Cavallo's trick

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2023-05-03.
Last modified on 2024-03-13.

module synthetic-homotopy-theory.cavallos-trick where
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.function-types
open import foundation.homotopies
open import foundation.identity-types
open import foundation.sections
open import foundation.universe-levels
open import foundation.whiskering-identifications-concatenation

open import structured-types.pointed-homotopies
open import structured-types.pointed-maps
open import structured-types.pointed-types


Cavallo's trick is a way of upgrading an unpointed homotopy between pointed maps to a pointed homotopy.

Originally, this trick was formulated by Evan Cavallo for homogeneous spaces, but it works as soon as the evaluation map (id ~ id) → Ω B has a section.


module _
  {l1 l2 : Level} {A : Pointed-Type l1} {B : Pointed-Type l2}

  htpy-cavallos-trick :
    (f g : A →∗ B)  section  (H : id ~ id)  H (point-Pointed-Type B)) 
    (map-pointed-map f ~ map-pointed-map g) 
    unpointed-htpy-pointed-map f g
  htpy-cavallos-trick (f , refl) (g , q) (K , α) H a =
    K (inv q  inv (H (point-Pointed-Type A))) (f a)  H a

  coherence-point-cavallos-trick :
    (f g : A →∗ B) (s : section  (H : id ~ id)  H (point-Pointed-Type B))) 
    (H : map-pointed-map f ~ map-pointed-map g) 
    coherence-point-unpointed-htpy-pointed-Π f g
      ( htpy-cavallos-trick f g s H)
  coherence-point-cavallos-trick (f , refl) (g , q) (K , α) H =
      ( ( right-whisker-concat
          ( ( right-whisker-concat (α _) (H _)) 
            ( is-section-inv-concat' (H _) (inv q)))
          ( q)) 
        ( left-inv q))

  cavallos-trick :
    (f g : A →∗ B)  section  (H : id ~ id)  H (point-Pointed-Type B)) 
    (map-pointed-map f ~ map-pointed-map g)  f ~∗ g
  pr1 (cavallos-trick f g s H) = htpy-cavallos-trick f g s H
  pr2 (cavallos-trick f g s H) = coherence-point-cavallos-trick f g s H


  • Cavallo's trick was originally formalized in the cubical agda library.
  • The above generalization was found by Buchholtz, Christensen, Rijke, and Taxerås Flaten, in [BCFR23].
Ulrik Buchholtz, J. Daniel Christensen, Jarl G. Taxerås Flaten, and Egbert Rijke. Central H-spaces and banded types. 02 2023. arXiv:2301.02636.

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