The replacement axiom for type theory

module foundation.replacement where

Imports

open import foundation.images
open import foundation.locally-small-types

open import foundation-core.small-types
open import foundation-core.universe-levels

Idea

The type theoretic replacement axiom asserts that the image of a map f : A → B from a small type A into a locally small type B is small.

Definition

Replacement : (l : Level) → UUω
Replacement l =
  {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) →
  is-small l A → is-locally-small l B → is-small l (im f)

postulate replacement : {l : Level} → Replacement l

replacement-UU :
  {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) →
  is-locally-small l1 B → is-small l1 (im f)
replacement-UU {l1} {l2} {A} f = replacement f is-small'