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revert some merge conflicts
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fredrik-bakke committed Oct 19, 2024
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Expand Up @@ -51,7 +51,7 @@ Decidable-Total-Order-Large-Precategory =
( λ l l))
```

### The precategory or total orders of universe level `l`
### The precategory of total orders at a universe level

```agda
Decidable-Total-Order-Precategory : (l : Level) Precategory (lsuc l) l
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12 changes: 11 additions & 1 deletion src/order-theory/precategory-of-posets.lagda.md
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Expand Up @@ -28,6 +28,16 @@ consists of [posets](order-theory.posets.md) and

### The large precategory of posets

**Remark.** In this formalization we see a clear limit to our approach using
[large precategories](category-theory.large-precategories.md). Large
precategories are only designed to encapsulate structures that are universe
polymorphic in a single universe level variable. However, posets are polymorphic
in two universe level variables, leading to a shortcoming in the below
formalization. Namely, we cannot capture the structure of all posets and
morphisms between them. For instance, we can only capture morphisms between two
posets of the form `A : Poset (α l1) (β l1)` and `B : Poset (α l2) (β l2)`, and
not arbitrary ones of the form `A : Poset l1 l2` and `B : Poset l3 l4`.

```agda
parametric-Poset-Large-Precategory :
(α β : Level Level)
Expand Down Expand Up @@ -55,7 +65,7 @@ Poset-Large-Precategory : Large-Precategory lsuc (_⊔_)
Poset-Large-Precategory = parametric-Poset-Large-Precategory (λ l l) (λ l l)
```

### The precategory or posets of universe level `l`
### The precategory of posets at a universe level

```agda
Poset-Precategory : (l : Level) Precategory (lsuc l) l
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2 changes: 1 addition & 1 deletion src/order-theory/precategory-of-total-orders.lagda.md
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Expand Up @@ -48,7 +48,7 @@ Total-Order-Large-Precategory =
( parametric-Total-Order-Full-Large-Subprecategory (λ l l) (λ l l))
```

### The precategory or total orders of universe level `l`
### The precategory of total orders at a universe level

```agda
Total-Order-Precategory : (l : Level) Precategory (lsuc l) l
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