Fiberwise orthogonal maps
Content created by Fredrik Bakke.
Created on 2024-06-04.
Last modified on 2024-06-04.
module orthogonal-factorization-systems.fiberwise-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.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.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.homotopies open import foundation.morphisms-arrows open import foundation.postcomposition-functions open import foundation.postcomposition-pullbacks open import foundation.precomposition-functions open import foundation.products-pullbacks open import foundation.propositions open import foundation.pullbacks open import foundation.standard-pullbacks open import foundation.type-arithmetic-dependent-function-types 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.null-maps open import orthogonal-factorization-systems.orthogonal-maps open import orthogonal-factorization-systems.pullback-hom
Idea
The map f : A → B
is said to be
fiberwise left orthogonal¶
to g : X → Y
if every base change
of f
is left orthogonal to g
.
More concretely, f
is fiberwise left orthogonal to g
if for every
pullback square
A' -------> A
| ⌟ |
f'| | f
∨ ∨
B' -------> B,
the exponential square
- ∘ f'
B' → X -------> A' → X
| |
g ∘ - | | g ∘ -
∨ ∨
B' → Y -------> A' → Y
- ∘ f'
is also a pullback.
Definitions
The pullback condition for fiberwise 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-fiberwise-orthogonal-pullback-condition : UUω is-fiberwise-orthogonal-pullback-condition = {l5 l6 : Level} {A' : UU l5} {B' : UU l6} (f' : A' → B') (α : cartesian-hom-arrow f' f) → is-orthogonal-pullback-condition f' g
The universal property of fiberwise 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 universal-property-fiberwise-orthogonal-maps : UUω universal-property-fiberwise-orthogonal-maps = {l5 l6 : Level} {A' : UU l5} {B' : UU l6} (f' : A' → B') (α : cartesian-hom-arrow f' f) → universal-property-orthogonal-maps f' g
Properties
The pullback condition for fiberwise orthogonal maps is equivalent to the universal property
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-fiberwise-orthogonal-maps-is-fiberwise-orthogonal-pullback-condition : is-fiberwise-orthogonal-pullback-condition f g → universal-property-fiberwise-orthogonal-maps f g universal-property-fiberwise-orthogonal-maps-is-fiberwise-orthogonal-pullback-condition H {A' = A'} f' α = universal-property-pullback-is-pullback ( precomp f' Y) ( postcomp A' g) ( cone-pullback-hom f' g) ( H f' α) is-fiberwise-orthogonal-pullback-condition-universal-property-fiberwise-orthogonal-maps : universal-property-fiberwise-orthogonal-maps f g → is-fiberwise-orthogonal-pullback-condition f g is-fiberwise-orthogonal-pullback-condition-universal-property-fiberwise-orthogonal-maps H {A' = A'} f' α = is-pullback-universal-property-pullback ( precomp f' Y) ( postcomp A' g) ( cone-pullback-hom f' g) ( H f' α)
Fiberwise orthogonal maps are 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 is-orthogonal-pullback-condition-is-fiberwise-orthogonal-pullback-condition : is-fiberwise-orthogonal-pullback-condition f g → is-orthogonal-pullback-condition f g is-orthogonal-pullback-condition-is-fiberwise-orthogonal-pullback-condition H = H f id-cartesian-hom-arrow
Fiberwise orthogonality is preserved by homotopies
module _ {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4} where is-fiberwise-orthogonal-pullback-condition-htpy-left : {f f' : A → B} (F' : f' ~ f) (g : X → Y) → is-fiberwise-orthogonal-pullback-condition f g → is-fiberwise-orthogonal-pullback-condition f' g is-fiberwise-orthogonal-pullback-condition-htpy-left F' g H f'' α = H f'' (cartesian-hom-arrow-htpy refl-htpy F' α) is-fiberwise-orthogonal-pullback-condition-htpy-right : (f : A → B) {g g' : X → Y} (G : g ~ g') → is-fiberwise-orthogonal-pullback-condition f g → is-fiberwise-orthogonal-pullback-condition f g' is-fiberwise-orthogonal-pullback-condition-htpy-right f {g} G H f'' α = is-orthogonal-pullback-condition-htpy-right f'' g G (H f'' α) is-fiberwise-orthogonal-pullback-condition-htpy : {f f' : A → B} (F' : f' ~ f) {g g' : X → Y} (G : g ~ g') → is-fiberwise-orthogonal-pullback-condition f g → is-fiberwise-orthogonal-pullback-condition f' g' is-fiberwise-orthogonal-pullback-condition-htpy {f} {f'} F' {g} G H = is-fiberwise-orthogonal-pullback-condition-htpy-right f' G ( is-fiberwise-orthogonal-pullback-condition-htpy-left F' g H)
Equivalences are fiberwise left and right 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-fiberwise-orthogonal-pullback-condition-is-equiv-left : is-equiv f → is-fiberwise-orthogonal-pullback-condition f g is-fiberwise-orthogonal-pullback-condition-is-equiv-left F f' α = is-orthogonal-pullback-condition-is-equiv-left f' g ( is-equiv-source-is-equiv-target-cartesian-hom-arrow f' f α F) is-fiberwise-orthogonal-pullback-condition-is-equiv-right : is-equiv g → is-fiberwise-orthogonal-pullback-condition f g is-fiberwise-orthogonal-pullback-condition-is-equiv-right G f' _ = is-orthogonal-pullback-condition-is-equiv-right f' g G 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-fiberwise-orthogonal-pullback-condition-equiv-left : is-fiberwise-orthogonal-pullback-condition (map-equiv f) g is-fiberwise-orthogonal-pullback-condition-equiv-left = is-fiberwise-orthogonal-pullback-condition-is-equiv-left ( map-equiv f) ( g) ( is-equiv-map-equiv 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-fiberwise-orthogonal-pullback-condition-equiv-right : is-fiberwise-orthogonal-pullback-condition f (map-equiv g) is-fiberwise-orthogonal-pullback-condition-equiv-right = is-fiberwise-orthogonal-pullback-condition-is-equiv-right ( f) ( map-equiv g) ( is-equiv-map-equiv g)
If g
is fiberwise right orthogonal to f
then it is null at the fibers of 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-null-map-fibers-is-fiberwise-orthogonal-pullback-condition : is-fiberwise-orthogonal-pullback-condition f g → (y : B) → is-null-map (fiber f y) g is-null-map-fibers-is-fiberwise-orthogonal-pullback-condition H y = is-null-map-is-orthogonal-pullback-condition-terminal-map ( fiber f y) ( g) ( H (terminal-map (fiber f y)) (fiber-cartesian-hom-arrow f y))
Closure properties of right fiberwise orthogonal maps
The right class is closed under composition and left cancellation
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-fiberwise-orthogonal-pullback-condition-right-comp : is-fiberwise-orthogonal-pullback-condition f h → is-fiberwise-orthogonal-pullback-condition f g → is-fiberwise-orthogonal-pullback-condition f (h ∘ g) is-fiberwise-orthogonal-pullback-condition-right-comp H G f' α = is-orthogonal-pullback-condition-right-comp f' g h (H f' α) (G f' α) is-fiberwise-orthogonal-pullback-condition-right-right-factor : is-fiberwise-orthogonal-pullback-condition f h → is-fiberwise-orthogonal-pullback-condition f (h ∘ g) → is-fiberwise-orthogonal-pullback-condition f g is-fiberwise-orthogonal-pullback-condition-right-right-factor H HG f' α = is-orthogonal-pullback-condition-right-right-factor f' g h ( H f' α) ( HG f' α)
The right class is closed under 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-fiberwise-orthogonal-pullback-condition-right-Π : ((i : I) → is-fiberwise-orthogonal-pullback-condition f (g i)) → is-fiberwise-orthogonal-pullback-condition f (map-Π g) is-fiberwise-orthogonal-pullback-condition-right-Π G f' α = is-orthogonal-pullback-condition-right-Π f' g (λ i → G i f' α)
The right class is closed under exponentiation
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-fiberwise-orthogonal-pullback-condition-right-postcomp : is-fiberwise-orthogonal-pullback-condition f g → is-fiberwise-orthogonal-pullback-condition f (postcomp S g) is-fiberwise-orthogonal-pullback-condition-right-postcomp G f' α = is-orthogonal-pullback-condition-right-postcomp S f' g (G f' α)
The right class is closed under cartesian products
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-fiberwise-orthogonal-pullback-condition-right-product : is-fiberwise-orthogonal-pullback-condition f g → is-fiberwise-orthogonal-pullback-condition f g' → is-fiberwise-orthogonal-pullback-condition f (map-product g g') is-fiberwise-orthogonal-pullback-condition-right-product G G' f' α = is-orthogonal-pullback-condition-right-product f' g g' (G f' α) (G' f' α)
The right class is closed under base change
Given a base change of g
X' -----> X
| ⌟ |
g'| | g
∨ ∨
Y' -----> Y,
if g
is fiberwise right orthogonal to f
, then g'
is fiberwise right
orthogonal to f
.
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') (β : cartesian-hom-arrow g' g) where is-fiberwise-orthogonal-pullback-condition-right-base-change : is-fiberwise-orthogonal-pullback-condition f g → is-fiberwise-orthogonal-pullback-condition f g' is-fiberwise-orthogonal-pullback-condition-right-base-change G f' α = is-orthogonal-pullback-condition-right-base-change f' g g' β (G f' α)
Closure properties of left fiberwise orthogonal maps
The left class is closed under composition and have the right cancellation property
This remains to be formalized.
The left class is closed under coproducts
This remains to be formalized.
The left class is preserved under base change
module _ {l1 l2 l3 l4 l5 l6 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4} {A' : UU l5} {B' : UU l6} (f : A → B) (g : X → Y) (f' : A' → B') (α : cartesian-hom-arrow f' f) where is-fiberwise-orthogonal-pullback-condition-left-base-change : is-fiberwise-orthogonal-pullback-condition f g → is-fiberwise-orthogonal-pullback-condition f' g is-fiberwise-orthogonal-pullback-condition-left-base-change H f'' α' = H f'' (comp-cartesian-hom-arrow f'' f' f α α')
The left class is closed under cobase change
This remains to be formalized.
The left class is closed transfininte composition
This remains to be formalized.
The left class is closed under cartesian products
This remains to be formalized.
The left class is closed under taking image inclusions
This remains to be formalized.
References
- Reid Barton’s note on Internal cd-structures (GitHub upload)
Recent changes
- 2024-06-04. Fredrik Bakke. Fiberwise orthogonal maps and closure properties of the right class (#1152).