Morphisms of twisted arrows

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2023-11-09.
Last modified on 2023-11-09.

module foundation.morphisms-twisted-arrows where
open import foundation.dependent-pair-types
open import foundation.universe-levels

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


A morphism of twisted arrows from f : A → B to g : X → Y is a triple (i , j , H) consisting of

  • a map i : X → A

  • a map j : B → Y, and

  • a homotopy H : j ∘ f ∘ i ~ g witnessing that the square

        A <----- X
        |        |
      f |        | g
        V        V
        B -----> Y


Thus, a morphism of twisted arrows can also be understood as a factorization of g through f.


module _
  {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4}
  (f : A  B) (g : X  Y)

  hom-twisted-arrow : UU (l1  l2  l3  l4)
  hom-twisted-arrow =
    Σ (X  A)  i  Σ (B  Y)  j  j  f  i ~ g))

  module _
    (α : hom-twisted-arrow)

    map-domain-hom-twisted-arrow : X  A
    map-domain-hom-twisted-arrow = pr1 α

    map-codomain-hom-twisted-arrow : B  Y
    map-codomain-hom-twisted-arrow = pr1 (pr2 α)

    coh-hom-twisted-arrow :
      map-codomain-hom-twisted-arrow  f  map-domain-hom-twisted-arrow ~ g
    coh-hom-twisted-arrow = pr2 (pr2 α)

See also

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