Move properties of categories to Cat.Category.Properties

This commit is contained in:
Frederik Hanghøj Iversen 2018-01-21 15:01:01 +01:00
parent 316de7e4f9
commit 793fc30534
2 changed files with 63 additions and 71 deletions

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@ -53,68 +53,26 @@ record Category ( ' : Level) : Set (lsuc (' ⊔ )) where
open Category open Category
module _ { ' : Level} { : Category '} { A B : .Object } where module _ { ' : Level} { : Category '} where
private module _ { A B : .Object } where
open module = Category
_+_ = ._⊕_
Isomorphism : (f : .Arrow A B) Set ' Isomorphism : (f : .Arrow A B) Set '
Isomorphism f = Σ[ g .Arrow B A ] g .⊕ f .𝟙 × f + g .𝟙 Isomorphism f = Σ[ g .Arrow B A ] ._⊕_ g f .𝟙 × ._⊕_ f g .𝟙
Epimorphism : {X : .Object } (f : .Arrow A B) Set ' Epimorphism : {X : .Object } (f : .Arrow A B) Set '
Epimorphism {X} f = ( g₀ g₁ : .Arrow B X ) g₀ + f g₁ + f g₀ g₁ Epimorphism {X} f = ( g₀ g₁ : .Arrow B X ) ._⊕_ g₀ f ._⊕_ g₁ f g₀ g₁
Monomorphism : {X : .Object} (f : .Arrow A B) Set ' Monomorphism : {X : .Object} (f : .Arrow A B) Set '
Monomorphism {X} f = ( g₀ g₁ : .Arrow X A ) f + g₀ f + g₁ g₀ g₁ Monomorphism {X} f = ( g₀ g₁ : .Arrow X A ) ._⊕_ f g₀ ._⊕_ f g₁ g₀ g₁
iso-is-epi : {X} (f : .Arrow A B) Isomorphism f Epimorphism {X = X} f
iso-is-epi f (f- , left-inv , right-inv) g₀ g₁ eq =
begin
g₀ ≡⟨ sym (fst ident)
g₀ + .𝟙 ≡⟨ cong (_+_ g₀) (sym right-inv)
g₀ + (f + f-) ≡⟨ assoc
(g₀ + f) + f- ≡⟨ cong (λ x x + f-) eq
(g₁ + f) + f- ≡⟨ sym assoc
g₁ + (f + f-) ≡⟨ cong (_+_ g₁) right-inv
g₁ + .𝟙 ≡⟨ fst ident
g₁
where
open IsCategory .isCategory
iso-is-mono : {X} (f : .Arrow A B ) Isomorphism f Monomorphism {X = X} f
iso-is-mono f (f- , (left-inv , right-inv)) g₀ g₁ eq =
begin
g₀ ≡⟨ sym (snd ident)
.𝟙 + g₀ ≡⟨ cong (λ x x + g₀) (sym left-inv)
(f- + f) + g₀ ≡⟨ sym assoc
f- + (f + g₀) ≡⟨ cong (_+_ f-) eq
f- + (f + g₁) ≡⟨ assoc
(f- + f) + g₁ ≡⟨ cong (λ x x + g₁) left-inv
.𝟙 + g₁ ≡⟨ snd ident
g₁
where
open IsCategory .isCategory
iso-is-epi-mono : {X} (f : .Arrow A B ) Isomorphism f Epimorphism {X = X} f × Monomorphism {X = X} f
iso-is-epi-mono f iso = iso-is-epi f iso , iso-is-mono f iso
{-
epi-mono-is-not-iso : { '} ¬ (( : Category {} {'}) {A B X : Object } (f : Arrow A B ) Epimorphism { = } {X = X} f Monomorphism { = } {X = X} f Isomorphism { = } f)
epi-mono-is-not-iso f =
let k = f {!!} {!!} {!!} {!!}
in {!!}
-}
-- Isomorphism of objects -- Isomorphism of objects
_≅_ : { '} { : Category '} (A B : Object ) Set ' _≅_ : (A B : Object ) Set '
_≅_ { = } A B = Σ[ f .Arrow A B ] (Isomorphism { = } f) _≅_ A B = Σ[ f .Arrow A B ] (Isomorphism f)
IsProduct : { '} ( : Category ') {A B obj : Object } (π₁ : Arrow obj A) (π₂ : Arrow obj B) Set ( ') module _ { ' : Level} ( : Category ') {A B obj : Object } where
IsProduct {A = A} {B = B} π₁ π₂ IsProduct : (π₁ : Arrow obj A) (π₂ : Arrow obj B) Set ( ')
IsProduct π₁ π₂
= {X : .Object} (x₁ : .Arrow X A) (x₂ : .Arrow X B) = {X : .Object} (x₁ : .Arrow X A) (x₂ : .Arrow X B)
∃![ x ] (π₁ .⊕ x x₁ × π₂ .⊕ x x₂) ∃![ x ] ( ._⊕_ π₁ x x₁ × ._⊕_ π₂ x x₂)
where
open module = Category
-- Tip from Andrea; Consider this style for efficiency: -- Tip from Andrea; Consider this style for efficiency:
-- record IsProduct { ' : Level} ( : Category {} {'}) -- record IsProduct { ' : Level} ( : Category {} {'})
@ -131,19 +89,7 @@ record Product { ' : Level} { : Category '} (A B : .Object)
proj₂ : .Arrow obj B proj₂ : .Arrow obj B
{{isProduct}} : IsProduct proj₁ proj₂ {{isProduct}} : IsProduct proj₁ proj₂
-- Two pairs are equal if their components are equal.
eqpair : {a b} {A : Set a} {B : Set b} {a a' : A} {b b' : B}
a a' b b' (a , b) (a' , b')
eqpair eqa eqb i = eqa i , eqb i
module _ { ' : Level} ( : Category ') where module _ { ' : Level} ( : Category ') where
private
instance
_ : IsCategory ( .Object) (flip ( .Arrow)) ( .𝟙) (flip ( ._⊕_))
_ = record { assoc = sym assoc ; ident = swap ident }
where
open IsCategory ( .isCategory)
Opposite : Category ' Opposite : Category '
Opposite = Opposite =
record record
@ -151,7 +97,10 @@ module _ { ' : Level} ( : Category ') where
; Arrow = flip ( .Arrow) ; Arrow = flip ( .Arrow)
; 𝟙 = .𝟙 ; 𝟙 = .𝟙
; _⊕_ = flip ( ._⊕_) ; _⊕_ = flip ( ._⊕_)
; isCategory = record { assoc = sym assoc ; ident = swap ident }
} }
where
open IsCategory ( .isCategory)
-- A consequence of no-eta-equality; `Opposite-is-involution` is no longer -- A consequence of no-eta-equality; `Opposite-is-involution` is no longer
-- definitional - i.e.; you must match on the fields: -- definitional - i.e.; you must match on the fields:

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@ -2,10 +2,53 @@
module Cat.Category.Properties where module Cat.Category.Properties where
open import Agda.Primitive
open import Data.Product
open import Cubical.PathPrelude
open import Cat.Category open import Cat.Category
open import Cat.Functor open import Cat.Functor
open import Cat.Categories.Sets open import Cat.Categories.Sets
module _ { ' : Level} { : Category '} { A B : .Category.Object } {X : .Category.Object} (f : .Category.Arrow A B) where
open Category
open IsCategory (isCategory)
iso-is-epi : Isomorphism { = } f Epimorphism { = } {X = X} f
iso-is-epi (f- , left-inv , right-inv) g₀ g₁ eq =
begin
g₀ ≡⟨ sym (proj₁ ident)
g₀ 𝟙 ≡⟨ cong (_⊕_ g₀) (sym right-inv)
g₀ (f f-) ≡⟨ assoc
(g₀ f) f- ≡⟨ cong (λ φ φ f-) eq
(g₁ f) f- ≡⟨ sym assoc
g₁ (f f-) ≡⟨ cong (_⊕_ g₁) right-inv
g₁ 𝟙 ≡⟨ proj₁ ident
g₁
iso-is-mono : Isomorphism { = } f Monomorphism { = } {X = X} f
iso-is-mono (f- , (left-inv , right-inv)) g₀ g₁ eq =
begin
g₀ ≡⟨ sym (proj₂ ident)
𝟙 g₀ ≡⟨ cong (λ φ φ g₀) (sym left-inv)
(f- f) g₀ ≡⟨ sym assoc
f- (f g₀) ≡⟨ cong (_⊕_ f-) eq
f- (f g₁) ≡⟨ assoc
(f- f) g₁ ≡⟨ cong (λ φ φ g₁) left-inv
𝟙 g₁ ≡⟨ proj₂ ident
g₁
iso-is-epi-mono : Isomorphism { = } f Epimorphism { = } {X = X} f × Monomorphism { = } {X = X} f
iso-is-epi-mono iso = iso-is-epi iso , iso-is-mono iso
{-
epi-mono-is-not-iso : { '} ¬ (( : Category {} {'}) {A B X : Object } (f : Arrow A B ) Epimorphism { = } {X = X} f Monomorphism { = } {X = X} f Isomorphism { = } f)
epi-mono-is-not-iso f =
let k = f {!!} {!!} {!!} {!!}
in {!!}
-}
module _ {a a' b b'} where module _ {a a' b b'} where
Exponential : Category a a' Category b b' Category {!!} {!!} Exponential : Category a a' Category b b' Category {!!} {!!}
Exponential A B = record Exponential A B = record
@ -13,7 +56,7 @@ module _ {a a' b b'} where
; Arrow = {!!} ; Arrow = {!!}
; 𝟙 = {!!} ; 𝟙 = {!!}
; _⊕_ = {!!} ; _⊕_ = {!!}
; isCategory = ? ; isCategory = {!!}
} }
_⇑_ = Exponential _⇑_ = Exponential