The replacement axiom for type theory

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

Created on 2022-03-02.
Last modified on 2023-06-08.

module foundation.replacement where
open import foundation.images
open import foundation.locally-small-types
open import foundation.universe-levels

open import foundation-core.small-types


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.


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 f = replacement f is-small'


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