Small types

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

Created on 2022-02-17.
Last modified on 2023-10-21.

module foundation.small-types where

open import foundation-core.small-types public
Imports
open import foundation.dependent-pair-types
open import foundation.images
open import foundation.locally-small-types
open import foundation.replacement
open import foundation.surjective-maps
open import foundation.uniqueness-image
open import foundation.universal-property-image
open import foundation.universe-levels

open import foundation-core.embeddings
open import foundation-core.homotopies

Properties

If f is a surjective map from a small type into a locally small type, then replacement implies that the codomain is small

is-small-is-surjective :
  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {f : A  B} 
  is-surjective f  is-small l3 A  is-locally-small l3 B 
  is-small l3 B
is-small-is-surjective {f = f} H K L =
  is-small-equiv'
    ( im f)
    ( equiv-equiv-slice-uniqueness-im f id-emb
      ( pair f refl-htpy)
      ( is-image-is-surjective f id-emb (pair f refl-htpy) H))
    ( replacement f K L)

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