# Hilbert's ε-operators

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

Created on 2022-03-30.

module foundation.hilberts-epsilon-operators where

Imports
open import foundation.functoriality-propositional-truncation
open import foundation.propositional-truncations
open import foundation.universe-levels

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


## Idea

Hilbert's -operator at a type A is a map type-trunc-Prop A → A. Contrary to Hilbert, we will not assume that such an operator exists for each type A.

## Definition

ε-operator-Hilbert : {l : Level} → UU l → UU l
ε-operator-Hilbert A = type-trunc-Prop A → A


## Properties

### The existence of Hilbert's ε-operators is invariant under equivalences

ε-operator-equiv :
{l1 l2 : Level} {X : UU l1} {Y : UU l2} (e : X ≃ Y) →
ε-operator-Hilbert X → ε-operator-Hilbert Y
ε-operator-equiv e f =
(map-equiv e ∘ f) ∘ (map-trunc-Prop (map-inv-equiv e))

ε-operator-equiv' :
{l1 l2 : Level} {X : UU l1} {Y : UU l2} (e : X ≃ Y) →
ε-operator-Hilbert Y → ε-operator-Hilbert X
ε-operator-equiv' e f =
(map-inv-equiv e ∘ f) ∘ (map-trunc-Prop (map-equiv e))