Embeddings

Content created by Fredrik Bakke, Egbert Rijke and Jonathan Prieto-Cubides.

Created on 2022-02-07.
Last modified on 2023-11-24.

module foundation-core.embeddings where
Imports 
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.universe-levels

open import foundation-core.equivalences
open import foundation-core.function-types
open import foundation-core.identity-types

Idea

An embedding from one type into another is a map that induces equivalences on identity types. In other words, the identitifications (f x) ＝ (f y) for an embedding f : A → B are in one-to-one correspondence with the identitifications x ＝ y. Embeddings are better behaved homotopically than injective maps, because the condition of being an equivalence is a property under function extensionality.

Definition

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2}
  where

  is-emb : (A → B) → UU (l1 ⊔ l2)
  is-emb f = (x y : A) → is-equiv (ap f {x} {y})

  equiv-ap-is-emb :
    {f : A → B} (e : is-emb f) {x y : A} → (x ＝ y) ≃ (f x ＝ f y)
  pr1 (equiv-ap-is-emb {f} e) = ap f
  pr2 (equiv-ap-is-emb {f} e {x} {y}) = e x y

  inv-equiv-ap-is-emb :
    {f : A → B} (e : is-emb f) {x y : A} → (f x ＝ f y) ≃ (x ＝ y)
  inv-equiv-ap-is-emb e = inv-equiv (equiv-ap-is-emb e)

infix 5 _↪_
_↪_ :
  {l1 l2 : Level} → UU l1 → UU l2 → UU (l1 ⊔ l2)
A ↪ B = Σ (A → B) is-emb

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2}
  where

  map-emb : A ↪ B → A → B
  map-emb = pr1

  is-emb-map-emb : (f : A ↪ B) → is-emb (map-emb f)
  is-emb-map-emb = pr2

  equiv-ap-emb :
    (e : A ↪ B) {x y : A} → (x ＝ y) ≃ (map-emb e x ＝ map-emb e y)
  equiv-ap-emb e = equiv-ap-is-emb (is-emb-map-emb e)

  inv-equiv-ap-emb :
    (e : A ↪ B) {x y : A} → (map-emb e x ＝ map-emb e y) ≃ (x ＝ y)
  inv-equiv-ap-emb e = inv-equiv (equiv-ap-emb e)

Examples

The identity map is an embedding

module _
  {l : Level} {A : UU l}
  where

  is-emb-id : is-emb (id {A = A})
  is-emb-id x y = is-equiv-htpy id ap-id is-equiv-id

  id-emb : A ↪ A
  pr1 id-emb = id
  pr2 id-emb = is-emb-id

To prove that a map is an embedding, a point in the domain may be assumed

module _
  {l : Level} {A : UU l} {l2 : Level} {B : UU l2} {f : A → B}
  where

  abstract
    is-emb-is-emb : (A → is-emb f) → is-emb f
    is-emb-is-emb H x y = H x x y

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