# Large symmetric globular types

Content created by Fredrik Bakke.

Created on 2024-03-20.

{-# OPTIONS --guardedness #-}

module structured-types.large-symmetric-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.symmetric-globular-types


## Idea

We say a large globular type is symmetric if there is a symmetry action on its -cells for positive , mapping -cells from x to y to -cells from y to x.

## Definition

### Symmetry structure on a large globular structure

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

open is-symmetric-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-symmetric-large-globular-structure G)
where

sym-1-cell-is-symmetric-large-globular-structure :
{l1 l2 : Level} {x : A l1} {y : A l2} →
1-cell-large-globular-structure G x y →
1-cell-large-globular-structure G y x
sym-1-cell-is-symmetric-large-globular-structure {x = x} {y} =
is-symmetric-1-cell-is-symmetric-large-globular-structure r x y

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