The precategory of total orders

Content created by Fredrik Bakke.

Created on 2023-10-16.
Last modified on 2024-09-23.

module order-theory.precategory-of-total-orders where
Imports 
open import category-theory.full-large-subprecategories
open import category-theory.large-precategories
open import category-theory.precategories

open import foundation.universe-levels

open import order-theory.precategory-of-posets
open import order-theory.total-orders

Idea

The (large) precategory of total orders consists of total orders and order preserving maps and is exhibited as a full subprecategory of the precategory of posets.

Definitions

The large precategory of total orders

parametric-Total-Order-Full-Large-Subprecategory :
  (α β : Level → Level) →
  Full-Large-Subprecategory
    ( λ l → α l ⊔ β l)
    ( parametric-Poset-Large-Precategory α β)
parametric-Total-Order-Full-Large-Subprecategory α β = is-total-Poset-Prop

Total-Order-Large-Precategory :
  Large-Precategory lsuc (_⊔_)
Total-Order-Large-Precategory =
  large-precategory-Full-Large-Subprecategory
    ( Poset-Large-Precategory)
    ( parametric-Total-Order-Full-Large-Subprecategory (λ l → l) (λ l → l))

The precategory of total orders at a universe level

Total-Order-Precategory : (l : Level) → Precategory (lsuc l) l
Total-Order-Precategory =
  precategory-Large-Precategory Total-Order-Large-Precategory

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