The precategory of finite total orders

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2023-10-16.
Last modified on 2023-10-20.

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

Idea

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

Definitions

The large precategory of finite total orders

parametric-Total-Order-𝔽-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-𝔽-Full-Large-Subprecategory α β =
  is-finite-total-order-Poset-Prop

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

The precategory of finite total orders of universe level l

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

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