Connected maps

Content created by Egbert Rijke, Fredrik Bakke, Jonathan Prieto-Cubides, Daniel Gratzer, Elisabeth Stenholm, Raymond Baker, Tom de Jong and Victor Blanchi.

Created on 2022-07-17.
Last modified on 2024-04-11.

module foundation.connected-maps where
open import foundation.connected-types
open import foundation.dependent-pair-types
open import foundation.function-extensionality
open import foundation.fundamental-theorem-of-identity-types
open import foundation.homotopy-induction
open import foundation.precomposition-dependent-functions
open import foundation.structure-identity-principle
open import foundation.subtype-identity-principle
open import foundation.truncated-types
open import foundation.truncation-levels
open import foundation.truncations
open import foundation.univalence
open import foundation.universal-property-family-of-fibers-of-maps
open import foundation.universe-levels

open import foundation-core.contractible-maps
open import foundation-core.embeddings
open import foundation-core.equivalences
open import foundation-core.fibers-of-maps
open import foundation-core.function-types
open import foundation-core.functoriality-dependent-pair-types
open import foundation-core.homotopies
open import foundation-core.identity-types
open import foundation-core.propositions
open import foundation-core.subtypes
open import foundation-core.torsorial-type-families
open import foundation-core.truncated-maps


A map is said to be k-connected if its fibers are k-connected types.


Connected maps

is-connected-map-Prop :
  {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2}  (A  B)  Prop (l1  l2)
is-connected-map-Prop k {B = B} f =
  Π-Prop B  b  is-connected-Prop k (fiber f b))

is-connected-map :
  {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2}  (A  B)  UU (l1  l2)
is-connected-map k f = type-Prop (is-connected-map-Prop k f)

is-prop-is-connected-map :
  {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} (f : A  B) 
  is-prop (is-connected-map k f)
is-prop-is-connected-map k f = is-prop-type-Prop (is-connected-map-Prop k f)

The type of connected maps between two types

connected-map : {l1 l2 : Level} (k : 𝕋)  UU l1  UU l2  UU (l1  l2)
connected-map k A B = type-subtype (is-connected-map-Prop k {A} {B})

module _
  {l1 l2 : Level} {k : 𝕋} {A : UU l1} {B : UU l2}

  map-connected-map : connected-map k A B  A  B
  map-connected-map = inclusion-subtype (is-connected-map-Prop k)

  is-connected-map-connected-map :
    (f : connected-map k A B)  is-connected-map k (map-connected-map f)
  is-connected-map-connected-map =
    is-in-subtype-inclusion-subtype (is-connected-map-Prop k)

  emb-inclusion-connected-map : connected-map k A B  (A  B)
  emb-inclusion-connected-map = emb-subtype (is-connected-map-Prop k)

  htpy-connected-map : (f g : connected-map k A B)  UU (l1  l2)
  htpy-connected-map f g = (map-connected-map f) ~ (map-connected-map g)

  refl-htpy-connected-map : (f : connected-map k A B)  htpy-connected-map f f
  refl-htpy-connected-map f = refl-htpy

  is-torsorial-htpy-connected-map :
    (f : connected-map k A B)  is-torsorial (htpy-connected-map f)
  is-torsorial-htpy-connected-map f =
      ( is-torsorial-htpy (map-connected-map f))
      ( is-prop-is-connected-map k)
      ( map-connected-map f)
      ( refl-htpy-connected-map f)
      ( is-connected-map-connected-map f)

  htpy-eq-connected-map :
    (f g : connected-map k A B)  f  g  htpy-connected-map f g
  htpy-eq-connected-map f .f refl = refl-htpy-connected-map f

  is-equiv-htpy-eq-connected-map :
    (f g : connected-map k A B)  is-equiv (htpy-eq-connected-map f g)
  is-equiv-htpy-eq-connected-map f =
      ( is-torsorial-htpy-connected-map f)
      ( htpy-eq-connected-map f)

  extensionality-connected-map :
    (f g : connected-map k A B)  (f  g)  htpy-connected-map f g
  pr1 (extensionality-connected-map f g) = htpy-eq-connected-map f g
  pr2 (extensionality-connected-map f g) = is-equiv-htpy-eq-connected-map f g

  eq-htpy-connected-map :
    (f g : connected-map k A B)  htpy-connected-map f g  (f  g)
  eq-htpy-connected-map f g =
    map-inv-equiv (extensionality-connected-map f g)

The type of connected maps into a type

Connected-Map :
  {l1 : Level} (l2 : Level) (k : 𝕋) (A : UU l1)  UU (l1  lsuc l2)
Connected-Map l2 k A = Σ (UU l2)  X  connected-map k A X)

module _
  {l1 l2 : Level} {k : 𝕋} {A : UU l1} (f : Connected-Map l2 k A)

  type-Connected-Map : UU l2
  type-Connected-Map = pr1 f

  connected-map-Connected-Map : connected-map k A type-Connected-Map
  connected-map-Connected-Map = pr2 f

  map-Connected-Map : A  type-Connected-Map
  map-Connected-Map = map-connected-map connected-map-Connected-Map

  is-connected-map-Connected-Map : is-connected-map k map-Connected-Map
  is-connected-map-Connected-Map =
    is-connected-map-connected-map connected-map-Connected-Map

The type of connected maps into a truncated type

Connected-Map-Into-Truncated-Type :
  {l1 : Level} (l2 : Level) (k l : 𝕋) (A : UU l1)  UU (l1  lsuc l2)
Connected-Map-Into-Truncated-Type l2 k l A =
  Σ (Truncated-Type l2 l)  X  connected-map k A (type-Truncated-Type X))

module _
  {l1 l2 : Level} {k l : 𝕋} {A : UU l1}
  (f : Connected-Map-Into-Truncated-Type l2 k l A)

  truncated-type-Connected-Map-Into-Truncated-Type : Truncated-Type l2 l
  truncated-type-Connected-Map-Into-Truncated-Type = pr1 f

  type-Connected-Map-Into-Truncated-Type : UU l2
  type-Connected-Map-Into-Truncated-Type =
    type-Truncated-Type truncated-type-Connected-Map-Into-Truncated-Type

  is-trunc-type-Connected-Map-Into-Truncated-Type :
    is-trunc l type-Connected-Map-Into-Truncated-Type
  is-trunc-type-Connected-Map-Into-Truncated-Type =

  connected-map-Connected-Map-Into-Truncated-Type :
    connected-map k A type-Connected-Map-Into-Truncated-Type
  connected-map-Connected-Map-Into-Truncated-Type = pr2 f

  map-Connected-Map-Into-Truncated-Type :
    A  type-Connected-Map-Into-Truncated-Type
  map-Connected-Map-Into-Truncated-Type =
    map-connected-map connected-map-Connected-Map-Into-Truncated-Type

  is-connected-map-Connected-Map-Into-Truncated-Type :
    is-connected-map k map-Connected-Map-Into-Truncated-Type
  is-connected-map-Connected-Map-Into-Truncated-Type =


All maps are (-2)-connected

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A  B)

  is-neg-two-connected-map : is-connected-map neg-two-𝕋 f
  is-neg-two-connected-map b = is-neg-two-connected (fiber f b)

Equivalences are k-connected for any k

module _
  {l1 l2 : Level} {k : 𝕋} {A : UU l1} {B : UU l2}

  is-connected-map-is-equiv :
    {f : A  B}  is-equiv f  is-connected-map k f
  is-connected-map-is-equiv H b =
    is-connected-is-contr k (is-contr-map-is-equiv H b)

  is-connected-map-equiv :
    (e : A  B)  is-connected-map k (map-equiv e)
  is-connected-map-equiv e =
    is-connected-map-is-equiv (is-equiv-map-equiv e)

  connected-map-equiv :
    (A  B)  connected-map k A B
  pr1 (connected-map-equiv e) = map-equiv e
  pr2 (connected-map-equiv e) = is-connected-map-equiv e

A (k+1)-connected map is k-connected

module _
  {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} {f : A  B}

  is-connected-map-is-connected-map-succ-𝕋 :
    is-connected-map (succ-𝕋 k) f  is-connected-map k f
  is-connected-map-is-connected-map-succ-𝕋 H b =
    is-connected-is-connected-succ-𝕋 k (H b)

The composition of two k-connected maps is k-connected

module _
  {l1 l2 l3 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} {C : UU l3}

  is-connected-map-comp :
    {g : B  C} {f : A  B} 
    is-connected-map k g  is-connected-map k f  is-connected-map k (g  f)
  is-connected-map-comp K H c =
      ( compute-fiber-comp _ _ c)
      ( is-connected-Σ k (K c) (H  pr1))

  comp-connected-map :
    connected-map k B C  connected-map k A B  connected-map k A C
  pr1 (comp-connected-map g f) =
    map-connected-map g  map-connected-map f
  pr2 (comp-connected-map g f) =
      ( is-connected-map-connected-map g)
      ( is-connected-map-connected-map f)

The total map induced by a family of maps is k-connected if and only if all maps in the family are k-connected

module _
  {l1 l2 l3 : Level} (k : 𝕋) {A : UU l1} {B : A  UU l2} {C : A  UU l3}
  (f : (x : A)  B x  C x)

  is-connected-map-tot-is-fiberwise-connected-map :
    ((x : A)  is-connected-map k (f x)) 
    is-connected-map k (tot f)
  is-connected-map-tot-is-fiberwise-connected-map H (x , y) =
    is-connected-equiv (compute-fiber-tot f (x , y)) (H x y)

  is-fiberwise-connected-map-is-connected-map-tot :
    is-connected-map k (tot f) 
    (x : A)  is-connected-map k (f x)
  is-fiberwise-connected-map-is-connected-map-tot H x y =
    is-connected-equiv (inv-compute-fiber-tot f (x , y)) (H (x , y))

Dependent universal property for connected maps

module _
  {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} (f : A  B)

  dependent-universal-property-connected-map : UUω
  dependent-universal-property-connected-map =
    {l3 : Level} (P : B  Truncated-Type l3 k) 
    is-equiv (precomp-Π f  b  type-Truncated-Type (P b)))

module _
  {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} {f : A  B}

  dependent-universal-property-is-connected-map :
    is-connected-map k f  dependent-universal-property-connected-map k f
  dependent-universal-property-is-connected-map H P =
      ( λ b  is-equiv-diagonal-exponential-is-connected (P b) (H b))

module _
  {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} (f : connected-map k A B)

  equiv-dependent-universal-property-is-connected-map :
    {l3 : Level} (P : B  Truncated-Type l3 k) 
    ((b : B)  type-Truncated-Type (P b)) 
    ((a : A)  type-Truncated-Type (P (map-connected-map f a)))
  pr1 (equiv-dependent-universal-property-is-connected-map P) =
    precomp-Π (map-connected-map f)  b  type-Truncated-Type (P b))
  pr2 (equiv-dependent-universal-property-is-connected-map P) =
    dependent-universal-property-is-connected-map k
      ( is-connected-map-connected-map f)
      ( P)

A map that satisfies the dependent universal property for connected maps is a connected map

Proof: Consider a map f : A → B such that the precomposition function

  - ∘ f : ((b : B) → P b) → ((a : A) → P (f a))

is an equivalence for every family P of k-truncated types. Then it follows that the precomposition function

  - ∘ f : ((b : B) → ∥fiber f b∥_k) → ((a : A) → ∥fiber f (f a)∥_k)

is an equivalence. In particular, the element λ a → η (a , refl) in the codomain of this equivalence induces an element c b : ∥fiber f b∥_k for each b : B. We take these elements as our centers of contraction. Note that by construction, we have an identification c (f a) = η (a , refl).

To construct a contraction of ∥fiber f b∥_k for each b : B, we have to show that

  (b : B) (u : ∥fiber f b∥_k) → c b = u.

Since the type c b = u is k-truncated, this type is equivalent to the type (b : B) (u : fiber f b) → c b = η u. By reduction of the universal quantification over the fibers we see that this type is equivalent to the type

  (a : A) → c (f a) = η (a , refl).

This identification holds by construction of c.

module _
  {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} {f : A  B}
  (H : dependent-universal-property-connected-map k f)

  center-is-connected-map-dependent-universal-property-connected-map :
    (b : B)  type-trunc k (fiber f b)
  center-is-connected-map-dependent-universal-property-connected-map =
      ( H  b  trunc k (fiber f b)))
      ( λ a  unit-trunc (a , refl))

  compute-center-is-connected-map-dependent-universal-property-connected-map :
    (a : A) 
    center-is-connected-map-dependent-universal-property-connected-map (f a) 
    unit-trunc (a , refl)
  compute-center-is-connected-map-dependent-universal-property-connected-map =
      ( is-section-map-inv-is-equiv
        ( H  b  trunc k (fiber f b)))
        ( λ a  unit-trunc (a , refl)))

  contraction-is-connected-map-dependent-universal-property-connected-map :
    (b : B) (u : type-trunc k (fiber f b)) 
    center-is-connected-map-dependent-universal-property-connected-map b  u
  contraction-is-connected-map-dependent-universal-property-connected-map =
      ( λ b 
          ( Id-Truncated-Type' (trunc k (fiber f b)) _))
      ( extend-lift-family-of-elements-fiber f
        ( λ b u  _  unit-trunc u)
        ( compute-center-is-connected-map-dependent-universal-property-connected-map))

    is-connected-map-dependent-universal-property-connected-map :
      is-connected-map k f
    pr1 (is-connected-map-dependent-universal-property-connected-map b) =
      center-is-connected-map-dependent-universal-property-connected-map b
    pr2 (is-connected-map-dependent-universal-property-connected-map b) =
      contraction-is-connected-map-dependent-universal-property-connected-map b

The map unit-trunc {k} is k-connected

module _
  {l1 : Level} (k : 𝕋) {A : UU l1}

  is-connected-map-unit-trunc :
    is-connected-map k (unit-trunc {k = k} {A = A})
  is-connected-map-unit-trunc =
    is-connected-map-dependent-universal-property-connected-map k

A map f : A → B is k-connected if and only if precomposing dependent functions into k + n-truncated types is an n-2-truncated map for all n : ℕ

is-trunc-map-precomp-Π-is-connected-map :
  {l1 l2 l3 : Level} (k l n : 𝕋)  k +𝕋 (succ-𝕋 (succ-𝕋 n))  l 
  {A : UU l1} {B : UU l2} {f : A  B}  is-connected-map k f 
  (P : B  Truncated-Type l3 l) 
    ( n)
    ( precomp-Π f  b  type-Truncated-Type (P b)))
  {l1} {l2} {l3} k ._ neg-two-𝕋 refl {A} {B} H P =
    ( dependent-universal-property-is-connected-map k H
      ( λ b 
          ( type-Truncated-Type (P b))
          ( is-trunc-eq
            ( right-unit-law-add-𝕋 k)
            ( is-trunc-type-Truncated-Type (P b)))))
is-trunc-map-precomp-Π-is-connected-map k ._ (succ-𝕋 n) refl {A} {B} {f} H P =
    ( λ g h 
      is-trunc-map-precomp-Π-is-connected-map k _ n refl H
        ( λ b 
            ( eq-value g h b)
            ( is-trunc-eq
              ( right-successor-law-add-𝕋 k n)
              ( is-trunc-type-Truncated-Type (P b))
              ( g b)
              ( h b))))

Characterization of the identity type of Connected-Map l2 k A

equiv-Connected-Map :
  {l1 l2 : Level} {k : 𝕋} {A : UU l1} 
  (f g : Connected-Map l2 k A)  UU (l1  l2)
equiv-Connected-Map f g =
  Σ ( type-Connected-Map f  type-Connected-Map g)
    ( λ e  (map-equiv e  map-Connected-Map f) ~ map-Connected-Map g)

module _
  {l1 l2 : Level} {k : 𝕋} {A : UU l1} (f : Connected-Map l2 k A)

  id-equiv-Connected-Map : equiv-Connected-Map f f
  pr1 id-equiv-Connected-Map = id-equiv
  pr2 id-equiv-Connected-Map = refl-htpy

  is-torsorial-equiv-Connected-Map :
    is-torsorial (equiv-Connected-Map f)
  is-torsorial-equiv-Connected-Map =
      ( is-torsorial-equiv (type-Connected-Map f))
      ( pair (type-Connected-Map f) id-equiv)
      ( is-torsorial-htpy-connected-map (connected-map-Connected-Map f))

  equiv-eq-Connected-Map :
    (g : Connected-Map l2 k A)  (f  g)  equiv-Connected-Map f g
  equiv-eq-Connected-Map .f refl = id-equiv-Connected-Map

  is-equiv-equiv-eq-Connected-Map :
    (g : Connected-Map l2 k A)  is-equiv (equiv-eq-Connected-Map g)
  is-equiv-equiv-eq-Connected-Map =

  extensionality-Connected-Map :
    (g : Connected-Map l2 k A)  (f  g)  equiv-Connected-Map f g
  pr1 (extensionality-Connected-Map g) = equiv-eq-Connected-Map g
  pr2 (extensionality-Connected-Map g) = is-equiv-equiv-eq-Connected-Map g

  eq-equiv-Connected-Map :
    (g : Connected-Map l2 k A)  equiv-Connected-Map f g  (f  g)
  eq-equiv-Connected-Map g =
    map-inv-equiv (extensionality-Connected-Map g)

Characterization of the identity type of Connected-Map-Into-Truncated-Type l2 k A

equiv-Connected-Map-Into-Truncated-Type :
  {l1 l2 : Level} {k l : 𝕋} {A : UU l1} 
  (f g : Connected-Map-Into-Truncated-Type l2 k l A)  UU (l1  l2)
equiv-Connected-Map-Into-Truncated-Type f g =
  Σ ( type-Connected-Map-Into-Truncated-Type f 
      type-Connected-Map-Into-Truncated-Type g)
    ( λ e 
      ( map-equiv e  map-Connected-Map-Into-Truncated-Type f) ~
      ( map-Connected-Map-Into-Truncated-Type g))

module _
  {l1 l2 : Level} {k l : 𝕋} {A : UU l1}
  (f : Connected-Map-Into-Truncated-Type l2 k l A)

  id-equiv-Connected-Map-Into-Truncated-Type :
    equiv-Connected-Map-Into-Truncated-Type f f
  pr1 id-equiv-Connected-Map-Into-Truncated-Type = id-equiv
  pr2 id-equiv-Connected-Map-Into-Truncated-Type = refl-htpy

  is-torsorial-equiv-Connected-Map-Into-Truncated-Type :
    is-torsorial (equiv-Connected-Map-Into-Truncated-Type f)
  is-torsorial-equiv-Connected-Map-Into-Truncated-Type =
      ( is-torsorial-equiv-Truncated-Type
        ( truncated-type-Connected-Map-Into-Truncated-Type f))
      ( pair (truncated-type-Connected-Map-Into-Truncated-Type f) id-equiv)
      ( is-torsorial-htpy-connected-map
        ( connected-map-Connected-Map-Into-Truncated-Type f))

  equiv-eq-Connected-Map-Into-Truncated-Type :
    (g : Connected-Map-Into-Truncated-Type l2 k l A) 
    (f  g)  equiv-Connected-Map-Into-Truncated-Type f g
  equiv-eq-Connected-Map-Into-Truncated-Type .f refl =

  is-equiv-equiv-eq-Connected-Map-Into-Truncated-Type :
    (g : Connected-Map-Into-Truncated-Type l2 k l A) 
    is-equiv (equiv-eq-Connected-Map-Into-Truncated-Type g)
  is-equiv-equiv-eq-Connected-Map-Into-Truncated-Type =

  extensionality-Connected-Map-Into-Truncated-Type :
    (g : Connected-Map-Into-Truncated-Type l2 k l A) 
    (f  g)  equiv-Connected-Map-Into-Truncated-Type f g
  pr1 (extensionality-Connected-Map-Into-Truncated-Type g) =
    equiv-eq-Connected-Map-Into-Truncated-Type g
  pr2 (extensionality-Connected-Map-Into-Truncated-Type g) =
    is-equiv-equiv-eq-Connected-Map-Into-Truncated-Type g

  eq-equiv-Connected-Map-Into-Truncated-Type :
    (g : Connected-Map-Into-Truncated-Type l2 k l A) 
      equiv-Connected-Map-Into-Truncated-Type f g  (f  g)
  eq-equiv-Connected-Map-Into-Truncated-Type g =
    map-inv-equiv (extensionality-Connected-Map-Into-Truncated-Type g)

The type Connected-Map-Into-Truncated-Type l2 k k A is contractible

This remains to be shown. #733

The type Connected-Map-Into-Truncated-Type l2 k l A is k - l - 2-truncated

This remains to be shown. #733

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