Truncation images of maps

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

Created on 2022-07-19.
Last modified on 2023-09-06.

module foundation.truncation-images-of-maps where
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.truncations
open import foundation.universe-levels

open import foundation-core.fibers-of-maps
open import foundation-core.identity-types
open import foundation-core.truncation-levels


The k-truncation image of a map f : A → B is the type trunc-im k f that fits in the (k-connected,k-truncated) factorization of f. It is defined as the type

  trunc-im k f := Σ (y : B), type-trunc k (fiber f y)


module _
  {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} (f : A  B)

  trunc-im : UU (l1  l2)
  trunc-im = Σ B  y  type-trunc k (fiber f y))

  unit-trunc-im : A  trunc-im
  pr1 (unit-trunc-im x) = f x
  pr2 (unit-trunc-im x) = unit-trunc (pair x refl)

  projection-trunc-im : trunc-im  B
  projection-trunc-im = pr1


Characterization of the identity types of k+1-truncation images

module _
  {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} (f : A  B)

  Eq-unit-trunc-im : A  A  UU (l1  l2)
  Eq-unit-trunc-im x y = trunc-im k (ap f {x} {y})

  extensionality-trunc-im :
    (x y : A) →
    ( unit-trunc-im (succ-𝕋 k) f x = unit-trunc-im (succ-𝕋 k) f y) ≃
    ( Eq-unit-trunc-im x y)
  extensionality-trunc-im x y =
    ( equiv-tot
      ( λ q →
        {!!})) ∘e
    ( equiv-pair-eq-Σ
      ( unit-trunc-im (succ-𝕋 k) f x)
      ( unit-trunc-im (succ-𝕋 k) f y))

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