This is an archived version pinned as of the submission of my master's thesis.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 identity systems

module foundation.universal-property-identity-systems where
Imports 
open import foundation.dependent-pair-types
open import foundation.identity-systems
open import foundation.universal-property-contractible-types
open import foundation.universal-property-dependent-pair-types
open import foundation.universe-levels

open import foundation-core.contractible-types
open import foundation-core.equivalences
open import foundation-core.identity-types
open import foundation-core.torsorial-type-families

Idea

A (unary) identity system on a type A equipped with a point a : A consists of a type family B over A equipped with a point b : B a that satisfies an induction principle analogous to the induction principle of the identity type at a. The dependent universal property of identity types also follows for identity systems.

Definition

The universal property of identity systems

dependent-universal-property-identity-system :
  {l1 l2 : Level} {A : UU l1} (B : A → UU l2) {a : A} (b : B a) → UUω
dependent-universal-property-identity-system {A = A} B b =
  {l3 : Level} (P : (x : A) (y : B x) → UU l3) →
  is-equiv (ev-refl-identity-system b {P})

Properties

A type family satisfies the dependent universal property of identity systems if and only if it is torsorial

module _
  {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {a : A} (b : B a)
  where

  dependent-universal-property-identity-system-is-torsorial :
    is-torsorial B →
    dependent-universal-property-identity-system B b
  dependent-universal-property-identity-system-is-torsorial
    H P =
    is-equiv-left-factor
      ( ev-refl-identity-system b)
      ( ev-pair)
      ( dependent-universal-property-contr-is-contr
        ( a , b)
        ( H)
        ( λ u → P (pr1 u) (pr2 u)))
      ( is-equiv-ev-pair)

  equiv-dependent-universal-property-identity-system-is-torsorial :
    is-torsorial B →
    {l : Level} {C : (x : A) → B x → UU l} →
    ((x : A) (y : B x) → C x y) ≃ C a b
  pr1 (equiv-dependent-universal-property-identity-system-is-torsorial H) =
    ev-refl-identity-system b
  pr2 (equiv-dependent-universal-property-identity-system-is-torsorial H) =
    dependent-universal-property-identity-system-is-torsorial H _

  is-torsorial-dependent-universal-property-identity-system :
    dependent-universal-property-identity-system B b →
    is-torsorial B
  pr1 (is-torsorial-dependent-universal-property-identity-system H) = (a , b)
  pr2 (is-torsorial-dependent-universal-property-identity-system H) u =
    map-inv-is-equiv
      ( H (λ x y → (a , b) ＝ (x , y)))
      ( refl)
      ( pr1 u)
      ( pr2 u)

A type family satisfies the dependent universal property of identity systems if and only if it is an identity system

module _
  {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {a : A} (b : B a)
  where

  dependent-universal-property-identity-system-is-identity-system :
    is-identity-system B a b →
    dependent-universal-property-identity-system B b
  dependent-universal-property-identity-system-is-identity-system H =
    dependent-universal-property-identity-system-is-torsorial b
      ( is-torsorial-is-identity-system a b H)

  is-identity-system-dependent-universal-property-identity-system :
    dependent-universal-property-identity-system B b →
    is-identity-system B a b
  is-identity-system-dependent-universal-property-identity-system H P =
    section-is-equiv (H P)