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Categories of propositions and sieves #1163

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Categories of propositions and sieves #1163

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8cb8f11
PROP & SIEVE.
anuyts Oct 28, 2024
42939d5
SieveOn.
anuyts Oct 29, 2024
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3 changes: 3 additions & 0 deletions Cubical/Categories/Instances/Functors.agda
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Expand Up @@ -23,6 +23,7 @@ open import Cubical.Categories.Functor.Base
open import Cubical.Categories.Morphism
open import Cubical.Categories.NaturalTransformation.Base
open import Cubical.Categories.NaturalTransformation.Properties
open import Cubical.Categories.Thin

private
variable
Expand Down Expand Up @@ -138,3 +139,5 @@ module _ (C : Category ℓC ℓC') (D : Category ℓD ℓD') where
≡⟨ D .⋆Assoc _ _ _ ⟩
(β .N-ob e ∘⟨ D ⟩ G .F-hom g) ∘⟨ D ⟩ (α .N-ob d ∘⟨ D ⟩ F .F-hom f) ∎

isThinFUNCTOR : isThin D → isThin FUNCTOR
isThinFUNCTOR thinD F G = isThin→isPropNatTrans thinD
28 changes: 28 additions & 0 deletions Cubical/Categories/Instances/Propositions.agda
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{-# OPTIONS --safe #-}

module Cubical.Categories.Instances.Propositions where

open import Cubical.Foundations.Prelude
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Function

open import Cubical.Categories.Category
open Category
open import Cubical.Categories.Thin

private
variable
ℓ : Level

PROP : ∀ (ℓ : Level) → Category (ℓ-suc ℓ) ℓ
ob (PROP ℓ) = hProp ℓ
Hom[_,_] (PROP ℓ) (P , _) (Q , _) = P → Q
id (PROP ℓ) = idfun _
_⋆_ (PROP ℓ) f g x = g (f x)
⋆IdL (PROP ℓ) {P , _} {Q , propQ} f = isProp→ propQ _ _
⋆IdR (PROP ℓ) {P , _} {Q , propQ} f = isProp→ propQ _ _
⋆Assoc (PROP ℓ) {P , _} {Q , _} {R , _} {S , propS} f g h = isPropΠ (λ _ → propS) _ _
isSetHom (PROP ℓ) {P , _} {Q , propQ} = isProp→isSet (isProp→ propQ)

isThinPROP : isThin (PROP ℓ)
isThinPROP (P , propP) (Q , propQ) = isProp→ propQ
37 changes: 37 additions & 0 deletions Cubical/Categories/Instances/SievesAsFunctors.agda
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{-# OPTIONS --safe #-}

module Cubical.Categories.Instances.SievesAsFunctors where

-- There are also sieves in Cubical.Site.Sieve
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I think it would be good to also have a pointer there.
Maybe it would also be good to mention that the definitions are equivalent.


open import Cubical.Foundations.Prelude
open import Cubical.Foundations.HLevels

open import Cubical.Categories.Category
open Category
open import Cubical.Categories.Instances.Functors
open import Cubical.Categories.Instances.Propositions
open import Cubical.Categories.Constructions.Slice
open import Cubical.Categories.Thin

private
variable
ℓ ℓ' ℓP : Level
C : Category ℓ ℓ'

SIEVE : (C : Category ℓ ℓ') → (ℓP : Level)
→ Category (ℓ-max (ℓ-max ℓ ℓ') (ℓ-suc ℓP)) (ℓ-max (ℓ-max ℓ ℓ') ℓP)
SIEVE C ℓP = FUNCTOR (C ^op) (PROP ℓP)

Sieve : (C : Category ℓ ℓ') → (ℓP : Level) → Type (ℓ-max (ℓ-max ℓ ℓ') (ℓ-suc ℓP))
Sieve C ℓP = SIEVE C ℓP .ob

isThinSIEVE : isThin (SIEVE C ℓP)
isThinSIEVE = isThinFUNCTOR _ _ isThinPROP

SIEVE_ON : (C : Category ℓ ℓ') → (c : C .ob) → (ℓP : Level)
→ Category (ℓ-max (ℓ-max ℓ ℓ') (ℓ-suc ℓP)) (ℓ-max (ℓ-max ℓ ℓ') ℓP)
SIEVE_ON C c ℓP = SIEVE (SliceCat C c) ℓP

SieveOn : (C : Category ℓ ℓ') → (c : C .ob) → (ℓP : Level) → Type (ℓ-max (ℓ-max ℓ ℓ') (ℓ-suc ℓP))
SieveOn C c ℓP = SIEVE_ON C c ℓP .ob
3 changes: 3 additions & 0 deletions Cubical/Categories/NaturalTransformation/Properties.agda
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Expand Up @@ -16,6 +16,7 @@ open import Cubical.Categories.Functor.Properties
open import Cubical.Categories.Morphism
open import Cubical.Categories.Isomorphism
open import Cubical.Categories.NaturalTransformation.Base
open import Cubical.Categories.Thin

private
variable
Expand Down Expand Up @@ -193,3 +194,5 @@ inv congNatIso^opFiso = _
rightInv congNatIso^opFiso _ = refl
leftInv congNatIso^opFiso _ = refl

isThin→isPropNatTrans : ∀ {F G : Functor C D} → isThin D → isProp (NatTrans F G)
isThin→isPropNatTrans thinD μ ν = makeNatTransPath (isPropΠ (λ x → thinD _ _) _ _)
12 changes: 12 additions & 0 deletions Cubical/Categories/Thin.agda
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{-# OPTIONS --safe #-}
module Cubical.Categories.Thin where

open import Cubical.Foundations.Prelude
open import Cubical.Categories.Category
open Category

private
variable ℓ ℓ' : Level

isThin : (C : Category ℓ ℓ') → Type (ℓ-max ℓ ℓ')
isThin C = (x y : C .ob) → isProp (C [ x , y ])
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