Reflexive globular types

Content created by Fredrik Bakke, Egbert Rijke and Vojtěch Štěpančík.

Created on 2024-03-20.
Last modified on 2024-09-03.

{-# OPTIONS --guardedness #-}

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

open import structured-types.globular-types

Idea

A globular type is reflexive if every -cell x comes with a choice of -cell from x to x.

Definition

Reflexivity structure on a globular structure

record
  is-reflexive-globular-structure
  {l1 l2 : Level} {A : UU l1} (G : globular-structure l2 A) : UU (l1 ⊔ l2)
  where
  coinductive
  field
    is-reflexive-1-cell-is-reflexive-globular-structure :
      is-reflexive (1-cell-globular-structure G)
    is-reflexive-globular-structure-1-cell-is-reflexive-globular-structure :
      (x y : A) →
      is-reflexive-globular-structure
        ( globular-structure-1-cell-globular-structure G x y)

open is-reflexive-globular-structure public

module _
  {l1 l2 : Level} {A : UU l1} {G : globular-structure l2 A}
  (r : is-reflexive-globular-structure G)
  where

  refl-1-cell-is-reflexive-globular-structure :
    {x : A} → 1-cell-globular-structure G x x
  refl-1-cell-is-reflexive-globular-structure {x} =
    is-reflexive-1-cell-is-reflexive-globular-structure r x

  refl-2-cell-is-reflexive-globular-structure :
    {x y : A} {f : 1-cell-globular-structure G x y} →
    2-cell-globular-structure G f f
  refl-2-cell-is-reflexive-globular-structure {x} {y} {f} =
    is-reflexive-1-cell-is-reflexive-globular-structure
      ( is-reflexive-globular-structure-1-cell-is-reflexive-globular-structure
        ( r)
        ( x)
        ( y))
      ( f)

  is-reflexive-globular-structure-2-cell-is-reflexive-globular-structure :
    {x y : A}
    (f g : 1-cell-globular-structure G x y) →
    is-reflexive-globular-structure
      ( globular-structure-2-cell-globular-structure G f g)
  is-reflexive-globular-structure-2-cell-is-reflexive-globular-structure
    { x} {y} =
    is-reflexive-globular-structure-1-cell-is-reflexive-globular-structure
      ( is-reflexive-globular-structure-1-cell-is-reflexive-globular-structure
        ( r)
        ( x)
        ( y))

  refl-3-cell-is-reflexive-globular-structure :
    {x y : A}
    {f g : 1-cell-globular-structure G x y}
    {H : 2-cell-globular-structure G f g} →
    3-cell-globular-structure G H H
  refl-3-cell-is-reflexive-globular-structure {x} {y} {f} {g} {H} =
    is-reflexive-1-cell-is-reflexive-globular-structure
      ( is-reflexive-globular-structure-2-cell-is-reflexive-globular-structure
        ( f)
        ( g))
      ( H)

The type of reflexive globular structures

reflexive-globular-structure :
  {l1 : Level} (l2 : Level) (A : UU l1) → UU (l1 ⊔ lsuc l2)
reflexive-globular-structure l2 A =
  Σ (globular-structure l2 A) (is-reflexive-globular-structure)

Reflexivity structure on a globular type

module _
  {l1 l2 : Level} (G : Globular-Type l1 l2)
  where

  is-reflexive-Globular-Type : UU (l1 ⊔ l2)
  is-reflexive-Globular-Type =
    is-reflexive-globular-structure (globular-structure-0-cell-Globular-Type G)

Reflexive globular types

record
  Reflexive-Globular-Type (l1 l2 : Level) : UU (lsuc l1 ⊔ lsuc l2)
  where

  field
    globular-type-Reflexive-Globular-Type : Globular-Type l1 l2

  0-cell-Reflexive-Globular-Type : UU l1
  0-cell-Reflexive-Globular-Type =
    0-cell-Globular-Type globular-type-Reflexive-Globular-Type

  1-cell-Reflexive-Globular-Type :
    0-cell-Reflexive-Globular-Type → 0-cell-Reflexive-Globular-Type → UU l2
  1-cell-Reflexive-Globular-Type =
    1-cell-Globular-Type globular-type-Reflexive-Globular-Type

  2-cell-Reflexive-Globular-Type :
    {x x' : 0-cell-Reflexive-Globular-Type}
    (y y' : 1-cell-Reflexive-Globular-Type x x') → UU l2
  2-cell-Reflexive-Globular-Type =
    2-cell-Globular-Type globular-type-Reflexive-Globular-Type

  globular-structure-Reflexive-Globular-Type :
    globular-structure l2 0-cell-Reflexive-Globular-Type
  globular-structure-Reflexive-Globular-Type =
    globular-structure-0-cell-Globular-Type
      ( globular-type-Reflexive-Globular-Type)

  field
    refl-Reflexive-Globular-Type :
      is-reflexive-globular-structure globular-structure-Reflexive-Globular-Type

  refl-0-cell-Reflexive-Globular-Type :
    {x : 0-cell-Reflexive-Globular-Type} →
    1-cell-Reflexive-Globular-Type x x
  refl-0-cell-Reflexive-Globular-Type =
    is-reflexive-1-cell-is-reflexive-globular-structure
      ( refl-Reflexive-Globular-Type)
      ( _)

  1-cell-globular-type-Reflexive-Globular-Type :
    (x y : 0-cell-Reflexive-Globular-Type) → Globular-Type l2 l2
  1-cell-globular-type-Reflexive-Globular-Type =
    1-cell-globular-type-Globular-Type globular-type-Reflexive-Globular-Type

  1-cell-reflexive-globular-type-Reflexive-Globular-Type :
    (x y : 0-cell-Reflexive-Globular-Type) → Reflexive-Globular-Type l2 l2
  globular-type-Reflexive-Globular-Type
    ( 1-cell-reflexive-globular-type-Reflexive-Globular-Type x y) =
    1-cell-globular-type-Reflexive-Globular-Type x y
  refl-Reflexive-Globular-Type
    ( 1-cell-reflexive-globular-type-Reflexive-Globular-Type x y) =
    is-reflexive-globular-structure-1-cell-is-reflexive-globular-structure
      ( refl-Reflexive-Globular-Type)
      ( x)
      ( y)

open Reflexive-Globular-Type public

Examples

The reflexive globular structure on a type given by its identity types

is-reflexive-globular-structure-Id :
  {l : Level} (A : UU l) →
  is-reflexive-globular-structure (globular-structure-Id A)
is-reflexive-globular-structure-Id A =
  λ where
  .is-reflexive-1-cell-is-reflexive-globular-structure x →
    refl
  .is-reflexive-globular-structure-1-cell-is-reflexive-globular-structure x y →
    is-reflexive-globular-structure-Id (x ＝ y)

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