoc : ∀ {ℓ ℓ' ℓ''} {A : Type ℓ} {B : A → Type ℓ'} {C : (x : A) → B x → Type ℓ''} → (Σ[ x ∈ A ] Σ[ y ∈ B x ] C x y) ≃ (Σ[ x ∈ Σ _ B ] (C (x .fst) (x .snd))) Σ-assoc .fst (x , y , z) = (x , y) , z Σ-assoc .snd .is-eqv y .centre = strict-fibres (λ { ((x , y) , z) → x , y , z}) y .fst Σ-assoc .snd .is-eqv y .paths = strict-fibres (λ { ((x , y) , z) → x , y , z}) y .snd Σ-Π-distrib : ∀ {ℓ ℓ' ℓ''} {A : Type ℓ} {B : A → Type ℓ'} {C : (x : A) → B x → Type ℓ''} → ((x : A) → Σ[ y ∈ B x ] C x y) ≃ (Σ[ f ∈ ((x : A) → B x) ] ((x : A) → C x (f x))) Σ-Π-distrib .fst f = (λ x → f x .fst) , λ x → f x .snd Σ-Π-distrib .snd .is-eqv y .centre = strict-fibres (λ f x → f .fst x , f .snd x) y .fst Σ-Π-distrib .snd .is-eqv y .paths = strict-fibres (λ f x → f .fst x , f .snd x) y .snd
Paths in subtypes🔗
When P is a family of propositions, it is sound to
regard Σ[ x ∈ A ] (P x) as a subtype of
A. This is because identification in the subtype is
characterised uniquely by identification of the first projections:
Σ-prop-path : {B : A → Type ℓ} → (∀ x → is-prop (B x)) → {x y : Σ _ B} → (x .fst ≡ y .fst) → x ≡ y Σ-prop-path bp {x} {y} p i = p i , is-prop→pathp (λ i → bp (p i)) (x .snd) (y .snd) i
The proof that this is an equivalence uses a cubical argument, but
the gist of it is that since B is a family of propositions,
it really doesn’t matter where we get our equality of B-s
from - whether it was from the input, or from Σ≡Path.
Σ-prop-path-is-equiv : {B : A → Type ℓ} → (bp : ∀ x → is-prop (B x)) → {x y : Σ _ B} → is-equiv (Σ-prop-path bp {x} {y}) Σ-prop-path-is-equiv bp {x} {y} = is-iso→is-equiv isom where isom : is-iso _ isom .is-iso.inv = ap fst isom .is-iso.linv p = refl
The inverse is the action on paths of the first projection, which lets us
conclude x .fst ≡ y .fst from x ≡ y. This is a
left inverse to Σ-prop-path on the nose. For
the other direction, we have the aforementioned cubical argument:
isom .is-iso.rinv p i j = p j .fst , is-prop→pathp (λ k → Path-is-hlevel 1 (bp (p k .fst)) {x = Σ-prop-path bp {x} {y} (ap fst p) k .snd} {y = p k .snd}) refl refl j i
Since Σ-prop-path is an equivalence,
this implies that its inverse, ap fst, is also an equivalence; This is
precisely what it means for fst to be an embedding.
There is also a convenient packaging of the previous two definitions into an equivalence:
Σ-prop-path≃ : {B : A → Type ℓ} → (∀ x → is-prop (B x)) → {x y : Σ _ B} → (x .fst ≡ y .fst) ≃ (x ≡ y) Σ-prop-path≃ bp = Σ-prop-path bp , Σ-prop-path-is-equiv bp
Σ-prop-square : ∀ {ℓ ℓ'} {A : Type ℓ} {B : A → Type ℓ'} → {w x y z : Σ _ B} → (∀ x → is-prop (B x)) → {p : x ≡ w} {q : x ≡ y} {s : w ≡ z} {r : y ≡ z} → Square (ap fst p) (ap fst q) (ap fst s) (ap fst r) → Square p q s r Σ-prop-square Bprop sq i j .fst = sq i j Σ-prop-square Bprop {p} {q} {s} {r} sq i j .snd = is-prop→squarep (λ i j → Bprop (sq i j)) (ap snd p) (ap snd q) (ap snd s) (ap snd r) i j Σ-set-square : ∀ {ℓ ℓ'} {A : Type ℓ} {B : A → Type ℓ'} → {w x y z : Σ _ B} → (∀ x → is-set (B x)) → {p : x ≡ w} {q : x ≡ y} {s : w ≡ z} {r : y ≡ z} → Square (ap fst p) (ap fst q) (ap fst s) (ap fst r) → Square p q s r Σ-set-square Bset sq i j .fst = sq i j Σ-set-square Bset {p} {q} {s} {r} sq i j .snd = is-set→squarep (λ i j → Bset (sq i j)) (ap snd p) (ap snd q) (ap snd s) (ap snd r) i j
Dependent sums of contractibles🔗
If B is a family of contractible types, then
Σ B ≃ A:
Σ-contract : {B : A → Type ℓ} → (∀ x → is-contr (B x)) → Σ _ B ≃ A Σ-contract bcontr = Iso→Equiv the-iso where the-iso : Iso _ _ the-iso .fst (a , b) = a the-iso .snd .is-iso.inv x = x , bcontr _ .centre the-iso .snd .is-iso.rinv x = refl the-iso .snd .is-iso.linv (a , b) i = a , bcontr a .paths b i
Σ-map : (f : A → A') → ({x : A} → P x → Q (f x)) → Σ _ P → Σ _ Q Σ-map f g (x , y) = f x , g y Σ-map₂ : ({x : A} → P x → Q x) → Σ _ P → Σ _ Q Σ-map₂ f (x , y) = (x , f y) ⟨_,_⟩ : (X → Y) → (X → Z) → X → Y × Z ⟨ f , g ⟩ x = f x , g x ×-map : (A → A') → (X → X') → A × X → A' × X' ×-map f g (x , y) = (f x , g y) ×-map₁ : (A → A') → A × X → A' × X ×-map₁ f = ×-map f id ×-map₂ : (X → X') → A × X → A × X' ×-map₂ f = ×-map id f
_,ₚ_ = Σ-pathp infixr 4 _,ₚ_ Σ-prop-pathp : ∀ {ℓ ℓ'} {A : I → Type ℓ} {B : ∀ i → A i → Type ℓ'} → (∀ i x → is-prop (B i x)) → {x : Σ (A i0) (B i0)} {y : Σ (A i1) (B i1)} → PathP A (x .fst) (y .fst) → PathP (λ i → Σ (A i) (B i)) x y Σ-prop-pathp bp {x} {y} p i = p i , is-prop→pathp (λ i → bp i (p i)) (x .snd) (y .snd) i Σ-inj-set : ∀ {ℓ ℓ'} {A : Type ℓ} {B : A → Type ℓ'} {x y z} → is-set A → Path (Σ A B) (x , y) (x , z) → y ≡ z Σ-inj-set {B = B} {y = y} {z} aset path = subst (λ e → e ≡ z) (ap (λ e → transport (ap B e) y) (aset _ _ _ _) ∙ transport-refl y) (from-pathp (ap snd path)) Σ-swap₂ : ∀ {ℓ ℓ' ℓ''} {A : Type ℓ} {B : Type ℓ'} {C : A → B → Type ℓ''} → (Σ[ x ∈ A ] Σ[ y ∈ B ] (C x y)) ≃ (Σ[ y ∈ B ] Σ[ x ∈ A ] (C x y)) Σ-swap₂ .fst (x , y , f) = y , x , f Σ-swap₂ .snd .is-eqv y = contr (f .fst) (f .snd) where f = strict-fibres _ y -- agda can actually infer the inverse here, which is neat ×-swap : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} → (A × B) ≃ (B × A) ×-swap .fst (x , y) = y , x ×-swap .snd .is-eqv y = contr (f .fst) (f .snd) where f = strict-fibres _ y Σ-contr-eqv : ∀ {ℓ ℓ'} {A : Type ℓ} {B : A → Type ℓ'} → (c : is-contr A) → (Σ A B) ≃ B (c .centre) Σ-contr-eqv {B = B} c .fst (_ , p) = subst B (sym (c .paths _)) p Σ-contr-eqv {B = B} c .snd = is-iso→is-equiv λ where .is-iso.inv x → _ , x .is-iso.rinv x → ap (λ e → subst B e x) (is-contr→is-set c _ _ _ _) ∙ transport-refl x .is-iso.linv x → Σ-path (c .paths _) (transport⁻transport (ap B (sym (c .paths (x .fst)))) (x .snd))