# Binary equivalences

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

Created on 2022-03-10.

module foundation.binary-equivalences where

Imports
open import foundation.dependent-pair-types
open import foundation.universe-levels

open import foundation-core.cartesian-product-types
open import foundation-core.equivalences


## Idea

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

## Definitions

fix-left :
{l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} (f : A → B → C) →
A → B → C
fix-left f a = f a

fix-right :
{l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} (f : A → B → C) →
B → A → C
fix-right f b a = f a b

is-binary-equiv :
{l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} →
(A → B → C) → UU (l1 ⊔ l2 ⊔ l3)
is-binary-equiv {A = A} {B = B} f =
((b : B) → is-equiv (fix-right f b)) × ((a : A) → is-equiv (fix-left f a))

is-equiv-fix-left :
{l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} (f : A → B → C) →
is-binary-equiv f → {a : A} → is-equiv (fix-left f a)
is-equiv-fix-left f H {a} = pr2 H a

is-equiv-fix-right :
{l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} (f : A → B → C) →
is-binary-equiv f → {b : B} → is-equiv (fix-right f b)
is-equiv-fix-right f H {b} = pr1 H b