This is an archived version pinned as of the submission of my master's thesis. An up-to-date version may be found online.

The universal property of the unit type

module foundation.universal-property-unit-type where

open import foundation.dependent-pair-types
open import foundation.diagonal-maps-of-types
open import foundation.unit-type
open import foundation.universal-property-contractible-types
open import foundation.universal-property-equivalences
open import foundation.universe-levels

open import foundation-core.constant-maps
open import foundation-core.contractible-types
open import foundation-core.equivalences
open import foundation-core.homotopies
open import foundation-core.precomposition-functions

Idea

The universal property of the unit type characterizes maps out of the unit type. Similarly, the dependent universal property of the unit type characterizes dependent functions out of the unit type.

In foundation.contractible-types we have alread proven related universal properties of contractible types.

Properties

ev-star :
  {l : Level} (P : unit → UU l) → ((x : unit) → P x) → P star
ev-star P f = f star

ev-star' :
  {l : Level} (Y : UU l) → (unit → Y) → Y
ev-star' Y = ev-star (λ t → Y)

abstract
  dependent-universal-property-unit :
    {l : Level} (P : unit → UU l) → is-equiv (ev-star P)
  dependent-universal-property-unit =
    dependent-universal-property-contr-is-contr star is-contr-unit

equiv-dependent-universal-property-unit :
  {l : Level} (P : unit → UU l) → ((x : unit) → P x) ≃ P star
pr1 (equiv-dependent-universal-property-unit P) = ev-star P
pr2 (equiv-dependent-universal-property-unit P) =
  dependent-universal-property-unit P

abstract
  universal-property-unit :
    {l : Level} (Y : UU l) → is-equiv (ev-star' Y)
  universal-property-unit Y = dependent-universal-property-unit (λ t → Y)

equiv-universal-property-unit :
  {l : Level} (Y : UU l) → (unit → Y) ≃ Y
pr1 (equiv-universal-property-unit Y) = ev-star' Y
pr2 (equiv-universal-property-unit Y) = universal-property-unit Y

inv-equiv-universal-property-unit :
  {l : Level} (Y : UU l) → Y ≃ (unit → Y)
inv-equiv-universal-property-unit Y =
  inv-equiv (equiv-universal-property-unit Y)

abstract
  is-equiv-point-is-contr :
    {l1 : Level} {X : UU l1} (x : X) →
    is-contr X → is-equiv (point x)
  is-equiv-point-is-contr x is-contr-X =
    is-equiv-is-contr (point x) is-contr-unit is-contr-X

abstract
  is-equiv-point-universal-property-unit :
    {l1 : Level} (X : UU l1) (x : X) →
    ({l2 : Level} (Y : UU l2) → is-equiv (λ (f : X → Y) → f x)) →
    is-equiv (point x)
  is-equiv-point-universal-property-unit X x H =
    is-equiv-is-equiv-precomp
      ( point x)
      ( λ Y →
        is-equiv-right-factor
          ( ev-star' Y)
          ( precomp (point x) Y)
          ( universal-property-unit Y)
          ( H Y))

abstract
  universal-property-unit-is-equiv-point :
    {l1 : Level} {X : UU l1} (x : X) →
    is-equiv (point x) →
    ({l2 : Level} (Y : UU l2) → is-equiv (λ (f : X → Y) → f x))
  universal-property-unit-is-equiv-point x is-equiv-point Y =
    is-equiv-comp
      ( ev-star' Y)
      ( precomp (point x) Y)
      ( is-equiv-precomp-is-equiv (point x) is-equiv-point Y)
      ( universal-property-unit Y)

abstract
  universal-property-unit-is-contr :
    {l1 : Level} {X : UU l1} (x : X) →
    is-contr X →
    ({l2 : Level} (Y : UU l2) → is-equiv (λ (f : X → Y) → f x))
  universal-property-unit-is-contr x is-contr-X =
    universal-property-unit-is-equiv-point x
      ( is-equiv-point-is-contr x is-contr-X)

abstract
  is-equiv-diagonal-exponential-is-equiv-point :
    {l1 : Level} {X : UU l1} (x : X) →
    is-equiv (point x) →
    ({l2 : Level} (Y : UU l2) → is-equiv (diagonal-exponential Y X))
  is-equiv-diagonal-exponential-is-equiv-point x is-equiv-point Y =
    is-equiv-is-section
      ( universal-property-unit-is-equiv-point x is-equiv-point Y)
      ( refl-htpy)