Contractible types
Content created by Egbert Rijke, Fredrik Bakke, Jonathan Prieto-Cubides, Elisabeth Stenholm and Victor Blanchi.
Created on 2022-01-26.
Last modified on 2024-04-11.
module foundation.contractible-types where open import foundation-core.contractible-types public
Imports
open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.diagonal-maps-of-types open import foundation.function-extensionality open import foundation.logical-equivalences open import foundation.subuniverses open import foundation.unit-type open import foundation.universe-levels open import foundation-core.constant-maps open import foundation-core.contractible-maps open import foundation-core.equivalences open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes open import foundation-core.truncated-types open import foundation-core.truncation-levels
Definition
The proposition of being contractible
is-contr-Prop : {l : Level} → UU l → Prop l pr1 (is-contr-Prop A) = is-contr A pr2 (is-contr-Prop A) = is-property-is-contr
The subuniverse of contractible types
Contr : (l : Level) → UU (lsuc l) Contr l = type-subuniverse is-contr-Prop type-Contr : {l : Level} → Contr l → UU l type-Contr A = pr1 A abstract is-contr-type-Contr : {l : Level} (A : Contr l) → is-contr (type-Contr A) is-contr-type-Contr A = pr2 A equiv-Contr : {l1 l2 : Level} (X : Contr l1) (Y : Contr l2) → UU (l1 ⊔ l2) equiv-Contr X Y = type-Contr X ≃ type-Contr Y equiv-eq-Contr : {l1 : Level} (X Y : Contr l1) → X = Y → equiv-Contr X Y equiv-eq-Contr X Y = equiv-eq-subuniverse is-contr-Prop X Y abstract is-equiv-equiv-eq-Contr : {l1 : Level} (X Y : Contr l1) → is-equiv (equiv-eq-Contr X Y) is-equiv-equiv-eq-Contr X Y = is-equiv-equiv-eq-subuniverse is-contr-Prop X Y eq-equiv-Contr : {l1 : Level} {X Y : Contr l1} → equiv-Contr X Y → X = Y eq-equiv-Contr = eq-equiv-subuniverse is-contr-Prop abstract center-Contr : (l : Level) → Contr l center-Contr l = pair (raise-unit l) is-contr-raise-unit contraction-Contr : {l : Level} (A : Contr l) → center-Contr l = A contraction-Contr A = eq-equiv-Contr ( equiv-is-contr is-contr-raise-unit (is-contr-type-Contr A)) abstract is-contr-Contr : (l : Level) → is-contr (Contr l) is-contr-Contr l = pair (center-Contr l) contraction-Contr
The predicate that a subuniverse contains any contractible types
contains-contractible-types-subuniverse : {l1 l2 : Level} → subuniverse l1 l2 → UU (lsuc l1 ⊔ l2) contains-contractible-types-subuniverse {l1} P = (X : UU l1) → is-contr X → is-in-subuniverse P X
The predicate that a subuniverse is closed under the is-contr
predicate
We state a general form involving two universes, and a more traditional form using a single universe
is-closed-under-is-contr-subuniverses : {l1 l2 l3 : Level} (P : subuniverse l1 l2) (Q : subuniverse l1 l3) → UU (lsuc l1 ⊔ l2 ⊔ l3) is-closed-under-is-contr-subuniverses P Q = (X : type-subuniverse P) → is-in-subuniverse Q (is-contr (inclusion-subuniverse P X)) is-closed-under-is-contr-subuniverse : {l1 l2 : Level} (P : subuniverse l1 l2) → UU (lsuc l1 ⊔ l2) is-closed-under-is-contr-subuniverse P = is-closed-under-is-contr-subuniverses P P
Properties
If two types are equivalent then so are the propositions that they are contractible
equiv-is-contr-equiv : {l1 l2 : Level} {A : UU l1} {B : UU l2} → A ≃ B → is-contr A ≃ is-contr B equiv-is-contr-equiv {A = A} {B = B} e = equiv-iff-is-prop ( is-property-is-contr) ( is-property-is-contr) ( is-contr-retract-of A ( map-inv-equiv e , map-equiv e , is-section-map-inv-equiv e)) ( is-contr-retract-of B ( map-equiv e , map-inv-equiv e , is-retraction-map-inv-equiv e))
Contractible types are k
-truncated for any k
module _ {l : Level} {A : UU l} where abstract is-trunc-is-contr : (k : 𝕋) → is-contr A → is-trunc k A is-trunc-is-contr neg-two-𝕋 is-contr-A = is-contr-A is-trunc-is-contr (succ-𝕋 k) is-contr-A = is-trunc-succ-is-trunc k (is-trunc-is-contr k is-contr-A)
Contractibility of Σ-types where the dependent type is a proposition
module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (a : A) (b : B a) where is-contr-Σ-is-prop : ((x : A) → is-prop (B x)) → ((x : A) → B x → a = x) → is-contr (Σ A B) pr1 (is-contr-Σ-is-prop p f) = pair a b pr2 (is-contr-Σ-is-prop p f) (pair x y) = eq-type-subtype ( λ x' → pair (B x') (p x')) ( f x y)
The diagonal of contractible types
module _ {l1 : Level} {A : UU l1} where abstract is-equiv-self-diagonal-exponential-is-equiv-diagonal-exponential : ({l : Level} (X : UU l) → is-equiv (diagonal-exponential X A)) → is-equiv (diagonal-exponential A A) is-equiv-self-diagonal-exponential-is-equiv-diagonal-exponential H = H A abstract is-contr-is-equiv-self-diagonal-exponential : is-equiv (diagonal-exponential A A) → is-contr A is-contr-is-equiv-self-diagonal-exponential H = tot (λ x → htpy-eq) (center (is-contr-map-is-equiv H id)) abstract is-contr-is-equiv-diagonal-exponential : ({l : Level} (X : UU l) → is-equiv (diagonal-exponential X A)) → is-contr A is-contr-is-equiv-diagonal-exponential H = is-contr-is-equiv-self-diagonal-exponential ( is-equiv-self-diagonal-exponential-is-equiv-diagonal-exponential H) abstract is-equiv-diagonal-exponential-is-contr : is-contr A → {l : Level} (X : UU l) → is-equiv (diagonal-exponential X A) is-equiv-diagonal-exponential-is-contr H X = is-equiv-is-invertible ( ev-point' (center H)) ( λ f → eq-htpy (λ x → ap f (contraction H x))) ( λ x → refl) equiv-diagonal-exponential-is-contr : {l : Level} (X : UU l) → is-contr A → X ≃ (A → X) pr1 (equiv-diagonal-exponential-is-contr X H) = diagonal-exponential X A pr2 (equiv-diagonal-exponential-is-contr X H) = is-equiv-diagonal-exponential-is-contr H X
Recent changes
- 2024-04-11. Fredrik Bakke. Refactor diagonals (#1096).
- 2024-04-11. Fredrik Bakke and Egbert Rijke. Propositional operations (#1008).
- 2024-02-08. Fredrik Bakke. Computational identity types (#1015).
- 2024-02-06. Egbert Rijke and Fredrik Bakke. Refactor files about identity types and homotopies (#1014).
- 2023-12-21. Fredrik Bakke. Action on homotopies of functions (#973).