Orthogonal maps

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

Created on 2023-02-18.
Last modified on 2024-06-04.

module orthogonal-factorization-systems.orthogonal-maps where
Imports 
open import foundation.action-on-identifications-functions
open import foundation.cartesian-morphisms-arrows
open import foundation.cartesian-product-types
open import foundation.composition-algebra
open import foundation.contractible-maps
open import foundation.contractible-types
open import foundation.coproduct-types
open import foundation.coproducts-pullbacks
open import foundation.dependent-pair-types
open import foundation.dependent-products-pullbacks
open import foundation.dependent-sums-pullbacks
open import foundation.equivalences
open import foundation.equivalences-arrows
open import foundation.fibered-maps
open import foundation.fibers-of-maps
open import foundation.function-extensionality
open import foundation.function-types
open import foundation.functoriality-cartesian-product-types
open import foundation.functoriality-coproduct-types
open import foundation.functoriality-dependent-pair-types
open import foundation.functoriality-fibers-of-maps
open import foundation.homotopies
open import foundation.morphisms-arrows
open import foundation.postcomposition-functions
open import foundation.postcomposition-pullbacks
open import foundation.precomposition-dependent-functions
open import foundation.precomposition-functions
open import foundation.products-pullbacks
open import foundation.propositions
open import foundation.pullbacks
open import foundation.type-arithmetic-dependent-function-types
open import foundation.type-theoretic-principle-of-choice
open import foundation.unit-type
open import foundation.universal-property-cartesian-product-types
open import foundation.universal-property-coproduct-types
open import foundation.universal-property-dependent-pair-types
open import foundation.universal-property-equivalences
open import foundation.universal-property-pullbacks
open import foundation.universe-levels
open import foundation.whiskering-homotopies-composition

open import orthogonal-factorization-systems.lifting-structures-on-squares
open import orthogonal-factorization-systems.local-types
open import orthogonal-factorization-systems.pullback-hom

Idea

The map f : A → B is said to be orthogonal to g : X → Y if any of the following equivalent definitions hold

  1. Their pullback-hom is an equivalence.

  2. There is a unique lifting operation between f and g.

  3. The following is a pullback square:

                 - ∘ f
      B → X -------> A → X
        |              |
  g ∘ - |              | g ∘ -
        ∨              ∨
      B → Y -------> A → Y.
             - ∘ f

  4. The induced dependent precomposition map

      - ∘ f : ((x : B) → fiber g (h x)) --> ((x : A) → fiber g (h (f x)))

    is an equivalence for every h : B → Y.

If f is orthogonal to g, we say that f is left orthogonal to g, and g is right orthogonal to f, and may write f ⊥ g.

Definitions

The orthogonality predicate

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

  is-orthogonal : UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
  is-orthogonal = is-equiv (pullback-hom f g)

  _⊥_ = is-orthogonal

  is-prop-is-orthogonal : is-prop is-orthogonal
  is-prop-is-orthogonal = is-property-is-equiv (pullback-hom f g)

  is-orthogonal-Prop : Prop (l1 ⊔ l2 ⊔ l3 ⊔ l4)
  pr1 is-orthogonal-Prop = is-orthogonal
  pr2 is-orthogonal-Prop = is-prop-is-orthogonal

A term of is-right-orthogonal f g asserts that g is right orthogonal to f.

  is-right-orthogonal : UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
  is-right-orthogonal = is-orthogonal

A term of is-left-orthogonal f g asserts that g is left orthogonal to 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)
  where

  is-left-orthogonal : UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
  is-left-orthogonal = is-orthogonal g f

The pullback condition for orthogonal maps

The maps f and g are orthogonal if and only if the square

             - ∘ f
      B → X -------> A → X
        |              |
  g ∘ - |              | g ∘ -
        ∨              ∨
      B → Y -------> A → Y.
             - ∘ f

is a pullback.

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

  is-orthogonal-pullback-condition : UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
  is-orthogonal-pullback-condition =
    is-pullback (precomp f Y) (postcomp A g) (cone-pullback-hom f g)

  is-orthogonal-pullback-condition-Prop : Prop (l1 ⊔ l2 ⊔ l3 ⊔ l4)
  is-orthogonal-pullback-condition-Prop =
    is-pullback-Prop (precomp f Y) (postcomp A g) (cone-pullback-hom f g)

  is-prop-is-orthogonal-pullback-condition :
    is-prop is-orthogonal-pullback-condition
  is-prop-is-orthogonal-pullback-condition =
    is-prop-is-pullback (precomp f Y) (postcomp A g) (cone-pullback-hom f g)

The universal property of orthogonal pairs of maps

The universal property of orthogonal maps is the universal property associated to the pullback condition, which, as opposed to the pullback condition itself, is a large proposition.

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

  universal-property-orthogonal-maps : UUω
  universal-property-orthogonal-maps =
    universal-property-pullback
      ( precomp f Y)
      ( postcomp A g)
      ( cone-pullback-hom f g)

The fiber condition for orthogonal maps

The maps f and g are orthogonal if and only if the induced family of maps on fibers

                           (- ∘ f)
   ((x : B) → fiber g (h x)) --> ((x : A) → fiber g (h (f x)))
                      |               |
                      |               |
                      ∨    (- ∘ f)    ∨
                   (B → X) ------> (A → X)
                      |               |
              (g ∘ -) |               | (g ∘ -)
                      ∨    (- ∘ f)    ∨
               h ∈ (B → Y) ------> (A → Y)

is an equivalence for every h : B → Y.

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

  is-orthogonal-fiber-condition-right-map : UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
  is-orthogonal-fiber-condition-right-map =
    (h : B → Y) → is-equiv (precomp-Π f (fiber g ∘ h))

Properties

Being orthogonal means that the type of lifting squares is contractible

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

  unique-lifting-structure-squares-is-orthogonal :
    is-orthogonal f g →
    (h : hom-arrow f g) →
    is-contr (lifting-structure-square f g h)
  unique-lifting-structure-squares-is-orthogonal H h =
    is-contr-equiv
      ( fiber (pullback-hom f g) h)
      ( compute-fiber-pullback-hom f g h)
      ( is-contr-map-is-equiv H h)

  is-orthogonal-unique-lifting-structure-squares :
    ( (h : hom-arrow f g) → is-contr (lifting-structure-square f g h)) →
    is-orthogonal f g
  is-orthogonal-unique-lifting-structure-squares H =
    is-equiv-is-contr-map
      ( λ h →
        is-contr-equiv'
          ( lifting-structure-square f g h)
          ( compute-fiber-pullback-hom f g h)
          ( H h))

Being orthogonal means satisfying the pullback condition of being orthogonal maps

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

  is-orthogonal-pullback-condition-is-orthogonal :
    is-orthogonal f g → is-orthogonal-pullback-condition f g
  is-orthogonal-pullback-condition-is-orthogonal H =
    is-equiv-left-map-triangle
      ( gap-pullback-hom f g)
      ( map-compute-pullback-hom f g)
      ( pullback-hom f g)
      ( inv-htpy (triangle-pullback-hom f g))
      ( H)
      ( is-equiv-map-compute-pullback-hom f g)

  is-orthogonal-is-orthogonal-pullback-condition :
    is-orthogonal-pullback-condition f g → is-orthogonal f g
  is-orthogonal-is-orthogonal-pullback-condition =
    is-equiv-top-map-triangle
      ( gap-pullback-hom f g)
      ( map-compute-pullback-hom f g)
      ( pullback-hom f g)
      ( inv-htpy (triangle-pullback-hom f g))
      ( is-equiv-map-compute-pullback-hom f g)

Being orthogonal means satisfying the universal property of being orthogonal

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

  universal-property-orthogonal-maps-is-orthogonal :
    is-orthogonal f g → universal-property-orthogonal-maps f g
  universal-property-orthogonal-maps-is-orthogonal H =
    universal-property-pullback-is-pullback
      ( precomp f Y)
      ( postcomp A g)
      ( cone-pullback-hom f g)
      ( is-orthogonal-pullback-condition-is-orthogonal f g H)

  is-orthogonal-universal-property-orthogonal-maps :
    universal-property-orthogonal-maps f g → is-orthogonal f g
  is-orthogonal-universal-property-orthogonal-maps H =
    is-orthogonal-is-orthogonal-pullback-condition f g
      ( is-pullback-universal-property-pullback
        ( precomp f Y)
        ( postcomp A g)
        ( cone-pullback-hom f g)
        ( H))

Being orthogonal means satisfying the fiber condition for orthogonal maps

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

  is-orthogonal-fiber-condition-right-map-is-orthogonal-pullback-condition :
    is-orthogonal-pullback-condition f g →
    is-orthogonal-fiber-condition-right-map f g
  is-orthogonal-fiber-condition-right-map-is-orthogonal-pullback-condition H h =
    is-equiv-source-is-equiv-target-equiv-arrow
      ( precomp-Π f (fiber g ∘ h))
      ( map-fiber-vertical-map-cone
        ( precomp f Y)
        ( postcomp A g)
        ( cone-pullback-hom f g)
        ( h))
      ( compute-map-fiber-vertical-pullback-hom f g h)
      ( is-fiberwise-equiv-map-fiber-vertical-map-cone-is-pullback
        ( precomp f Y)
        ( postcomp A g)
        ( cone-pullback-hom f g)
        ( H)
        ( h))

  is-orthogonal-fiber-condition-right-map-is-orthogonal :
    is-orthogonal f g →
    is-orthogonal-fiber-condition-right-map f g
  is-orthogonal-fiber-condition-right-map-is-orthogonal H =
    is-orthogonal-fiber-condition-right-map-is-orthogonal-pullback-condition
      ( is-orthogonal-pullback-condition-is-orthogonal f g H)

  is-orthogonal-pullback-condition-is-orthogonal-fiber-condition-right-map :
    is-orthogonal-fiber-condition-right-map f g →
    is-orthogonal-pullback-condition f g
  is-orthogonal-pullback-condition-is-orthogonal-fiber-condition-right-map H =
    is-pullback-is-fiberwise-equiv-map-fiber-vertical-map-cone
      ( precomp f Y)
      ( postcomp A g)
      ( cone-pullback-hom f g)
      ( λ h →
        is-equiv-target-is-equiv-source-equiv-arrow
          ( precomp-Π f (fiber g ∘ h))
          ( map-fiber-vertical-map-cone
            ( precomp f Y)
            ( postcomp A g)
            ( cone-pullback-hom f g)
            ( h))
          ( compute-map-fiber-vertical-pullback-hom f g h)
          ( H h))

  is-orthogonal-is-orthogonal-fiber-condition-right-map :
    is-orthogonal-fiber-condition-right-map f g →
    is-orthogonal f g
  is-orthogonal-is-orthogonal-fiber-condition-right-map H =
    is-orthogonal-is-orthogonal-pullback-condition f g
      ( is-orthogonal-pullback-condition-is-orthogonal-fiber-condition-right-map
        ( H))

Orthogonality is preserved by homotopies

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

  is-orthogonal-pullback-condition-htpy :
    {f' : A → B} {g' : X → Y} → f ~ f' → g ~ g' →
    is-orthogonal-pullback-condition f g →
    is-orthogonal-pullback-condition f' g'
  is-orthogonal-pullback-condition-htpy F G =
    is-pullback-htpy'
      ( htpy-precomp F Y)
      ( htpy-postcomp A G)
      ( cone-pullback-hom f g)
      ( ( htpy-postcomp B G) ,
        ( htpy-precomp F X) ,
        ( ( commutative-htpy-postcomp-htpy-precomp F G) ∙h
          ( inv-htpy-right-unit-htpy)))

  is-orthogonal-pullback-condition-htpy-left :
    {f' : A → B} → f ~ f' →
    is-orthogonal-pullback-condition f g →
    is-orthogonal-pullback-condition f' g
  is-orthogonal-pullback-condition-htpy-left F =
    is-orthogonal-pullback-condition-htpy F refl-htpy

  is-orthogonal-pullback-condition-htpy-right :
    {g' : X → Y} → g ~ g' →
    is-orthogonal-pullback-condition f g →
    is-orthogonal-pullback-condition f g'
  is-orthogonal-pullback-condition-htpy-right =
    is-orthogonal-pullback-condition-htpy refl-htpy

Equivalences are orthogonal to every map

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

  is-orthogonal-pullback-condition-is-equiv-left :
    is-equiv f → is-orthogonal-pullback-condition f g
  is-orthogonal-pullback-condition-is-equiv-left is-equiv-f =
    is-pullback-is-equiv-horizontal-maps
      ( precomp f Y)
      ( postcomp A g)
      ( cone-pullback-hom f g)
      ( is-equiv-precomp-is-equiv f is-equiv-f Y)
      ( is-equiv-precomp-is-equiv f is-equiv-f X)

  is-orthogonal-is-equiv-left : is-equiv f → is-orthogonal f g
  is-orthogonal-is-equiv-left is-equiv-f =
    is-orthogonal-is-orthogonal-pullback-condition f g
      ( is-orthogonal-pullback-condition-is-equiv-left is-equiv-f)

  is-orthogonal-pullback-condition-is-equiv-right :
    is-equiv g → is-orthogonal-pullback-condition f g
  is-orthogonal-pullback-condition-is-equiv-right is-equiv-g =
    is-pullback-is-equiv-vertical-maps
      ( precomp f Y)
      ( postcomp A g)
      ( cone-pullback-hom f g)
      ( is-equiv-postcomp-is-equiv g is-equiv-g A)
      ( is-equiv-postcomp-is-equiv g is-equiv-g B)

  is-orthogonal-is-equiv-right : is-equiv g → is-orthogonal f g
  is-orthogonal-is-equiv-right is-equiv-g =
    is-orthogonal-is-orthogonal-pullback-condition f g
      ( is-orthogonal-pullback-condition-is-equiv-right is-equiv-g)

module _
  {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4}
  where

  is-orthogonal-pullback-condition-equiv-left :
    (f : A ≃ B) (g : X → Y) → is-orthogonal-pullback-condition (map-equiv f) g
  is-orthogonal-pullback-condition-equiv-left f g =
    is-orthogonal-pullback-condition-is-equiv-left
      ( map-equiv f)
      ( g)
      ( is-equiv-map-equiv f)

  is-orthogonal-equiv-left :
    (f : A ≃ B) (g : X → Y) → is-orthogonal (map-equiv f) g
  is-orthogonal-equiv-left f g =
    is-orthogonal-is-equiv-left (map-equiv f) g (is-equiv-map-equiv f)

  is-orthogonal-pullback-condition-equiv-right :
    (f : A → B) (g : X ≃ Y) → is-orthogonal-pullback-condition f (map-equiv g)
  is-orthogonal-pullback-condition-equiv-right f g =
    is-orthogonal-pullback-condition-is-equiv-right
      ( f)
      ( map-equiv g)
      ( is-equiv-map-equiv g)

  is-orthogonal-equiv-right :
    (f : A → B) (g : X ≃ Y) → is-orthogonal f (map-equiv g)
  is-orthogonal-equiv-right f g =
    is-orthogonal-is-equiv-right f (map-equiv g) (is-equiv-map-equiv g)

Right orthogonal maps are closed under composition and have the left cancellation property

Given two composable maps h after g, if f ⊥ h, then f ⊥ g if and only if f ⊥ (h ∘ g).

Proof: By the pullback condition of orthogonal maps, the top square in the below diagram is a pullback precisely when g is right orthogonal to f:

             - ∘ f
      B → X -------> A → X
        |              |
  g ∘ - |              | g ∘ -
        ∨              ∨
      B → Y -------> A → Y
        | ⌟            |
  h ∘ - |              | h ∘ -
        ∨              ∨
      B → Z -------> A → Z.
             - ∘ f

so the result is an instance of the vertical pasting property for pullbacks.

module _
  {l1 l2 l3 l4 l5 : Level}
  {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4} {Z : UU l5}
  (f : A → B) (g : X → Y) (h : Y → Z)
  where

  is-orthogonal-pullback-condition-right-comp :
    is-orthogonal-pullback-condition f h →
    is-orthogonal-pullback-condition f g →
    is-orthogonal-pullback-condition f (h ∘ g)
  is-orthogonal-pullback-condition-right-comp =
    is-pullback-rectangle-is-pullback-top-square
      ( precomp f Z)
      ( postcomp A h)
      ( postcomp A g)
      ( cone-pullback-hom f h)
      ( cone-pullback-hom f g)

  up-orthogonal-right-comp :
    universal-property-orthogonal-maps f h →
    universal-property-orthogonal-maps f g →
    universal-property-orthogonal-maps f (h ∘ g)
  up-orthogonal-right-comp =
    universal-property-pullback-rectangle-universal-property-pullback-top
      ( precomp f Z)
      ( postcomp A h)
      ( postcomp A g)
      ( cone-pullback-hom f h)
      ( cone-pullback-hom f g)

  is-orthogonal-right-comp :
    is-orthogonal f h → is-orthogonal f g → is-orthogonal f (h ∘ g)
  is-orthogonal-right-comp H G =
    is-orthogonal-is-orthogonal-pullback-condition f (h ∘ g)
      ( is-orthogonal-pullback-condition-right-comp
        ( is-orthogonal-pullback-condition-is-orthogonal f h H)
        ( is-orthogonal-pullback-condition-is-orthogonal f g G))

  is-orthogonal-pullback-condition-right-right-factor :
    is-orthogonal-pullback-condition f h →
    is-orthogonal-pullback-condition f (h ∘ g) →
    is-orthogonal-pullback-condition f g
  is-orthogonal-pullback-condition-right-right-factor =
    is-pullback-top-square-is-pullback-rectangle
      ( precomp f Z)
      ( postcomp A h)
      ( postcomp A g)
      ( cone-pullback-hom f h)
      ( cone-pullback-hom f g)

  up-orthogonal-right-right-factor :
    universal-property-orthogonal-maps f h →
    universal-property-orthogonal-maps f (h ∘ g) →
    universal-property-orthogonal-maps f g
  up-orthogonal-right-right-factor =
    universal-property-pullback-top-universal-property-pullback-rectangle
      ( precomp f Z)
      ( postcomp A h)
      ( postcomp A g)
      ( cone-pullback-hom f h)
      ( cone-pullback-hom f g)

  is-orthogonal-right-right-factor :
    is-orthogonal f h → is-orthogonal f (h ∘ g) → is-orthogonal f g
  is-orthogonal-right-right-factor H HG =
    is-orthogonal-is-orthogonal-pullback-condition f g
      ( is-orthogonal-pullback-condition-right-right-factor
        ( is-orthogonal-pullback-condition-is-orthogonal f h H)
        ( is-orthogonal-pullback-condition-is-orthogonal f (h ∘ g) HG))

Left orthogonal maps are closed under composition and have the right cancellation property

Given two composable maps h after f, if f ⊥ g, then h ⊥ g if and only if (h ∘ f) ⊥ g.

Proof: By the universal property of orthogonal maps, the right square in the below diagram is a pullback precisely when f is left orthogonal to g:

             - ∘ h          - ∘ f
      C → X -------> B → X -------> A → X
        |              | ⌟            |
  g ∘ - |              |              | g ∘ -
        ∨              ∨              ∨
      C → Y -------> B → Y -------> A → Y
             - ∘ h          - ∘ f

so the result is an instance of the horizontal pasting property for pullbacks.

module _
  {l1 l2 l3 l4 l5 : Level}
  {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4} {C : UU l5}
  (f : A → B) (h : B → C) (g : X → Y)
  where

  is-orthogonal-pullback-condition-left-comp :
    is-orthogonal-pullback-condition f g →
    is-orthogonal-pullback-condition h g →
    is-orthogonal-pullback-condition (h ∘ f) g
  is-orthogonal-pullback-condition-left-comp =
    is-pullback-rectangle-is-pullback-left-square
      ( precomp h Y)
      ( precomp f Y)
      ( postcomp A g)
      ( cone-pullback-hom f g)
      ( cone-pullback-hom h g)

  up-orthogonal-left-comp :
    universal-property-orthogonal-maps f g →
    universal-property-orthogonal-maps h g →
    universal-property-orthogonal-maps (h ∘ f) g
  up-orthogonal-left-comp =
    universal-property-pullback-rectangle-universal-property-pullback-left-square
      ( precomp h Y)
      ( precomp f Y)
      ( postcomp A g)
      ( cone-pullback-hom f g)
      ( cone-pullback-hom h g)

  is-orthogonal-left-comp :
    is-orthogonal f g → is-orthogonal h g → is-orthogonal (h ∘ f) g
  is-orthogonal-left-comp F H =
    is-orthogonal-is-orthogonal-pullback-condition (h ∘ f) g
      ( is-orthogonal-pullback-condition-left-comp
        ( is-orthogonal-pullback-condition-is-orthogonal f g F)
        ( is-orthogonal-pullback-condition-is-orthogonal h g H))

  is-orthogonal-pullback-condition-left-left-factor :
    is-orthogonal-pullback-condition f g →
    is-orthogonal-pullback-condition (h ∘ f) g →
    is-orthogonal-pullback-condition h g
  is-orthogonal-pullback-condition-left-left-factor =
    is-pullback-left-square-is-pullback-rectangle
      ( precomp h Y)
      ( precomp f Y)
      ( postcomp A g)
      ( cone-pullback-hom f g)
      ( cone-pullback-hom h g)

  up-orthogonal-left-left-factor :
    universal-property-orthogonal-maps f g →
    universal-property-orthogonal-maps (h ∘ f) g →
    universal-property-orthogonal-maps h g
  up-orthogonal-left-left-factor =
    universal-property-pullback-left-square-universal-property-pullback-rectangle
      ( precomp h Y)
      ( precomp f Y)
      ( postcomp A g)
      ( cone-pullback-hom f g)
      ( cone-pullback-hom h g)

  is-orthogonal-left-left-factor :
    is-orthogonal f g → is-orthogonal (h ∘ f) g → is-orthogonal h g
  is-orthogonal-left-left-factor F HF =
    is-orthogonal-is-orthogonal-pullback-condition h g
      ( is-orthogonal-pullback-condition-left-left-factor
        ( is-orthogonal-pullback-condition-is-orthogonal f g F)
        ( is-orthogonal-pullback-condition-is-orthogonal (h ∘ f) g HF))

Right orthogonality is preserved by dependent products

If f ⊥ gᵢ, for each i : I, then f ⊥ (map-Π g).

Proof: We need to show that the square

                          - ∘ f
         (B → Πᵢ Xᵢ) ---------------> (A → Πᵢ Xᵢ)
              |                           |
              |                           |
  map-Π g ∘ - |                           | map-Π g ∘ -
              |                           |
              ∨                           ∨
         (B → Πᵢ Yᵢ) ---------------> (A → Πᵢ Yᵢ)
                          - ∘ f

is a pullback. By swapping the arguments at each vertex, this square is equivalent to

                          map-Π (- ∘ f)
            (Πᵢ B → Xᵢ) ---------------> (Πᵢ A → Xᵢ)
                  |                           |
                  |                           |
   map-Π (gᵢ ∘ -) |                           | map-Π (gᵢ ∘ -)
                  |                           |
                  ∨                           ∨
            (Πᵢ B → Yᵢ) ---------------> (Πᵢ A → Yᵢ)
                          map-Π (- ∘ f)

which is a pullback by assumption since pullbacks are preserved by dependent products.

module _
  {l1 l2 l3 l4 l5 : Level}
  {I : UU l1} {A : UU l2} {B : UU l3} {X : I → UU l4} {Y : I → UU l5}
  (f : A → B) (g : (i : I) → X i → Y i)
  where

  is-orthogonal-pullback-condition-right-Π :
    ((i : I) → is-orthogonal-pullback-condition f (g i)) →
    is-orthogonal-pullback-condition f (map-Π g)
  is-orthogonal-pullback-condition-right-Π G =
    is-pullback-bottom-is-pullback-top-cube-is-equiv
      ( postcomp B (map-Π g))
      ( precomp f ((i : I) → X i))
      ( precomp f ((i : I) → Y i))
      ( postcomp A (map-Π g))
      ( map-Π (λ i → postcomp B (g i)))
      ( map-Π (λ i → precomp f (X i)))
      ( map-Π (λ i → precomp f (Y i)))
      ( map-Π (λ i → postcomp A (g i)))
      ( swap-Π)
      ( swap-Π)
      ( swap-Π)
      ( swap-Π)
      ( λ _ → eq-htpy refl-htpy)
      ( refl-htpy)
      ( refl-htpy)
      ( refl-htpy)
      ( refl-htpy)
      ( refl-htpy)
      ( ( ap swap-Π) ·l
        ( eq-htpy-refl-htpy ∘ map-Π (λ i → precomp f (Y i) ∘ postcomp B (g i))))
      ( is-equiv-swap-Π)
      ( is-equiv-swap-Π)
      ( is-equiv-swap-Π)
      ( is-equiv-swap-Π)
      ( is-pullback-Π
        ( λ i → precomp f (Y i))
        ( λ i → postcomp A (g i))
        ( λ i → cone-pullback-hom f (g i))
        ( G))

  is-orthogonal-right-Π :
    ((i : I) → is-orthogonal f (g i)) → is-orthogonal f (map-Π g)
  is-orthogonal-right-Π G =
    is-orthogonal-is-orthogonal-pullback-condition f (map-Π g)
      ( is-orthogonal-pullback-condition-right-Π
        ( λ i → is-orthogonal-pullback-condition-is-orthogonal f (g i) (G i)))

Any map that is left orthogonal to a map g is also left orthogonal to postcomposing by g

If f ⊥ g then f ⊥ postcomp S g for every type S.

Proof: This is a special case of the previous result by taking g to be constant over S.

module _
  {l1 l2 l3 l4 l5 : Level}
  {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4} (S : UU l5)
  (f : A → B) (g : X → Y)
  where

  is-orthogonal-pullback-condition-right-postcomp :
    is-orthogonal-pullback-condition f g →
    is-orthogonal-pullback-condition f (postcomp S g)
  is-orthogonal-pullback-condition-right-postcomp G =
    is-orthogonal-pullback-condition-right-Π f (λ _ → g) (λ _ → G)

  is-orthogonal-right-postcomp :
    is-orthogonal f g → is-orthogonal f (postcomp S g)
  is-orthogonal-right-postcomp G =
    is-orthogonal-right-Π f (λ _ → g) (λ _ → G)

Right orthogonality is preserved by products

If f ⊥ g and f ⊥ g', then f ⊥ (g × g').

module _
  {l1 l2 l3 l4 l5 l6 : Level}
  {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4} {X' : UU l5} {Y' : UU l6}
  (f : A → B) (g : X → Y) (g' : X' → Y')
  where

  is-orthogonal-pullback-condition-right-product :
    is-orthogonal-pullback-condition f g →
    is-orthogonal-pullback-condition f g' →
    is-orthogonal-pullback-condition f (map-product g g')
  is-orthogonal-pullback-condition-right-product G G' =
    is-pullback-top-is-pullback-bottom-cube-is-equiv
      ( map-product (postcomp B g) (postcomp B g'))
      ( map-product (precomp f X) (precomp f X'))
      ( map-product (precomp f Y) (precomp f Y'))
      ( map-product (postcomp A g) (postcomp A g'))
      ( postcomp B (map-product g g'))
      ( precomp f (X × X'))
      ( precomp f (Y × Y'))
      ( postcomp A (map-product g g'))
      ( map-up-product)
      ( map-up-product)
      ( map-up-product)
      ( map-up-product)
      ( refl-htpy)
      ( refl-htpy)
      ( refl-htpy)
      ( refl-htpy)
      ( refl-htpy)
      ( refl-htpy)
      ( refl-htpy)
      ( up-product)
      ( up-product)
      ( up-product)
      ( up-product)
      ( is-pullback-product-is-pullback
        ( precomp f Y)
        ( postcomp A g)
        ( precomp f Y')
        ( postcomp A g')
        ( cone-pullback-hom f g)
        ( cone-pullback-hom f g')
        ( G)
        ( G'))

  is-orthogonal-right-product :
    is-orthogonal f g → is-orthogonal f g' → is-orthogonal f (map-product g g')
  is-orthogonal-right-product G G' =
    is-orthogonal-is-orthogonal-pullback-condition f (map-product g g')
      ( is-orthogonal-pullback-condition-right-product
        ( is-orthogonal-pullback-condition-is-orthogonal f g G)
        ( is-orthogonal-pullback-condition-is-orthogonal f g' G'))

Left orthogonality is preserved by dependent sums

If fᵢ ⊥ g for every i, then (tot f) ⊥ g.

Proof: We need to show that the square

                  - ∘ (tot f)
  ((Σ I B) → X) ---------------> ((Σ I A) → X)
        |                               |
        |                               |
  g ∘ - |                               | g ∘ -
        |                               |
        ∨                               ∨
  ((Σ I B) → Y) ---------------> ((Σ I A) → Y)
                  - ∘ (tot f)

is a pullback. However, by the universal property of dependent pair types this square is equivalent to

                    Πᵢ (- ∘ fᵢ)
        Πᵢ (Bᵢ → X) -----------> Πᵢ (Aᵢ → X)
             |                        |
             |                        |
  Πᵢ (g ∘ -) |                        | Πᵢ (g ∘ -)
             |                        |
             ∨                        ∨
        Πᵢ (Bᵢ → Y) -----------> Πᵢ (Aᵢ → Y),
                    Πᵢ (- ∘ fᵢ)

which is a pullback by assumption and the fact that pullbacks are preserved under dependent products.

module _
  {l1 l2 l3 l4 l5 : Level}
  {I : UU l1} {A : I → UU l2} {B : I → UU l3} {X : UU l4} {Y : UU l5}
  (f : (i : I) → A i → B i) (g : X → Y)
  where

  is-orthogonal-pullback-condition-left-Σ :
    ((i : I) → is-orthogonal-pullback-condition (f i) g) →
    is-orthogonal-pullback-condition (tot f) g
  is-orthogonal-pullback-condition-left-Σ F =
    is-pullback-top-is-pullback-bottom-cube-is-equiv
      ( map-Π (λ i → postcomp (B i) g))
      ( map-Π (λ i → precomp (f i) X))
      ( map-Π (λ i → precomp (f i) Y))
      ( map-Π (λ i → postcomp (A i) g))
      ( postcomp (Σ I B) g)
      ( precomp (tot f) X)
      ( precomp (tot f) Y)
      ( postcomp (Σ I A) g)
      ( ev-pair)
      ( ev-pair)
      ( ev-pair)
      ( ev-pair)
      ( refl-htpy)
      ( refl-htpy)
      ( refl-htpy)
      ( refl-htpy)
      ( refl-htpy)
      ( λ _ → eq-htpy refl-htpy)
      ( inv-htpy
        ( ( right-unit-htpy) ∙h
          ( ( eq-htpy-refl-htpy) ·r
            ( ( map-Π (λ i → precomp (f i) Y)) ∘
              ( map-Π (λ i → postcomp (B i) g)) ∘
              ( ev-pair)))))
      ( is-equiv-ev-pair)
      ( is-equiv-ev-pair)
      ( is-equiv-ev-pair)
      ( is-equiv-ev-pair)
      ( is-pullback-Π
        ( λ i → precomp (f i) Y)
        ( λ i → postcomp (A i) g)
        ( λ i → cone-pullback-hom (f i) g)
        ( F))

  is-orthogonal-left-Σ :
    ((i : I) → is-orthogonal (f i) g) → is-orthogonal (tot f) g
  is-orthogonal-left-Σ F =
    is-orthogonal-is-orthogonal-pullback-condition (tot f) g
      ( is-orthogonal-pullback-condition-left-Σ
        ( λ i → is-orthogonal-pullback-condition-is-orthogonal (f i) g (F i)))

Left orthogonality is preserved by coproducts

If f ⊥ g and f' ⊥ g, then (f + f') ⊥ g.

Proof: We need to show that the square

                     - ∘ (f + f')
  ((B + B') → X) ---------------> ((A + A') → X)
        |                               |
        |                               |
  g ∘ - |                               | g ∘ -
        |                               |
        ∨                               ∨
  ((B + B') → Y) ---------------> ((A + A') → Y)
                   - ∘ (f + f')

is a pullback. However, by the universal property of coproducts this square is equivalent to

                            (- ∘ f) × (- ∘ f')
            (B → X) × (B' → X) -----------> (A → X) × (A' → X)
                    |                               |
                    |                               |
  (g ∘ -) × (g ∘ -) |                               | (g ∘ -) × (g ∘ -)
                    |                               |
                    ∨                               ∨
            (B → Y) × (B' → Y) -----------> (A → Y) × (A' → Y),
                            (- ∘ f) × (- ∘ f')

which is a pullback by assumption and the fact that pullbacks are preserved under products.

Note: This result can also be obtained as a special case of the previous one by taking the indexing type to be the two-element type.

module _
  {l1 l2 l3 l4 l5 l6 : Level}
  {A : UU l1} {B : UU l2} {A' : UU l3} {B' : UU l4} {X : UU l5} {Y : UU l6}
  (f : A → B) (f' : A' → B') (g : X → Y)
  where

  is-orthogonal-pullback-condition-left-coproduct :
    is-orthogonal-pullback-condition f g →
    is-orthogonal-pullback-condition f' g →
    is-orthogonal-pullback-condition (map-coproduct f f') g
  is-orthogonal-pullback-condition-left-coproduct F F' =
    is-pullback-top-is-pullback-bottom-cube-is-equiv
      ( map-product (postcomp B g) (postcomp B' g))
      ( map-product (precomp f X) (precomp f' X))
      ( map-product (precomp f Y) (precomp f' Y))
      ( map-product (postcomp A g) (postcomp A' g))
      ( postcomp (B + B') g)
      ( precomp (map-coproduct f f') X)
      ( precomp (map-coproduct f f') Y)
      ( postcomp (A + A') g)
      ( ev-inl-inr (λ _ → X))
      ( ev-inl-inr (λ _ → Y))
      ( ev-inl-inr (λ _ → X))
      ( ev-inl-inr (λ _ → Y))
      ( refl-htpy)
      ( refl-htpy)
      ( refl-htpy)
      ( refl-htpy)
      ( refl-htpy)
      ( refl-htpy)
      ( refl-htpy)
      ( universal-property-coproduct X)
      ( universal-property-coproduct Y)
      ( universal-property-coproduct X)
      ( universal-property-coproduct Y)
      ( is-pullback-product-is-pullback
        ( precomp f Y)
        ( postcomp A g)
        ( precomp f' Y)
        ( postcomp A' g)
        ( cone-pullback-hom f g)
        ( cone-pullback-hom f' g)
        ( F)
        ( F'))

  is-orthogonal-left-coproduct :
    is-orthogonal f g →
    is-orthogonal f' g →
    is-orthogonal (map-coproduct f f') g
  is-orthogonal-left-coproduct F F' =
    is-orthogonal-is-orthogonal-pullback-condition (map-coproduct f f') g
      ( is-orthogonal-pullback-condition-left-coproduct
        ( is-orthogonal-pullback-condition-is-orthogonal f g F)
        ( is-orthogonal-pullback-condition-is-orthogonal f' g F'))

Right orthogonality is preserved under base change

Given a pullback square

    X' -----> X
    | ⌟       |
  g'|         | g
    ∨         ∨
    Y' -----> Y,

if f ⊥ g, then f ⊥ g'.

module _
  {l1 l2 l3 l4 l5 l6 : Level}
  {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4} {X' : UU l5} {Y' : UU l6}
  (f : A → B) (g : X → Y) (g' : X' → Y') (α : cartesian-hom-arrow g' g)
  where

  is-orthogonal-pullback-condition-right-base-change :
    is-orthogonal-pullback-condition f g → is-orthogonal-pullback-condition f g'
  is-orthogonal-pullback-condition-right-base-change G =
    is-pullback-back-right-is-pullback-back-left-cube
      ( refl-htpy)
      ( htpy-postcomp B (coh-cartesian-hom-arrow g' g α))
      ( refl-htpy)
      ( refl-htpy)
      ( htpy-postcomp A (coh-cartesian-hom-arrow g' g α))
      ( refl-htpy)
      ( ( right-unit-htpy) ∙h
        ( right-unit-htpy) ∙h
        ( inv-htpy
          ( commutative-precomp-htpy-postcomp
            ( f)
            ( coh-cartesian-hom-arrow g' g α))))
      ( G)
      ( is-pullback-postcomp-is-pullback
        ( map-codomain-cartesian-hom-arrow g' g α)
        ( g)
        ( cone-cartesian-hom-arrow g' g α)
        ( is-cartesian-cartesian-hom-arrow g' g α)
        ( A))
      ( is-pullback-postcomp-is-pullback
        ( map-codomain-cartesian-hom-arrow g' g α)
        ( g)
        ( cone-cartesian-hom-arrow g' g α)
        ( is-cartesian-cartesian-hom-arrow g' g α)
        ( B))

  is-orthogonal-right-base-change : is-orthogonal f g → is-orthogonal f g'
  is-orthogonal-right-base-change G =
    is-orthogonal-is-orthogonal-pullback-condition f g'
      ( is-orthogonal-pullback-condition-right-base-change
        ( is-orthogonal-pullback-condition-is-orthogonal f g G))

A type is f-local if and only if its terminal map is f-orthogonal

Proof (forward direction): If the terminal map is right orthogonal to f, that means we have a pullback square

            - ∘ f
      B → X -----> A → X
        | ⌟          |
  ! ∘ - |            | ! ∘ -
        ∨            ∨
      B → 1 -----> A → 1.
            - ∘ f

which displays precomp f X as a pullback of an equivalence, hence it is itself an equivalence.

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

  is-local-is-orthogonal-pullback-condition-terminal-map :
    is-orthogonal-pullback-condition f (terminal-map X) → is-local f X
  is-local-is-orthogonal-pullback-condition-terminal-map =
    is-equiv-horizontal-map-is-pullback
      ( precomp f unit)
      ( postcomp A (terminal-map X))
      ( cone-pullback-hom f (terminal-map X))
      ( is-local-is-contr f unit is-contr-unit)

  is-local-is-orthogonal-terminal-map :
    is-orthogonal f (terminal-map X) → is-local f X
  is-local-is-orthogonal-terminal-map F =
    is-local-is-orthogonal-pullback-condition-terminal-map
      ( is-orthogonal-pullback-condition-is-orthogonal f (terminal-map X) F)

  is-orthogonal-pullback-condition-terminal-map-is-local :
    is-local f X → is-orthogonal-pullback-condition f (terminal-map X)
  is-orthogonal-pullback-condition-terminal-map-is-local =
    is-pullback-is-equiv-horizontal-maps
      ( precomp f unit)
      ( postcomp A (terminal-map X))
      ( cone-pullback-hom f (terminal-map X))
      ( is-local-is-contr f unit is-contr-unit)

  is-orthogonal-terminal-map-is-local :
    is-local f X → is-orthogonal f (terminal-map X)
  is-orthogonal-terminal-map-is-local F =
    is-orthogonal-is-orthogonal-pullback-condition f (terminal-map X)
      ( is-orthogonal-pullback-condition-terminal-map-is-local F)

If the codomain of g is f-local, then g is f-orthogonal if and only if the domain of g is f-local

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

  is-orthogonal-pullback-condition-is-local-domain-is-local-codomain :
    is-local f Y → is-local f X → is-orthogonal-pullback-condition f g
  is-orthogonal-pullback-condition-is-local-domain-is-local-codomain =
    is-pullback-is-equiv-horizontal-maps
      ( precomp f Y)
      ( postcomp A g)
      ( cone-pullback-hom f g)

  is-orthogonal-is-local-domain-is-local-codomain :
    is-local f Y → is-local f X → is-orthogonal f g
  is-orthogonal-is-local-domain-is-local-codomain H K =
    is-orthogonal-is-orthogonal-pullback-condition f g
      ( is-orthogonal-pullback-condition-is-local-domain-is-local-codomain H K)

  is-local-domain-is-orthogonal-pullback-condition-is-local-codomain :
    is-local f Y → is-orthogonal-pullback-condition f g → is-local f X
  is-local-domain-is-orthogonal-pullback-condition-is-local-codomain H G =
    is-local-is-orthogonal-pullback-condition-terminal-map f
      ( is-orthogonal-pullback-condition-right-comp f g (terminal-map Y)
        ( is-orthogonal-pullback-condition-terminal-map-is-local f H)
        ( G))

  is-local-domain-is-orthogonal-is-local-codomain :
    is-local f Y → is-orthogonal f g → is-local f X
  is-local-domain-is-orthogonal-is-local-codomain H G =
    is-local-domain-is-orthogonal-pullback-condition-is-local-codomain H
      ( is-orthogonal-pullback-condition-is-orthogonal f g G)

References

[BW23]
Ulrik Buchholtz and Jonathan Weinberger. Synthetic fibered -category theory. Higher Structures, 7:74–165, 05 2023. arXiv:2105.01724, doi:10.21136/HS.2023.04.

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