Add functors from categories to preorders to setoids

3 files changed, 140 insertions, 0 deletions

diff --git a/Functor/Forgetful/Instance/Preorder.agda b/Functor/Forgetful/Instance/Preorder.agda
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+++ b/Functor/Forgetful/Instance/Preorder.agda
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+{-# OPTIONS --without-K --safe #-}
+
+module Functor.Forgetful.Instance.Preorder where
+
+open import Categories.Category.Instance.Setoids using (Setoids)
+open import Categories.Functor using (Functor)
+open import Category.Instance.Preorders using (Preorders)
+open import Level using (Level)
+open import Relation.Binary.Bundles using (Preorder)
+open import Relation.Binary.Morphism using (PreorderHomomorphism)
+
+-- Get the underlying setoid of a preorder.
+-- This is not the same thing as the setoid induced by _≲_.
+
+Forget : {c ℓ e : Level} → Functor (Preorders c ℓ e) (Setoids c ℓ)
+Forget = let open Preorder.Eq in record
+    { F₀ = setoid
+    ; F₁ = λ F → let open PreorderHomomorphism F in record
+        { to = ⟦_⟧
+        ; cong = cong
+        }
+    ; identity = λ {A} → refl A
+    ; homomorphism = λ {_ _ Z} → refl Z
+    ; F-resp-≈ = λ f≗g → f≗g
+    }
diff --git a/Functor/Free/Instance/InducedSetoid.agda b/Functor/Free/Instance/InducedSetoid.agda
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+++ b/Functor/Free/Instance/InducedSetoid.agda
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+{-# OPTIONS --without-K --safe #-}
+
+module Functor.Free.Instance.InducedSetoid where
+
+-- The induced setoid of a (primitive) preorder
+
+open import Categories.Category.Instance.Setoids using (Setoids)
+open import Categories.Functor using (Functor)
+open import Category.Instance.Preorders.Primitive using (Preorders)
+open import Function using (Func)
+open import Level using (Level)
+open import Preorder.Primitive using (Preorder; module Isomorphism)
+open import Preorder.Primitive.MonotoneMap using (MonotoneMap)
+open import Relation.Binary using (Setoid)
+
+module _ {c ℓ : Level} where
+
+  module _ (P : Preorder c ℓ) where
+
+    open Preorder P
+    open Isomorphism P using (_≅_; ≅-isEquivalence)
+
+    ≅-setoid : Setoid c ℓ
+    ≅-setoid = record
+        { Carrier = Carrier
+        ; _≈_ = _≅_
+        ; isEquivalence = ≅-isEquivalence
+        }
+
+  func : {A B : Preorder c ℓ} → MonotoneMap A B → Func (≅-setoid A) (≅-setoid B)
+  func {A} {B} f = let open MonotoneMap f in record
+      { to = map
+      ; cong = map-resp-≅
+      }
+
+  open Isomorphism using (module ≅)
+
+  InducedSetoid : Functor (Preorders c ℓ) (Setoids c ℓ)
+  InducedSetoid = record
+      { F₀ = ≅-setoid
+      ; F₁ = func
+      ; identity = λ {P} → ≅.refl P
+      ; homomorphism = λ {Z = Z} → ≅.refl Z
+      ; F-resp-≈ = λ f≃g {x} → f≃g x
+      }
+
+  module InducedSetoid = Functor InducedSetoid
diff --git a/Functor/Free/Instance/Preorder.agda b/Functor/Free/Instance/Preorder.agda
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+++ b/Functor/Free/Instance/Preorder.agda
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+{-# OPTIONS --without-K --safe #-}
+
+module Functor.Free.Instance.Preorder where
+
+open import Categories.Category using (Category)
+open import Categories.Category.Instance.Cats using (Cats)
+open import Function using (flip)
+open import Categories.Functor using (Functor; _∘F_)
+open import Categories.NaturalTransformation.NaturalIsomorphism using (NaturalIsomorphism)
+open import Category.Instance.Preorders.Primitive using (Preorders)
+open import Level using (Level; _⊔_; suc)
+open import Preorder.Primitive using (Preorder; module Isomorphism)
+open import Preorder.Primitive.MonotoneMap using (MonotoneMap; _≃_; module ≃)
+ 
+-- The "free preorder" of a category
+-- i.e. (0,1)-truncation
+
+-- In this setting, a category is a proof-relevant preorder,
+-- i.e. each proof of A ≤ B is a distinct morphism.
+
+-- To get a preorder from a category,
+-- we ignore this distinction,
+-- making all morphisms A ⇒ B "the same".
+-- Identity and composition become reflexivity and transitivity.
+
+-- Likewise, we can get a monotone map from a functor,
+-- and a pointwise isomorphism of monotone maps from
+-- a natural isomorphism of functors, simply by
+-- forgetting morphism equalities.
+
+module _ {o ℓ e : Level} where
+
+  preorder : Category o ℓ e → Preorder o ℓ
+  preorder C = let open Category C in record
+      { Carrier = Obj
+      ; _≲_ = _⇒_
+      ; refl = id
+      ; trans = flip _∘_
+      }
+
+  module _ {A B : Category o ℓ e} where
+
+    monotoneMap : Functor A B → MonotoneMap (preorder A) (preorder B)
+    monotoneMap F = let open Functor F in record
+        { map = F₀
+        ; mono = F₁
+        }
+
+    open NaturalIsomorphism using (module ⇒; module ⇐)
+
+    pointwiseIsomorphism : {F G : Functor A B} → NaturalIsomorphism F G → monotoneMap F ≃ monotoneMap G
+    pointwiseIsomorphism F≃G X = record
+        { from = ⇒.η F≃G X
+        ; to = ⇐.η F≃G X
+        }
+
+Free : {o ℓ e : Level} → Functor (Cats o ℓ e) (Preorders o ℓ)
+Free = record
+    { F₀ = preorder
+    ; F₁ = monotoneMap
+    ; identity = ≃.refl {x = id}
+    ; homomorphism = λ {f = f} {h} → ≃.refl {x = monotoneMap (h ∘F f)}
+    ; F-resp-≈ = pointwiseIsomorphism
+    }
+  where
+    open Category (Preorders _ _) using (id)
+
+module Free {o ℓ e} = Functor (Free {o} {ℓ} {e})