open import 1Lab.Prelude

open import Order.Base

module Order.Instances.Lift where

Lifting posets across universes🔗

Suppose you have a poset whose elements are in $o$ and relation in $r\text{,}$ but you really need it to have elements or relationships in some larger universe $o\bigsqcup o^{\prime }$ and homs in $r\bigsqcup r^{\prime }\text{.}$ Fortunately you can uniformly lift the poset to this bigger universe.

Liftᵖ : ∀ {o r} o′ r′ → Poset o r → Poset (o ⊔ o′) (r ⊔ r′)
Liftᵖ o' r' X .Poset.Ob      = Lift o' ⌞ X ⌟
Liftᵖ o' r' X .Poset._≤_ x y = Lift r' $ (X .Poset._≤_) (Lift.lower x) (Lift.lower y)
Liftᵖ o' r' X .Poset.≤-thin  = Lift-is-hlevel  $ X .Poset.≤-thin
Liftᵖ o' r' X .Poset.≤-refl  = lift $ X .Poset.≤-refl
Liftᵖ o' r' X .Poset.≤-trans (lift p) (lift q)   = lift (X .Poset.≤-trans p q)
Liftᵖ o' r' X .Poset.≤-antisym (lift p) (lift q) = ap lift (X .Poset.≤-antisym p q)

Lift-monotone-l
  : ∀ {so sr} bo br {o r} {X : Poset so sr} {Y : Poset o r}
  → Monotone X Y
  → Monotone (Liftᵖ bo br X) Y
Lift-monotone-l bo br G = F where
  open Monotone
  F : Monotone _ _
  F .hom    (lift x) = G .hom x
  F .pres-≤ (lift α) = G .pres-≤ α

Lift-monotone-r
  : ∀ {so sr} bo br {o r} {X : Poset so sr} {Y : Poset o r}
  → Monotone X Y
  → Monotone X (Liftᵖ bo br Y)
Lift-monotone-r bo br G = F where
  open Monotone
  F : Monotone _ _
  F .hom    x = lift $ G .hom x
  F .pres-≤ α = lift $ G .pres-≤ α