Local maps
Content created by Fredrik Bakke and Egbert Rijke.
Created on 2023-09-11.
Last modified on 2024-01-25.
module orthogonal-factorization-systems.local-maps where
Imports
open import foundation.cartesian-morphisms-arrows open import foundation.fibers-of-maps open import foundation.propositions open import foundation.unit-type open import foundation.universe-levels open import orthogonal-factorization-systems.local-families-of-types open import orthogonal-factorization-systems.local-types open import orthogonal-factorization-systems.orthogonal-maps
Idea
A map g : A → B
is said to be
local¶ at
f : Y → X
, or
f
-local¶, if
all its fibers are
f
-local types.
Definition
module _ {l1 l2 l3 l4 : Level} {Y : UU l1} {X : UU l2} {A : UU l3} {B : UU l4} (f : Y → X) (g : A → B) where is-local-map : UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) is-local-map = is-local-family f (fiber g) is-property-is-local-map : is-prop is-local-map is-property-is-local-map = is-property-is-local-family f (fiber g) is-local-map-Prop : Prop (l1 ⊔ l2 ⊔ l3 ⊔ l4) is-local-map-Prop = is-local-family-Prop f (fiber g)
A type B
is f
-local if and only if the terminal map B → unit
is f
-local
module _ {l1 l2 l3 : Level} {Y : UU l1} {X : UU l2} {B : UU l3} (f : Y → X) where is-local-is-local-terminal-map : is-local-map f (terminal-map B) → is-local f B is-local-is-local-terminal-map H = is-local-equiv f (inv-equiv-fiber-terminal-map star) (H star) is-local-terminal-map-is-local : is-local f B → is-local-map f (terminal-map B) is-local-terminal-map-is-local H u = is-local-equiv f (equiv-fiber-terminal-map u) H
A map is f
-local if and only if it is f
-orthogonal
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) where is-local-map-is-orthogonal-pullback-condition : is-orthogonal-pullback-condition f g → is-local-map f g is-local-map-is-orthogonal-pullback-condition G y = is-local-is-orthogonal-pullback-condition-terminal-map f ( is-orthogonal-pullback-condition-right-base-change f g ( terminal-map (fiber g y)) ( fiber-cartesian-hom-arrow g y) ( G)) is-local-map-is-orthogonal : is-orthogonal f g → is-local-map f g is-local-map-is-orthogonal G y = is-local-is-orthogonal-terminal-map f ( is-orthogonal-right-base-change f g ( terminal-map (fiber g y)) ( fiber-cartesian-hom-arrow g y) ( G))
The converse remains to be formalized.
See also
Recent changes
- 2024-01-25. Fredrik Bakke. Basic properties of orthogonal maps (#979).
- 2023-09-12. Egbert Rijke. Beyond foundation (#751).
- 2023-09-11. Fredrik Bakke. Transport along and action on equivalences (#706).
- 2023-09-11. Fredrik Bakke and Egbert Rijke. Some computations for different notions of equivalence (#711).