Large reflexive globular types

Created on 2024-11-17.
Last modified on 2024-11-17.

{-# OPTIONS --guardedness #-}

module globular-types.large-reflexive-globular-types where
Imports 
open import foundation.large-binary-relations
open import foundation.universe-levels

open import globular-types.large-globular-types
open import globular-types.reflexive-globular-types

Idea

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

Definition

Reflexivity structure on a large globular structure

record
  is-reflexive-large-globular-structure
  {α : Level → Level} {β : Level → Level → Level}
  {A : (l : Level) → UU (α l)}
  (G : large-globular-structure β A) : UUω
  where

  field
    is-reflexive-1-cell-is-reflexive-large-globular-structure :
      is-reflexive-Large-Relation A (1-cell-large-globular-structure G)

    is-reflexive-globular-structure-1-cell-is-reflexive-large-globular-structure :
      {l1 l2 : Level} (x : A l1) (y : A l2) →
      is-reflexive-globular-structure
        ( globular-structure-1-cell-large-globular-structure G x y)

open is-reflexive-large-globular-structure public

module _
  {α : Level → Level} {β : Level → Level → Level}
  {A : (l : Level) → UU (α l)}
  {G : large-globular-structure β A}
  (r : is-reflexive-large-globular-structure G)
  where

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

  refl-2-cell-is-reflexive-large-globular-structure :
    {l1 l2 : Level} {x : A l1} {y : A l2}
    {f : 1-cell-large-globular-structure G x y} →
    2-cell-large-globular-structure G f f
  refl-2-cell-is-reflexive-large-globular-structure {x = x} {y} {f} =
    is-reflexive-1-cell-is-reflexive-globular-structure
      ( is-reflexive-globular-structure-1-cell-is-reflexive-large-globular-structure
        ( r)
        ( x)
        ( y))
      ( f)

  refl-3-cell-is-reflexive-large-globular-structure :
    {l1 l2 : Level} {x : A l1} {y : A l2}
    {f g : 1-cell-large-globular-structure G x y} →
    {H : 2-cell-large-globular-structure G f g} →
    3-cell-large-globular-structure G H H
  refl-3-cell-is-reflexive-large-globular-structure {x = x} {y} =
    refl-2-cell-is-reflexive-globular-structure
      ( is-reflexive-globular-structure-1-cell-is-reflexive-large-globular-structure
        ( r)
        ( x)
        ( y))

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