# Small maps

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

Created on 2022-02-17.

module foundation.small-maps where

Imports
open import foundation.dependent-pair-types
open import foundation.locally-small-types
open import foundation.retracts-of-maps
open import foundation.split-idempotent-maps
open import foundation.universe-levels

open import foundation-core.fibers-of-maps
open import foundation-core.propositions
open import foundation-core.small-types


## Idea

A map is said to be small if its fibers are small.

More specifically, a map f : A → B is small with respect to a universe 𝒰 if, for every b : B, the fiber of f over y

  fiber f b ≐ Σ (x : A), (f x ＝ b),


is equivalent to a type in 𝒰 that may vary depending on b.

## Definition

is-small-map :
(l : Level) {l1 l2 : Level} {A : UU l1} {B : UU l2} →
(A → B) → UU (lsuc l ⊔ l1 ⊔ l2)
is-small-map l {B = B} f = (b : B) → is-small l (fiber f b)


## Properties

### Any map between small types is small

abstract
is-small-fiber :
{l l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) →
is-small l A → is-small l B → (b : B) → is-small l (fiber f b)
is-small-fiber f H K b =
is-small-Σ H (λ a → is-locally-small-is-small K (f a) b)


### Being a small map is a property

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

abstract
is-prop-is-small-map : is-prop (is-small-map l f)
is-prop-is-small-map = is-prop-Π (λ x → is-prop-is-small l (fiber f x))

is-small-map-Prop : Prop (lsuc l ⊔ l1 ⊔ l2)
is-small-map-Prop = is-small-map l f , is-prop-is-small-map


### Small maps are closed under retracts

module _
{l1 l2 l3 l4 l : Level} {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4}
{f : A → B} {g : X → Y} (R : f retract-of-map g)
where

is-small-map-retract : is-small-map l g → is-small-map l f
is-small-map-retract is-small-g b =
is-small-retract
( is-small-g (map-codomain-inclusion-retract-map f g R b))
( retract-fiber-retract-map f g R b)