Contractible types

Content created by Egbert Rijke, Fredrik Bakke, Jonathan Prieto-Cubides, Elisabeth Bonnevier and Victor Blanchi.

Created on 2022-01-26.
Last modified on 2023-09-11.

module foundation.contractible-types where

open import foundation-core.contractible-types public
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.function-extensionality
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.singleton-induction
open import foundation-core.subtypes
open import foundation-core.truncated-types
open import foundation-core.truncation-levels


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

  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

  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

  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 =
      ( equiv-is-contr is-contr-raise-unit (is-contr-type-Contr A))

  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


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 =
    ( 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}

    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)

  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) =
      ( λ x'  pair (B x') (p x'))
      ( f x y)

Equivalent characterizations of contractible types

module _
  {l1 : Level} {A : UU l1}

  dependent-universal-property-contr : (l : Level) (a : A)  UU (l1  lsuc l)
  dependent-universal-property-contr l a =
    (P : A  UU l)  is-equiv (ev-point a {P})

  universal-property-contr : (l : Level) (a : A)  UU (l1  lsuc l)
  universal-property-contr l a =
    (X : UU l)  is-equiv (ev-point' a {X})

  universal-property-dependent-universal-property-contr :
    (a : A) 
    ({l : Level}  dependent-universal-property-contr l a) 
    ({l : Level}  universal-property-contr l a)
  universal-property-dependent-universal-property-contr a dup-contr {l} X =
    dup-contr {l}  x  X)

    is-equiv-ev-point-universal-property-contr :
      (a : A)  ({l : Level}  universal-property-contr l a) 
      is-equiv (ev-point' a {A})
    is-equiv-ev-point-universal-property-contr a up-contr =
      up-contr A

    is-contr-is-equiv-ev-point :
      (a : A)  is-equiv (ev-point' a {A})  is-contr A
    pr1 (is-contr-is-equiv-ev-point a H) = a
    pr2 (is-contr-is-equiv-ev-point a H) =
        ( ap
          ( pr1)
          ( eq-is-contr'
            ( is-contr-map-is-equiv H a)
            ( pair  x  a) refl)
            ( pair id refl)))

    is-contr-universal-property-contr :
      (a : A) 
      ({l : Level}  universal-property-contr l a)  is-contr A
    is-contr-universal-property-contr a up-contr =
      is-contr-is-equiv-ev-point a
        ( is-equiv-ev-point-universal-property-contr a up-contr)

    is-contr-dependent-universal-property-contr :
      (a : A) 
      ({l : Level}  dependent-universal-property-contr l a)  is-contr A
    is-contr-dependent-universal-property-contr a dup-contr =
      is-contr-universal-property-contr a
        ( universal-property-dependent-universal-property-contr a dup-contr)

    dependent-universal-property-contr-is-contr :
      (a : A)  is-contr A 
      {l : Level}  dependent-universal-property-contr l a
    dependent-universal-property-contr-is-contr a H {l} P =
        ( ind-singleton-is-contr a H P)
        ( compute-ind-singleton-is-contr a H P)
        ( λ f 
            ( ind-singleton-is-contr a H
              ( λ x  ind-singleton-is-contr a H P (f a) x  f x)
              ( compute-ind-singleton-is-contr a H P (f a))))

  equiv-dependent-universal-property-contr :
    (a : A)  is-contr A  {l : Level} (B : A  UU l)  ((x : A)  B x)  B a
  pr1 (equiv-dependent-universal-property-contr a H P) = ev-point a
  pr2 (equiv-dependent-universal-property-contr a H P) =
    dependent-universal-property-contr-is-contr a H P

  apply-dependent-universal-property-contr :
    (a : A)  is-contr A  {l : Level} (B : A  UU l)  (B a  ((x : A)  B x))
  apply-dependent-universal-property-contr a H P =
    map-inv-equiv (equiv-dependent-universal-property-contr a H P)

    universal-property-contr-is-contr :
      (a : A)  is-contr A  {l : Level}  universal-property-contr l a
    universal-property-contr-is-contr a H =
      universal-property-dependent-universal-property-contr a
        ( dependent-universal-property-contr-is-contr a H)

  equiv-universal-property-contr :
    (a : A)  is-contr A  {l : Level} (X : UU l)  (A  X)  X
  pr1 (equiv-universal-property-contr a H X) = ev-point' a
  pr2 (equiv-universal-property-contr a H X) =
    universal-property-contr-is-contr a H X

  apply-universal-property-contr :
    (a : A)  is-contr A  {l : Level} (X : UU l)  X  (A  X)
  apply-universal-property-contr a H X =
    map-inv-equiv (equiv-universal-property-contr a H X)

    is-equiv-self-diagonal-is-equiv-diagonal :
      ({l : Level} (X : UU l)  is-equiv  x  const A X x)) 
      is-equiv  x  const A A x)
    is-equiv-self-diagonal-is-equiv-diagonal H = H A

    is-contr-is-equiv-self-diagonal :
      is-equiv  x  const A A x)  is-contr A
    is-contr-is-equiv-self-diagonal H =
      tot  x  htpy-eq) (center (is-contr-map-is-equiv H id))

    is-contr-is-equiv-diagonal :
      ({l : Level} (X : UU l)  is-equiv  x  const A X x))  is-contr A
    is-contr-is-equiv-diagonal H =
        ( is-equiv-self-diagonal-is-equiv-diagonal H)

    is-equiv-diagonal-is-contr :
      is-contr A 
      {l : Level} (X : UU l)  is-equiv  x  const A X x)
    is-equiv-diagonal-is-contr H X =
        ( ev-point' (center H))
        ( λ f  eq-htpy  x  ap f (contraction H x)))
        ( λ x  refl)

  equiv-diagonal-is-contr :
    {l : Level} (X : UU l)  is-contr A  X  (A  X)
  pr1 (equiv-diagonal-is-contr X H) = const A X
  pr2 (equiv-diagonal-is-contr X H) = is-equiv-diagonal-is-contr H X

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