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Add notion of pre-category

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Frederik Hanghøj Iversen 2018-04-05 14:37:25 +02:00
parent 8276deb4aa
commit 6c5b68a8ac
8 changed files with 272 additions and 211 deletions
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12
src/Cat/Categories/Cat.agda
View file

@ -72,16 +72,20 @@ module CatProduct {ℓ ℓ' : Level} (ℂ 𝔻 : Category ℓ ℓ') where
= Σ≡ (fst ℂ.isIdentity) (fst 𝔻.isIdentity)
, Σ≡ (snd ℂ.isIdentity) (snd 𝔻.isIdentity)
isPreCategory : IsPreCategory rawProduct
IsPreCategory.isAssociative isPreCategory = Σ≡ ℂ.isAssociative 𝔻.isAssociative
IsPreCategory.isIdentity isPreCategory = isIdentity
IsPreCategory.arrowsAreSets isPreCategory = arrowsAreSets
postulate univalent : Univalence.Univalent isIdentity
instance
isCategory : IsCategory rawProduct
IsCategory.isAssociative isCategory = Σ≡ ℂ.isAssociative 𝔻.isAssociative
IsCategory.isIdentity isCategory = isIdentity
IsCategory.arrowsAreSets isCategory = arrowsAreSets
IsCategory.isPreCategory isCategory = isPreCategory
IsCategory.univalent isCategory = univalent
object : Category ℓ ℓ'
Category.raw object = rawProduct
Category.isCategory object = isCategory
fstF : Functor object ℂ
fstF = record

26
src/Cat/Categories/Fam.agda
View file

@ -35,15 +35,15 @@ module _ (ℓa ℓb : Level) where
open import Cubical.NType.Properties
open import Cubical.Sigma
instance
isCategory : IsCategory RawFam
isCategory = record
{ isAssociative = λ {A} {B} {C} {D} {f} {g} {h} → isAssociative {A} {B} {C} {D} {f} {g} {h}
; isIdentity = λ {A} {B} {f} → isIdentity {A} {B} {f = f}
; arrowsAreSets = λ {
{((A , hA) , famA)}
{((B , hB) , famB)}
→ setSig
isPreCategory : IsPreCategory RawFam
IsPreCategory.isAssociative isPreCategory
{A} {B} {C} {D} {f} {g} {h} = isAssociative {A} {B} {C} {D} {f} {g} {h}
IsPreCategory.isIdentity isPreCategory
{A} {B} {f} = isIdentity {A} {B} {f = f}
IsPreCategory.arrowsAreSets isPreCategory
{(A , hA) , famA} {(B , hB) , famB}
= setSig
{sA = setPi λ _ → hB}
{sB = λ f →
let
@ -55,9 +55,11 @@ module _ (ℓa ℓb : Level) where
res = {!!}
in res
}
}
; univalent = {!!}
}
isCategory : IsCategory RawFam
IsCategory.isPreCategory isCategory = isPreCategory
IsCategory.univalent isCategory = {!!}
Fam : Category (lsuc (ℓa ⊔ ℓb)) (ℓa ⊔ ℓb)
Category.raw Fam = RawFam
Category.isCategory Fam = isCategory

10
src/Cat/Categories/Free.agda
View file

@ -61,6 +61,12 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
arrowsAreSets : isSet (Path ℂ.Arrow A B)
arrowsAreSets a b p q = {!!}
isPreCategory : IsPreCategory RawFree
IsPreCategory.isAssociative isPreCategory {f = f} {g} {h} = isAssociative {r = f} {g} {h}
IsPreCategory.isIdentity isPreCategory = isIdentity
IsPreCategory.arrowsAreSets isPreCategory = arrowsAreSets
module _ {A B : ℂ.Object} where
eqv : isEquiv (A ≡ B) (A ≅ B) (Univalence.id-to-iso isIdentity A B)
eqv = {!!}
@ -68,9 +74,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
univalent = eqv
isCategory : IsCategory RawFree
IsCategory.isAssociative isCategory {f = f} {g} {h} = isAssociative {r = f} {g} {h}
IsCategory.isIdentity isCategory = isIdentity
IsCategory.arrowsAreSets isCategory = arrowsAreSets
IsCategory.isPreCategory isCategory = isPreCategory
IsCategory.univalent isCategory = univalent
Free : Category _ _

68
src/Cat/Categories/Fun.agda
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@ -10,11 +10,11 @@ import Cat.Category.NaturalTransformation
module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : Category ℓd ℓd') where
open NaturalTransformation ℂ 𝔻 public hiding (module Properties)
open NaturalTransformation.Properties ℂ 𝔻
private
module ℂ = Category ℂ
module 𝔻 = Category 𝔻
module _ where
-- Functor categories. Objects are functors, arrows are natural transformations.
raw : RawCategory (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') (ℓc ⊔ ℓc' ⊔ ℓd')
RawCategory.Object raw = Functor ℂ 𝔻
@ -22,8 +22,16 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
RawCategory.identity raw {F} = identity F
RawCategory._∘_ raw {F} {G} {H} = NT[_∘_] {F} {G} {H}
module _ where
open RawCategory raw hiding (identity)
open Univalence (λ {A} {B} {f} → isIdentity {F = A} {B} {f})
open NaturalTransformation.Properties ℂ 𝔻
isPreCategory : IsPreCategory raw
IsPreCategory.isAssociative isPreCategory {A} {B} {C} {D} = isAssociative {A} {B} {C} {D}
IsPreCategory.isIdentity isPreCategory {A} {B} = isIdentity {A} {B}
IsPreCategory.arrowsAreSets isPreCategory {F} {G} = naturalTransformationIsSet {F} {G}
open IsPreCategory isPreCategory hiding (identity)
module _ (F : Functor ℂ 𝔻) where
center : Σ[ G ∈ Object ] (F ≅ G)
@ -167,17 +175,15 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
re-ve : (x : A ≡ B) → reverse (obverse x) ≡ x
re-ve = {!!}
done : isEquiv (A ≡ B) (A ≅ B) (Univalence.id-to-iso (λ { {A} {B} → isIdentity {F = A} {B}}) A B)
done : isEquiv (A ≡ B) (A ≅ B) (id-to-iso A B)
done = {!gradLemma obverse reverse ve-re re-ve!}
-- univalent : Univalent
-- univalent = done
univalent : Univalent
univalent = {!done!}
isCategory : IsCategory raw
IsCategory.isAssociative isCategory {A} {B} {C} {D} = isAssociative {A} {B} {C} {D}
IsCategory.isIdentity isCategory {A} {B} = isIdentity {A} {B}
IsCategory.arrowsAreSets isCategory {F} {G} = naturalTransformationIsSet {F} {G}
IsCategory.univalent isCategory = {!!}
IsCategory.isPreCategory isCategory = isPreCategory
IsCategory.univalent isCategory = univalent
Fun : Category (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') (ℓc ⊔ ℓc' ⊔ ℓd')
Category.raw Fun = raw
@ -199,26 +205,26 @@ module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
; _∘_ = λ {F G H} → NT[_∘_] {F = F} {G = G} {H = H}
}
isCategory : IsCategory raw
isCategory = record
{ isAssociative =
λ{ {A} {B} {C} {D} {f} {g} {h}
→ F.isAssociative {A} {B} {C} {D} {f} {g} {h}
}
; isIdentity =
λ{ {A} {B} {f}
→ F.isIdentity {A} {B} {f}
}
; arrowsAreSets =
λ{ {A} {B}
→ F.arrowsAreSets {A} {B}
}
; univalent =
λ{ {A} {B}
→ F.univalent {A} {B}
}
}
-- isCategory : IsCategory raw
-- isCategory = record
-- { isAssociative =
-- λ{ {A} {B} {C} {D} {f} {g} {h}
-- → F.isAssociative {A} {B} {C} {D} {f} {g} {h}
-- }
-- ; isIdentity =
-- λ{ {A} {B} {f}
-- → F.isIdentity {A} {B} {f}
-- }
-- ; arrowsAreSets =
-- λ{ {A} {B}
-- → F.arrowsAreSets {A} {B}
-- }
-- ; univalent =
-- λ{ {A} {B}
-- → F.univalent {A} {B}
-- }
-- }
Presh : Category (ℓ ⊔ lsuc ℓ') (ℓ ⊔ ℓ')
Category.raw Presh = raw
Category.isCategory Presh = isCategory
-- Presh : Category (ℓ ⊔ lsuc ℓ') (ℓ ⊔ ℓ')
-- Category.raw Presh = raw
-- Category.isCategory Presh = isCategory

16
src/Cat/Categories/Rel.agda
View file

@ -157,10 +157,12 @@ RawRel = record
; _∘_ = λ {A B C} S R → λ {( a , c ) → Σ[ b ∈ B ] ( (a , b) ∈ R × (b , c) ∈ S )}
}
RawIsCategoryRel : IsCategory RawRel
RawIsCategoryRel = record
{ isAssociative = funExt is-isAssociative
; isIdentity = funExt ident-l , funExt ident-r
; arrowsAreSets = {!!}
; univalent = {!!}
}
isPreCategory : IsPreCategory RawRel
IsPreCategory.isAssociative isPreCategory = funExt is-isAssociative
IsPreCategory.isIdentity isPreCategory = funExt ident-l , funExt ident-r
IsPreCategory.arrowsAreSets isPreCategory = {!!}
Rel : PreCategory _ _
PreCategory.raw Rel = RawRel
PreCategory.isPreCategory Rel = isPreCategory

15
src/Cat/Categories/Sets.agda
View file

@ -34,17 +34,24 @@ module _ (ℓ : Level) where
RawCategory.identity SetsRaw = Function.id
RawCategory._∘_ SetsRaw = Function._∘′_
module _ where
private
open RawCategory SetsRaw hiding (_∘_)
isIdentity : IsIdentity Function.id
fst isIdentity = funExt λ _ → refl
snd isIdentity = funExt λ _ → refl
open Univalence (λ {A} {B} {f} → isIdentity {A} {B} {f})
arrowsAreSets : ArrowsAreSets
arrowsAreSets {B = (_ , s)} = setPi λ _ → s
isPreCat : IsPreCategory SetsRaw
IsPreCategory.isAssociative isPreCat = refl
IsPreCategory.isIdentity isPreCat {A} {B} = isIdentity {A} {B}
IsPreCategory.arrowsAreSets isPreCat {A} {B} = arrowsAreSets {A} {B}
open IsPreCategory isPreCat hiding (_∘_)
isIso = Eqv.Isomorphism
module _ {hA hB : hSet ℓ} where
open Σ hA renaming (fst to A ; snd to sA)
@ -255,9 +262,7 @@ module _ (ℓ : Level) where
univalent = from[Contr] univalent[Contr]
SetsIsCategory : IsCategory SetsRaw
IsCategory.isAssociative SetsIsCategory = refl
IsCategory.isIdentity SetsIsCategory {A} {B} = isIdentity {A} {B}
IsCategory.arrowsAreSets SetsIsCategory {A} {B} = arrowsAreSets {A} {B}
IsCategory.isPreCategory SetsIsCategory = isPreCat
IsCategory.univalent SetsIsCategory = univalent
𝓢𝓮𝓽 Sets : Category (lsuc ℓ) ℓ

116
src/Cat/Category.agda
View file

@ -203,15 +203,13 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
--
-- Sans `univalent` this would be what is referred to as a pre-category in
-- [HoTT].
record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc (ℓa ⊔ ℓb)) where
record IsPreCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc (ℓa ⊔ ℓb)) where
open RawCategory ℂ public
field
isAssociative : IsAssociative
isIdentity : IsIdentity identity
arrowsAreSets : ArrowsAreSets
open Univalence isIdentity public
field
univalent : Univalent
leftIdentity : {A B : Object} {f : Arrow A B} → identity ∘ f ≡ f
leftIdentity {A} {B} {f} = fst (isIdentity {A = A} {B} {f})
@ -251,19 +249,6 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
iso→epi×mono : Isomorphism f → Epimorphism {X = X} f × Monomorphism {X = X} f
iso→epi×mono iso = iso→epi iso , iso→mono iso
-- | The formulation of univalence expressed with _≃_ is trivially admissable -
-- just "forget" the equivalence.
univalent≃ : Univalent≃
univalent≃ = _ , univalent
module _ {A B : Object} where
open import Cat.Equivalence using (module Equiv≃)
iso-to-id : (A ≅ B) → (A ≡ B)
iso-to-id = fst (Equiv≃.toIso _ _ univalent)
-- | All projections are propositions.
module Propositionality where
propIsAssociative : isProp IsAssociative
propIsAssociative x y i = arrowsAreSets _ _ x y i
@ -298,8 +283,20 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
identity ∘ g' ≡⟨ leftIdentity ⟩
g' ∎
propUnivalent : isProp Univalent
propUnivalent a b i = propPi (λ iso → propIsContr) a b i
propIsInitial : ∀ I → isProp (IsInitial I)
propIsInitial I x y i {X} = res X i
where
module _ (X : Object) where
open Σ (x {X}) renaming (fst to fx ; snd to cx)
open Σ (y {X}) renaming (fst to fy ; snd to cy)
fp : fx ≡ fy
fp = cx fy
prop : (x : Arrow I X) → isProp (∀ f → x ≡ f)
prop x = propPi (λ y → arrowsAreSets x y)
cp : (λ i → ∀ f → fp i ≡ f) [ cx ≡ cy ]
cp = lemPropF prop fp
res : (fx , cx) ≡ (fy , cy)
res i = fp i , cp i
propIsTerminal : ∀ T → isProp (IsTerminal T)
propIsTerminal T x y i {X} = res X i
@ -316,6 +313,35 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
res : (fx , cx) ≡ (fy , cy)
res i = fp i , cp i
record PreCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
field
raw : RawCategory ℓa ℓb
isPreCategory : IsPreCategory raw
open IsPreCategory isPreCategory public
record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc (ℓa ⊔ ℓb)) where
field
isPreCategory : IsPreCategory ℂ
open IsPreCategory isPreCategory public
field
univalent : Univalent
-- | The formulation of univalence expressed with _≃_ is trivially admissable -
-- just "forget" the equivalence.
univalent≃ : Univalent≃
univalent≃ = _ , univalent
module _ {A B : Object} where
open import Cat.Equivalence using (module Equiv≃)
iso-to-id : (A ≅ B) → (A ≡ B)
iso-to-id = fst (Equiv≃.toIso _ _ univalent)
-- | All projections are propositions.
module Propositionality where
propUnivalent : isProp Univalent
propUnivalent a b i = propPi (λ iso → propIsContr) a b i
-- | Terminal objects are propositional - a.k.a uniqueness of terminal
-- | objects.
--
@ -353,20 +379,6 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
res i = p0 i , p1 i
-- Merely the dual of the above statement.
propIsInitial : ∀ I → isProp (IsInitial I)
propIsInitial I x y i {X} = res X i
where
module _ (X : Object) where
open Σ (x {X}) renaming (fst to fx ; snd to cx)
open Σ (y {X}) renaming (fst to fy ; snd to cy)
fp : fx ≡ fy
fp = cx fy
prop : (x : Arrow I X) → isProp (∀ f → x ≡ f)
prop x = propPi (λ y → arrowsAreSets x y)
cp : (λ i → ∀ f → fp i ≡ f) [ cx ≡ cy ]
cp = lemPropF prop fp
res : (fx , cx) ≡ (fy , cy)
res i = fp i , cp i
propInitial : isProp Initial
propInitial Xi Yi = res
@ -404,6 +416,23 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
open RawCategory ℂ
open Univalence
private
module _ (x y : IsPreCategory ℂ) where
module x = IsPreCategory x
module y = IsPreCategory y
-- In a few places I use the result of propositionality of the various
-- projections of `IsCategory` - Here I arbitrarily chose to use this
-- result from `x : IsCategory C`. I don't know which (if any) possibly
-- adverse effects this may have.
-- module Prop = X.Propositionality
propIsPreCategory : x ≡ y
IsPreCategory.isAssociative (propIsPreCategory i)
= x.propIsAssociative x.isAssociative y.isAssociative i
IsPreCategory.isIdentity (propIsPreCategory i)
= x.propIsIdentity x.isIdentity y.isIdentity i
IsPreCategory.arrowsAreSets (propIsPreCategory i)
= x.propArrowIsSet x.arrowsAreSets y.arrowsAreSets i
module _ (x y : IsCategory ℂ) where
module X = IsCategory x
module Y = IsCategory y
@ -414,7 +443,7 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
module Prop = X.Propositionality
isIdentity : (λ _ → IsIdentity identity) [ X.isIdentity ≡ Y.isIdentity ]
isIdentity = Prop.propIsIdentity X.isIdentity Y.isIdentity
isIdentity = X.propIsIdentity X.isIdentity Y.isIdentity
U : ∀ {a : IsIdentity identity}
→ (λ _ → IsIdentity identity) [ X.isIdentity ≡ a ]
@ -434,10 +463,13 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
eqUni : U isIdentity Y.univalent
eqUni = helper Y.univalent
done : x ≡ y
IsCategory.isAssociative (done i) = Prop.propIsAssociative X.isAssociative Y.isAssociative i
IsCategory.isIdentity (done i) = isIdentity i
IsCategory.arrowsAreSets (done i) = Prop.propArrowIsSet X.arrowsAreSets Y.arrowsAreSets i
IsCategory.isPreCategory (done i)
= propIsPreCategory X.isPreCategory Y.isPreCategory i
IsCategory.univalent (done i) = eqUni i
-- IsCategory.isAssociative (done i) = Prop.propIsAssociative X.isAssociative Y.isAssociative i
-- IsCategory.isIdentity (done i) = isIdentity i
-- IsCategory.arrowsAreSets (done i) = Prop.propArrowIsSet X.arrowsAreSets Y.arrowsAreSets i
-- IsCategory.univalent (done i) = eqUni i
propIsCategory : isProp (IsCategory ℂ)
propIsCategory = done
@ -486,6 +518,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
module Opposite {ℓa ℓb : Level} where
module _ (ℂ : Category ℓa ℓb) where
private
module _ where
module ℂ = Category ℂ
opRaw : RawCategory ℓa ℓb
RawCategory.Object opRaw = ℂ.Object
@ -498,7 +531,12 @@ module Opposite {ℓa ℓb : Level} where
isIdentity : IsIdentity identity
isIdentity = swap ℂ.isIdentity
open Univalence isIdentity
isPreCategory : IsPreCategory opRaw
IsPreCategory.isAssociative isPreCategory = sym ℂ.isAssociative
IsPreCategory.isIdentity isPreCategory = isIdentity
IsPreCategory.arrowsAreSets isPreCategory = ℂ.arrowsAreSets
open IsPreCategory isPreCategory
module _ {A B : ℂ.Object} where
open import Cat.Equivalence as Equivalence hiding (_≅_)
@ -571,9 +609,7 @@ module Opposite {ℓa ℓb : Level} where
univalent = Equiv≃.fromIso _ _ h
isCategory : IsCategory opRaw
IsCategory.isAssociative isCategory = sym ℂ.isAssociative
IsCategory.isIdentity isCategory = isIdentity
IsCategory.arrowsAreSets isCategory = ℂ.arrowsAreSets
IsCategory.isPreCategory isCategory = isPreCategory
IsCategory.univalent isCategory = univalent
opposite : Category ℓa ℓb

20
src/Cat/Category/Product.agda
View file

@ -122,14 +122,15 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
}
}
module _ where
open RawCategory raw
propEqs : ∀ {X' : Object}{Y' : Object} (let X , xa , xb = X') (let Y , ya , yb = Y')
→ (xy : ℂ.Arrow X Y) → isProp (ℂ [ ya ∘ xy ] ≡ xa × ℂ [ yb ∘ xy ] ≡ xb)
propEqs xs = propSig (ℂ.arrowsAreSets _ _) (\ _ → ℂ.arrowsAreSets _ _)
isAssocitaive : IsAssociative
isAssocitaive {A'@(A , a0 , a1)} {B , _} {C , c0 , c1} {D'@(D , d0 , d1)} {ff@(f , f0 , f1)} {gg@(g , g0 , g1)} {hh@(h , h0 , h1)} i
isAssociative : IsAssociative
isAssociative {A'@(A , a0 , a1)} {B , _} {C , c0 , c1} {D'@(D , d0 , d1)} {ff@(f , f0 , f1)} {gg@(g , g0 , g1)} {hh@(h , h0 , h1)} i
= s0 i , lemPropF propEqs s0 {P.snd l} {P.snd r} i
where
l = hh ∘ (gg ∘ ff)
@ -163,7 +164,12 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
arrowsAreSets {X , x0 , x1} {Y , y0 , y1}
= sigPresNType {n = ⟨0⟩} ℂ.arrowsAreSets λ a → propSet (propEqs _)
open Univalence isIdentity
isPreCat : IsPreCategory raw
IsPreCategory.isAssociative isPreCat = isAssociative
IsPreCategory.isIdentity isPreCat = isIdentity
IsPreCategory.arrowsAreSets isPreCat = arrowsAreSets
open IsPreCategory isPreCat
-- module _ (X : Object) where
-- center : Σ Object (X ≅_)
@ -327,12 +333,8 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
res = Equiv≃.fromIso _ _ iso
isCat : IsCategory raw
isCat = record
{ isAssociative = isAssocitaive
; isIdentity = isIdentity
; arrowsAreSets = arrowsAreSets
; univalent = univalent
}
IsCategory.isPreCategory isCat = isPreCat
IsCategory.univalent isCat = univalent
cat : Category _ _
cat = record