Large reflexive globular types

Content created by Fredrik Bakke.

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

{-# OPTIONS --guardedness #-}

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

open import structured-types.large-globular-types
open import structured-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)

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