Singleton induction

module foundation-core.singleton-induction where
Imports
open import foundation-core.contractible-types
open import foundation-core.dependent-pair-types
open import foundation-core.functions
open import foundation-core.homotopies
open import foundation-core.identity-types
open import foundation-core.sections
open import foundation-core.universe-levels

Idea

Singleton induction on a type A equipped with a point a : A is a principle analogous to the induction principle of the unit type. A type satisfies singleton induction if and only if it is contractible.

Definition

is-singleton :
  (l1 : Level) {l2 : Level} (A : UU l2)  A  UU (lsuc l1  l2)
is-singleton l A a = (B : A  UU l)  sec (ev-point a {B})

ind-is-singleton :
  {l1 l2 : Level} {A : UU l1} (a : A) 
  ({l : Level}  is-singleton l A a)  (B : A  UU l2) 
  B a  (x : A)  B x
ind-is-singleton a is-sing-A B = pr1 (is-sing-A B)

compute-ind-is-singleton :
  {l1 l2 : Level} {A : UU l1} (a : A) (H : {l : Level}  is-singleton l A a) 
  (B : A  UU l2)  (ev-point a {B}  ind-is-singleton a H B) ~ id
compute-ind-is-singleton a H B = pr2 (H B)

Properties

A type satisfies singleton induction if and only if it is contractible

ind-singleton-is-contr :
  {l1 l2 : Level} {A : UU l1} (a : A) (is-contr-A : is-contr A)
  (B : A  UU l2)  B a  (x : A)  B x
ind-singleton-is-contr a is-contr-A B b x =
  tr B ((inv (contraction is-contr-A a))  (contraction is-contr-A x)) b

compute-ind-singleton-is-contr :
  {l1 l2 : Level} {A : UU l1} (a : A) (is-contr-A : is-contr A)
  (B : A  UU l2) 
  ((ev-point a {B})  (ind-singleton-is-contr a is-contr-A B)) ~ id
compute-ind-singleton-is-contr a is-contr-A B b =
  ap  ω  tr B ω b) (left-inv (contraction is-contr-A a))

is-singleton-is-contr :
  {l1 l2 : Level} {A : UU l1} (a : A)  is-contr A  is-singleton l2 A a
pr1 (is-singleton-is-contr a is-contr-A B) =
  ind-singleton-is-contr a is-contr-A B
pr2 (is-singleton-is-contr a is-contr-A B) =
  compute-ind-singleton-is-contr a is-contr-A B

abstract
  is-contr-ind-singleton :
    {l1 : Level} (A : UU l1) (a : A) 
    ({l2 : Level} (P : A  UU l2)  P a  (x : A)  P x)  is-contr A
  pr1 (is-contr-ind-singleton A a S) = a
  pr2 (is-contr-ind-singleton A a S) = S  x  a  x) refl

abstract
  is-contr-is-singleton :
    {l1 : Level} (A : UU l1) (a : A) 
    ({l2 : Level}  is-singleton l2 A a)  is-contr A
  is-contr-is-singleton A a S = is-contr-ind-singleton A a  P  pr1 (S P))

Examples

The total space of an identity type satisfies singleton induction

abstract
  is-singleton-total-path :
    {l1 l2 : Level} (A : UU l1) (a : A) 
    is-singleton l2 (Σ A  x  a  x)) (pair a refl)
  pr1 (is-singleton-total-path A a B) = ind-Σ  (ind-Id a _)
  pr2 (is-singleton-total-path A a B) = refl-htpy