Pushout-products

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2023-11-23.
Last modified on 2024-04-25.

module synthetic-homotopy-theory.pushout-products where
Imports 
open import foundation.cartesian-product-types
open import foundation.contractible-types
open import foundation.dependent-pair-types
open import foundation.function-types
open import foundation.functoriality-cartesian-product-types
open import foundation.homotopies
open import foundation.universe-levels

open import synthetic-homotopy-theory.cocones-under-spans
open import synthetic-homotopy-theory.pushouts
open import synthetic-homotopy-theory.universal-property-pushouts

Idea

Consider two maps f : A → X and g : B → Y. The pushout-product f □ g of f and g is defined as the cogap map of the commuting square

              f × id
       A × B --------> X × B
         |               |
  id × g |      H ⇗      | id × g
         ∨               ∨
       A × Y --------> X × Y.
              f × id

In other words, the pushout-product is the unique map

  f □ g : (X × B) ⊔_{A × B} (A × Y) → X × Y

equipped with homotopies

  K : (f □ g) ∘ inl ~ f × id
  L : (f □ g) ∘ inr ~ id × g

and a homotopy M witnessing that the square of homotopies

                                 K ·r (id × g)
       (f □ g) ∘ inl ∘ (id × g) ---------------> (f × id) ∘ (id × g)
                  |                                       |
  (f □ g) ·l glue |                                       | H
                  |                                       |
                  ∨                                       ∨
       (f □ g) ∘ inr ∘ (f × id) ---------------> (id × g) ∘ (f × id)
                                 L ·r (f × id)

commutes. The pushout-products is often called the fiberwise join, because for each (x , y) : X × Y we have an equivalence

  fiber (f □ g) (x , y) ≃ (fiber f x) * (fiber g y).

from the fibers of f □ g to the join of the fibers of f and g.

There is an “adjoint relation” between the pushout-product and the pullback-hom: For any three maps f, g, and h we have a homotopy

  ⟨f □ g , h⟩ ~ ⟨f , ⟨g , h⟩⟩.

Definitions

The pushout-product

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

  domain-pushout-product : UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
  domain-pushout-product =
    pushout (map-product id g) (map-product f id)

  cocone-pushout-product : cocone (map-product id g) (map-product f id) (X × Y)
  pr1 cocone-pushout-product = map-product f id
  pr1 (pr2 cocone-pushout-product) = map-product id g
  pr2 (pr2 cocone-pushout-product) = coherence-square-map-product f g

  abstract
    uniqueness-pushout-product :
      is-contr
        ( Σ ( domain-pushout-product → X × Y)
            ( λ h →
              htpy-cocone
                ( map-product id g)
                ( map-product f id)
                ( cocone-map
                  ( map-product id g)
                  ( map-product f id)
                  ( cocone-pushout (map-product id g) (map-product f id))
                  ( h))
                ( cocone-pushout-product)))
    uniqueness-pushout-product =
      uniqueness-map-universal-property-pushout
        ( map-product id g)
        ( map-product f id)
        ( cocone-pushout (map-product id g) (map-product f id))
        ( up-pushout (map-product id g) (map-product f id))
        ( cocone-pushout-product)

  abstract
    pushout-product : domain-pushout-product → X × Y
    pushout-product = pr1 (center uniqueness-pushout-product)

    compute-inl-pushout-product :
      pushout-product ∘ inl-pushout (map-product id g) (map-product f id) ~
      map-product f id
    compute-inl-pushout-product =
      pr1 (pr2 (center uniqueness-pushout-product))

    compute-inr-pushout-product :
      pushout-product ∘ inr-pushout (map-product id g) (map-product f id) ~
      map-product id g
    compute-inr-pushout-product =
      pr1 (pr2 (pr2 (center uniqueness-pushout-product)))

    compute-glue-pushout-product :
      statement-coherence-htpy-cocone
        ( map-product id g)
        ( map-product f id)
        ( cocone-map
          ( map-product id g)
          ( map-product f id)
          ( cocone-pushout (map-product id g) (map-product f id))
          ( pushout-product))
        ( cocone-pushout-product)
        ( compute-inl-pushout-product)
        ( compute-inr-pushout-product)
    compute-glue-pushout-product =
      pr2 (pr2 (pr2 (center uniqueness-pushout-product)))

See also

A wikidata identifier for this concept is not available.

Recent changes