This is an archived version pinned as of the submission of my master's thesis.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

Reflexive relations

module foundation.reflexive-relations where
Imports 
open import foundation.binary-relations
open import foundation.dependent-pair-types
open import foundation.universe-levels

open import foundation-core.identity-types

Idea

A reflexive relation on a type A is a type-valued binary relation R : A → A → 𝒰 equipped with a proof r : (x : A) → R x x.

Definitions

Reflexive relations

Reflexive-Relation :
  {l1 : Level} (l2 : Level) → UU l1 → UU (l1 ⊔ lsuc l2)
Reflexive-Relation l2 A = Σ (Relation l2 A) (λ R → is-reflexive R)

module _
  {l1 l2 : Level} {A : UU l1} (R : Reflexive-Relation l2 A)
  where

  rel-Reflexive-Relation : Relation l2 A
  rel-Reflexive-Relation = pr1 R

  is-reflexive-Reflexive-Relation : is-reflexive rel-Reflexive-Relation
  is-reflexive-Reflexive-Relation = pr2 R

The identity reflexive relation on a type

Id-Reflexive-Relation : {l : Level} (A : UU l) → Reflexive-Relation l A
Id-Reflexive-Relation A = (Id , (λ x → refl))