# Localizations at maps

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2023-06-28.
Last modified on 2024-03-12.

module orthogonal-factorization-systems.localizations-maps where

Imports
open import foundation.cartesian-product-types
open import foundation.dependent-pair-types
open import foundation.universe-levels

open import orthogonal-factorization-systems.local-types
open import orthogonal-factorization-systems.localizations-subuniverses


## Idea

Let f be a map of type A → B and let X be a type. The localization of X at f, or f-localization, is an f-local type Y together with a map η : X → Y with the property that every type that is f-local is also η-local.

## Definition

### The predicate of being a localization at a map

is-localization :
{l1 l2 l3 l4 : Level} (l5 : Level)
{A : UU l1} {B : UU l2} (f : A → B)
(X : UU l3) (Y : UU l4) (η : X → Y) →
UU (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ lsuc l5)
is-localization l5 f X Y η =
( is-local f Y) ×
( (Z : UU l5) → is-local f Z → is-local η Z)

module _
{l1 l2 l3 l4 l5 : Level}
{A : UU l1} {B : UU l2} {f : A → B}
{X : UU l3} {Y : UU l4} {η : X → Y}
(is-localization-Y : is-localization l5 f X Y η)
where

is-local-is-localization : is-local f Y
is-local-is-localization = pr1 is-localization-Y


### The type of localizations of a type with respect to a map

localization :
{l1 l2 l3 : Level} (l4 l5 : Level)
{A : UU l1} {B : UU l2} (f : A → B)
(X : UU l3) → UU (l1 ⊔ l2 ⊔ l3 ⊔ lsuc l4 ⊔ lsuc l5)
localization l4 l5 f X =
Σ (UU l4) (λ Y → Σ (X → Y) (is-localization l5 f X Y))


## Properties

### Localizations at a map are localizations at a subuniverse

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

is-subuniverse-localization-is-localization :
is-localization l4 f X Y η →
is-subuniverse-localization (is-local-Prop f) X Y
pr1 (is-subuniverse-localization-is-localization is-localization-Y) =
pr1 is-localization-Y
pr1 (pr2 (is-subuniverse-localization-is-localization is-localization-Y)) = η
pr2 (pr2 (is-subuniverse-localization-is-localization is-localization-Y))
Z is-local-Z =
pr2 is-localization-Y Z is-local-Z


It remains to construct a converse.

## References

[Rij19]
Egbert Rijke. Classifying Types. PhD thesis, Carnegie Mellon University, 06 2019. arXiv:1906.09435.
[RSS20]
Egbert Rijke, Michael Shulman, and Bas Spitters. Modalities in homotopy type theory. Logical Methods in Computer Science, 01 2020. URL: https://lmcs.episciences.org/6015, arXiv:1706.07526, doi:10.23638/LMCS-16(1:2)2020.