# The precategory of total orders

Content created by Fredrik Bakke.

Created on 2023-10-16.

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 or total orders of universe level l

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