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refactor for exponential algebras

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Leon Vatthauer 2024-02-04 18:49:12 +01:00
parent 38c29332fc
commit 17c023c944
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1
agda/src/Algebra/Elgot/Properties.lagda.md
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@ -1,4 +1,5 @@
```agda ```agda
{-# OPTIONS --allow-unsolved-metas #-}
open import Level open import Level
open import Categories.FreeObjects.Free open import Categories.FreeObjects.Free
open import Categories.Functor.Core open import Categories.Functor.Core

2
agda/src/Category/Construction/PreElgotMonads.lagda.md
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@ -18,7 +18,7 @@ open import Categories.Category.Core
module Category.Construction.PreElgotMonads {o ℓ e} (ambient : Ambient o ℓ e) where module Category.Construction.PreElgotMonads {o ℓ e} (ambient : Ambient o ℓ e) where
open Ambient ambient open Ambient ambient
open import Monad.PreElgot ambient open import Monad.PreElgot ambient
open import Algebra.Elgot ambient open import Algebra.Elgot cocartesian
open HomReasoning open HomReasoning
open Equiv open Equiv
open M C open M C

2
agda/src/Category/Construction/StrongPreElgotMonads.lagda.md
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@ -20,7 +20,7 @@ open import Data.Product using (_,_)
module Category.Construction.StrongPreElgotMonads {o ℓ e} (ambient : Ambient o ℓ e) where module Category.Construction.StrongPreElgotMonads {o ℓ e} (ambient : Ambient o ℓ e) where
open Ambient ambient open Ambient ambient
open import Monad.PreElgot ambient open import Monad.PreElgot ambient
open import Algebra.Elgot ambient open import Algebra.Elgot cocartesian
open HomReasoning open HomReasoning
open Equiv open Equiv
open M C open M C

56
agda/src/FreeObject.lagda.md
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@ -1,56 +0,0 @@
<!--
```agda
open import Level
open import Categories.Category
open import Categories.Functor
```
-->
```agda
module FreeObject where
private
variable
o ℓ e o' ℓ' e' : Level
```
# Free objects
The notion of free object has been formalized in agda-categories:
```agda
open import Categories.FreeObjects.Free
```
but we need a predicate expressing that an object 'is free':
```agda
record IsFreeObject {C : Category o ℓ e} {D : Category o' ℓ' e'} (U : Functor D C) (X : Category.Obj C) (FX : Category.Obj D) : Set (suc (e ⊔ e' ⊔ o ⊔ ℓ ⊔ o' ⊔ ℓ')) where
private
module C = Category C using (Obj; id; identityʳ; identityˡ; _⇒_; _∘_; equiv; module Equiv)
module D = Category D using (Obj; _⇒_; _∘_; equiv)
module U = Functor U using (₀; ₁)
field
η : C [ X , U.₀ FX ]
_* : ∀ {A : D.Obj} → C [ X , U.₀ A ] → D [ FX , A ]
*-lift : ∀ {A : D.Obj} (f : C [ X , U.₀ A ]) → C [ (U.₁ (f *) C.∘ η) ≈ f ]
*-uniq : ∀ {A : D.Obj} (f : C [ X , U.₀ A ]) (g : D [ FX , A ]) → C [ (U.₁ g) C.∘ η ≈ f ] → D [ g ≈ f * ]
```
and some way to convert between these notions:
```agda
IsFreeObject⇒FreeObject : ∀ {C : Category o ℓ e} {D : Category o' ℓ' e'} (U : Functor D C) (X : Category.Obj C) (FX : Category.Obj D) → IsFreeObject U X FX → FreeObject U X
IsFreeObject⇒FreeObject U X FX isFO = record
{ FX = FX
; η = η
; _* = _*
; *-lift = *-lift
; *-uniq = *-uniq
}
where open IsFreeObject isFO
-- TODO FreeObject⇒IsFreeObject
```

8
agda/src/Monad/Instance/K.lagda.md
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@ -17,9 +17,11 @@ open import Categories.Monad.Construction.Kleisli
```agda ```agda
module Monad.Instance.K {o ℓ e} (ambient : Ambient o ℓ e) where module Monad.Instance.K {o ℓ e} (ambient : Ambient o ℓ e) where
open Ambient ambient open Ambient ambient
open import Category.Construction.ElgotAlgebras ambient using (Elgot-Algebras) open import Category.Construction.ElgotAlgebras {C = C}
open import Algebra.Elgot ambient using (Elgot-Algebra) open Cat cocartesian using (Elgot-Algebras)
open import Algebra.Elgot.Free ambient using (FreeElgotAlgebra; elgotForgetfulF; IsStableFreeElgotAlgebra) open import Algebra.Elgot cocartesian using (Elgot-Algebra)
open import Algebra.Elgot.Free cocartesian using (FreeElgotAlgebra; elgotForgetfulF)
open import Algebra.Elgot.Stable distributive using (IsStableFreeElgotAlgebra)
open Equiv open Equiv
open MR C open MR C

8
agda/src/Monad/Instance/K/Commutative.lagda.md
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@ -20,9 +20,11 @@ open MIK ambient
open MonadK MK open MonadK MK
open import Monad.Instance.K.Strong ambient MK open import Monad.Instance.K.Strong ambient MK
open import Monad.Instance.K.EquationalLifting ambient MK open import Monad.Instance.K.EquationalLifting ambient MK
open import Category.Construction.ElgotAlgebras ambient open import Category.Construction.ElgotAlgebras {C = C}
open import Algebra.Elgot ambient open Cat cocartesian
open import Algebra.Elgot.Free ambient using (FreeElgotAlgebra; elgotForgetfulF; IsStableFreeElgotAlgebra) open import Algebra.Elgot cocartesian
open import Algebra.Elgot.Free cocartesian using (FreeElgotAlgebra; elgotForgetfulF)
open import Algebra.Elgot.Stable distributive using (IsStableFreeElgotAlgebra)
open Equiv open Equiv
open HomReasoning open HomReasoning

7
agda/src/Monad/Instance/K/EquationalLifting.lagda.md
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@ -18,9 +18,10 @@ open Ambient ambient
open MIK ambient open MIK ambient
open MonadK MK open MonadK MK
open import Monad.Instance.K.Strong ambient MK open import Monad.Instance.K.Strong ambient MK
open import Category.Construction.ElgotAlgebras ambient open import Category.Construction.ElgotAlgebras {C = C}
open import Algebra.Elgot ambient open Cat cocartesian
open import Algebra.Elgot.Free ambient using (IsStableFreeElgotAlgebra) open import Algebra.Elgot cocartesian
open import Algebra.Elgot.Stable distributive using (IsStableFreeElgotAlgebra)
open HomReasoning open HomReasoning
open Equiv open Equiv

5
agda/src/Monad/Instance/K/PreElgot.lagda.md
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@ -18,13 +18,14 @@ module Monad.Instance.K.PreElgot {o ℓ e} (ambient : Ambient o ℓ e) (MK : MIK
open Ambient ambient open Ambient ambient
open MIK ambient open MIK ambient
open MonadK MK open MonadK MK
open import Algebra.Elgot ambient open import Algebra.Elgot cocartesian
open import Monad.PreElgot ambient open import Monad.PreElgot ambient
open import Monad.Instance.K ambient open import Monad.Instance.K ambient
open import Monad.Instance.K.Commutative ambient MK open import Monad.Instance.K.Commutative ambient MK
open import Monad.Instance.K.Strong ambient MK open import Monad.Instance.K.Strong ambient MK
open import Category.Construction.PreElgotMonads ambient open import Category.Construction.PreElgotMonads ambient
open import Category.Construction.ElgotAlgebras ambient open import Category.Construction.ElgotAlgebras {C = C}
open Cat cocartesian
open Equiv open Equiv
open HomReasoning open HomReasoning

8
agda/src/Monad/Instance/K/Strong.lagda.md
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@ -22,9 +22,11 @@ import Monad.Instance.K as MIK
```agda ```agda
module Monad.Instance.K.Strong {o ℓ e} (ambient : Ambient o ℓ e) (MK : MIK.MonadK ambient) where module Monad.Instance.K.Strong {o ℓ e} (ambient : Ambient o ℓ e) (MK : MIK.MonadK ambient) where
open Ambient ambient open Ambient ambient
open import Category.Construction.ElgotAlgebras ambient open import Category.Construction.ElgotAlgebras {C = C}
open import Algebra.Elgot ambient open Cat cocartesian
open import Algebra.Elgot.Free ambient using (FreeElgotAlgebra; elgotForgetfulF; IsStableFreeElgotAlgebra) open import Algebra.Elgot cocartesian
open import Algebra.Elgot.Free cocartesian using (FreeElgotAlgebra; elgotForgetfulF)
open import Algebra.Elgot.Stable distributive using (IsStableFreeElgotAlgebra)
open MIK ambient open MIK ambient
open MonadK MK open MonadK MK

8
agda/src/Monad/Instance/K/StrongPreElgot.lagda.md
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@ -21,15 +21,17 @@ module Monad.Instance.K.StrongPreElgot {o ℓ e} (ambient : Ambient o ℓ e) (MK
open Ambient ambient open Ambient ambient
open MIK ambient open MIK ambient
open MonadK MK open MonadK MK
open import Algebra.Elgot ambient open import Algebra.Elgot cocartesian
open import Algebra.Elgot.Free ambient open import Algebra.Elgot.Free cocartesian
open import Algebra.Elgot.Stable distributive
open import Monad.PreElgot ambient open import Monad.PreElgot ambient
open import Monad.Instance.K ambient open import Monad.Instance.K ambient
open import Monad.Instance.K.Commutative ambient MK open import Monad.Instance.K.Commutative ambient MK
open import Monad.Instance.K.Strong ambient MK open import Monad.Instance.K.Strong ambient MK
open import Monad.Instance.K.PreElgot ambient MK open import Monad.Instance.K.PreElgot ambient MK
open import Category.Construction.StrongPreElgotMonads ambient open import Category.Construction.StrongPreElgotMonads ambient
open import Category.Construction.ElgotAlgebras ambient open import Category.Construction.ElgotAlgebras {C = C}
open Cat cocartesian
open Equiv open Equiv
open HomReasoning open HomReasoning

13
agda/src/Monad/Instance/Setoids/K.lagda.md
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@ -12,13 +12,14 @@ open import Function.Construct.Setoid renaming (setoid to _⇨_; _∙_ to _∘_)
open import Function using (_∘′_; id) renaming (_∘_ to _∘f_) open import Function using (_∘′_; id) renaming (_∘_ to _∘f_)
import Relation.Binary.PropositionalEquality as Eq import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_) renaming (sym to ≣-sym; refl to ≣-refl; trans to ≣-trans) open Eq using (_≡_) renaming (sym to ≣-sym; refl to ≣-refl; trans to ≣-trans)
open import FreeObject
open import Categories.FreeObjects.Free open import Categories.FreeObjects.Free
open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_) open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
open import Data.Product.Function.NonDependent.Setoid using (proj₁ₛ) open import Data.Product.Function.NonDependent.Setoid using (proj₁ₛ)
open import Categories.Category.Instance.Properties.Setoids.Choice using () open import Categories.Category.Instance.Properties.Setoids.Choice using ()
open import Data.Product.Relation.Binary.Pointwise.NonDependent open import Data.Product.Relation.Binary.Pointwise.NonDependent
open import Data.Product open import Data.Product
open import Category.Ambient using (Ambient)
open import Categories.Category.Instance.Setoids
``` ```
--> -->
@ -27,12 +28,14 @@ open import Data.Product
```agda ```agda
module Monad.Instance.Setoids.K {ℓ : Level} where module Monad.Instance.Setoids.K {ℓ : Level} where
open _⟶_ using (cong) open _⟶_ using (cong)
open import Monad.Instance.Setoids.Delay {ℓ} {ℓ}
open import Category.Ambient.Setoids open import Category.Ambient.Setoids
open Ambient (setoidAmbient {ℓ}) using (cocartesian; distributive)
open import Monad.Instance.Setoids.Delay {ℓ} {ℓ}
open import Monad.Instance.K (setoidAmbient {ℓ}) open import Monad.Instance.K (setoidAmbient {ℓ})
open import Algebra.Elgot (setoidAmbient {ℓ}) open import Algebra.Elgot cocartesian
open import Algebra.Elgot.Free (setoidAmbient {ℓ}) open import Algebra.Elgot.Free cocartesian
open import Category.Construction.ElgotAlgebras (setoidAmbient {ℓ}) open import Category.Construction.ElgotAlgebras {C = Setoids ℓ ℓ}
open Cat cocartesian
open import Monad.PreElgot (setoidAmbient {ℓ}) open import Monad.PreElgot (setoidAmbient {ℓ})
open Bisimilarity renaming (_≈_ to [_][_≈_]; _≈′_ to [_][_≈′_]; _∼_ to [_][_∼_]; _∼′_ to [_][_∼′_]; _↓_ to [_][_↓_]) open Bisimilarity renaming (_≈_ to [_][_≈_]; _≈′_ to [_][_≈′_]; _∼_ to [_][_∼_]; _∼′_ to [_][_∼′_]; _↓_ to [_][_↓_])
open DelayMonad open DelayMonad

2
agda/src/Monad/PreElgot.lagda.md
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@ -17,7 +17,7 @@ module Monad.PreElgot {o ℓ e} (ambient : Ambient o ℓ e) where
open HomReasoning open HomReasoning
open MR C open MR C
open Equiv open Equiv
open import Algebra.Elgot ambient open import Algebra.Elgot cocartesian
``` ```
# (strong) pre-Elgot monads # (strong) pre-Elgot monads