Binary embeddings

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

Created on 2022-03-10.
Last modified on 2023-11-09.

module foundation.binary-embeddings where
open import foundation.action-on-identifications-binary-functions
open import foundation.action-on-identifications-functions
open import foundation.binary-equivalences
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.universe-levels

open import foundation-core.embeddings
open import foundation-core.equivalences


A binary operation f : A → B → C is said to be a binary embedding if the functions λ x → f x b and λ y → f a y are embeddings for each a : A and b : B respectively.


is-binary-emb :
  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} 
  (A  B  C)  UU (l1  l2  l3)
is-binary-emb {A = A} {B = B} f =
  {x x' : A} {y y' : B} 
    is-binary-equiv  (p : x  x') (q : y  y')  ap-binary f p q)


Any binary equivalence is a binary embedding

is-emb-fix-left-is-binary-equiv :
  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} (f : A  B  C) 
  is-binary-equiv f  {a : A}  is-emb (fix-left f a)
is-emb-fix-left-is-binary-equiv f H {a} =
  is-emb-is-equiv (is-equiv-fix-left f H)

is-emb-fix-right-is-binary-equiv :
  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} (f : A  B  C) 
  is-binary-equiv f  {b : B}  is-emb (fix-right f b)
is-emb-fix-right-is-binary-equiv f H {b} =
  is-emb-is-equiv (is-equiv-fix-right f H)

is-binary-emb-is-binary-equiv :
  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {f : A  B  C} 
  is-binary-equiv f  is-binary-emb f
is-binary-emb-is-binary-equiv {f = f} H {x} {x'} {y} {y'} =
    ( λ q 
        ( λ p  ap-binary f p q)
        ( concat' (f x y) (ap (fix-left f x') q))
        ( λ p  ap (fix-right f y) p)
        ( λ p  triangle-ap-binary f p q)
        ( is-emb-fix-right-is-binary-equiv f H x x')
        ( is-equiv-concat' (f x y) (ap (fix-left f x') q)))
    ( λ p 
        ( λ q  ap-binary f p q)
        ( concat (ap (fix-right f y) p) (f x' y'))
        ( λ q  ap (fix-left f x') q)
        ( λ q  triangle-ap-binary f p q)
        ( is-emb-fix-left-is-binary-equiv f H y y')
        ( is-equiv-concat (ap (fix-right f y) p) (f x' y')))

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