Move product, exponential and cart closed to own file

This commit is contained in:
Frederik Hanghøj Iversen 2018-02-05 14:08:30 +01:00
parent 8022ed349d
commit 20dc9d26ac
6 changed files with 107 additions and 77 deletions

View file

@ -3,10 +3,15 @@ module Cat where
import Cat.Category
import Cat.Functor
import Cat.CwF
import Cat.CartesianClosed
import Cat.Exponential
import Cat.Product
import Cat.Category.Pathy
import Cat.Category.Bij
import Cat.Category.Free
import Cat.Category.Properties
import Cat.Categories.Sets
-- import Cat.Categories.Cat
import Cat.Categories.Rel

View file

@ -0,0 +1,12 @@
module Cat.CartesianClosed where
open import Agda.Primitive
open import Cat.Category
open import Cat.Product
open import Cat.Exponential
record CartesianClosed { ' : Level} ( : Category ') : Set ( ') where
field
{{hasProducts}} : HasProducts
{{hasExponentials}} : HasExponentials

View file

@ -8,6 +8,7 @@ import Function
open import Cat.Category
open import Cat.Functor
open import Cat.Product
open Category
module _ { : Level} where

View file

@ -136,49 +136,6 @@ module Category {a b : Level} ( : Category a b) where
open Category using ( Object ; _[_,_] ; _[_∘_])
-- open RawCategory
module _ { ' : Level} ( : Category ') {A B obj : Object } where
IsProduct : (π₁ : [ obj , A ]) (π₂ : [ obj , B ]) Set ( ')
IsProduct π₁ π₂
= {X : Object } (x₁ : [ X , A ]) (x₂ : [ X , B ])
∃![ x ] ( [ π₁ x ] x₁ × [ π₂ x ] x₂)
-- Tip from Andrea; Consider this style for efficiency:
-- record IsProduct { ' : Level} ( : Category {} {'})
-- {A B obj : Object } (π₁ : Arrow obj A) (π₂ : Arrow obj B) : Set (') where
-- field
-- isProduct : ∀ {X : .Object} (x₁ : .Arrow X A) (x₂ : .Arrow X B)
-- → ∃![ x ] ( ._⊕_ π₁ x ≡ x₁ × . _⊕_ π₂ x ≡ x₂)
record Product { ' : Level} { : Category '} (A B : Object ) : Set ( ') where
no-eta-equality
field
obj : Object
proj₁ : [ obj , A ]
proj₂ : [ obj , B ]
{{isProduct}} : IsProduct proj₁ proj₂
arrowProduct : {X} (π₁ : [ X , A ]) (π₂ : [ X , B ])
[ X , obj ]
arrowProduct π₁ π₂ = fst (isProduct π₁ π₂)
record HasProducts { ' : Level} ( : Category ') : Set ( ') where
field
product : (A B : Object ) Product { = } A B
open Product
objectProduct : (A B : Object ) Object
objectProduct A B = Product.obj (product A B)
-- The product mentioned in awodey in Def 6.1 is not the regular product of arrows.
-- It's a "parallel" product
parallelProduct : {A A' B B' : Object } [ A , A' ] [ B , B' ]
[ objectProduct A B , objectProduct A' B' ]
parallelProduct {A = A} {A' = A'} {B = B} {B' = B'} a b = arrowProduct (product A' B')
( [ a (product A B) .proj₁ ])
( [ b (product A B) .proj₂ ])
module _ {a b : Level} ( : Category a b) where
private
open Category
@ -212,40 +169,6 @@ module _ {a b : Level} ( : Category a b) where
-- assoc (Opposite-is-involution i) = {!!}
-- ident (Opposite-is-involution i) = {!!}
module _ { '} ( : Category ') {{hasProducts : HasProducts }} where
open HasProducts hasProducts
open Product hiding (obj)
private
_×p_ : (A B : Object ) Object
_×p_ A B = Product.obj (product A B)
module _ (B C : Object ) where
IsExponential : (Cᴮ : Object ) [ Cᴮ ×p B , C ] Set ( ')
IsExponential Cᴮ eval = (A : Object ) (f : [ A ×p B , C ])
∃![ f~ ] ( [ eval parallelProduct f~ (Category.𝟙 )] f)
record Exponential : Set ( ') where
field
-- obj ≡ Cᴮ
obj : Object
eval : [ obj ×p B , C ]
{{isExponential}} : IsExponential obj eval
-- If I make this an instance-argument then the instance resolution
-- algorithm goes into an infinite loop. Why?
exponentialsHaveProducts : HasProducts
exponentialsHaveProducts = hasProducts
transpose : (A : Object ) [ A ×p B , C ] [ A , obj ]
transpose A f = fst (isExponential A f)
record HasExponentials { ' : Level} ( : Category ') {{_ : HasProducts }} : Set ( ') where
field
exponent : (A B : Object ) Exponential A B
record CartesianClosed { ' : Level} ( : Category ') : Set ( ') where
field
{{hasProducts}} : HasProducts
{{hasExponentials}} : HasExponentials
module _ {a b : Level} ( : Category a b) where
unique = isContr

39
src/Cat/Exponential.agda Normal file
View file

@ -0,0 +1,39 @@
module Cat.Exponential where
open import Agda.Primitive
open import Data.Product
open import Cubical
open import Cat.Category
open import Cat.Product
open Category
module _ { '} ( : Category ') {{hasProducts : HasProducts }} where
open HasProducts hasProducts
open Product hiding (obj)
private
_×p_ : (A B : Object ) Object
_×p_ A B = Product.obj (product A B)
module _ (B C : Object ) where
IsExponential : (Cᴮ : Object ) [ Cᴮ ×p B , C ] Set ( ')
IsExponential Cᴮ eval = (A : Object ) (f : [ A ×p B , C ])
∃![ f~ ] ( [ eval parallelProduct f~ (Category.𝟙 )] f)
record Exponential : Set ( ') where
field
-- obj ≡ Cᴮ
obj : Object
eval : [ obj ×p B , C ]
{{isExponential}} : IsExponential obj eval
-- If I make this an instance-argument then the instance resolution
-- algorithm goes into an infinite loop. Why?
exponentialsHaveProducts : HasProducts
exponentialsHaveProducts = hasProducts
transpose : (A : Object ) [ A ×p B , C ] [ A , obj ]
transpose A f = proj₁ (isExponential A f)
record HasExponentials { ' : Level} ( : Category ') {{_ : HasProducts }} : Set ( ') where
field
exponent : (A B : Object ) Exponential A B

50
src/Cat/Product.agda Normal file
View file

@ -0,0 +1,50 @@
module Cat.Product where
open import Agda.Primitive
open import Data.Product
open import Cubical
open import Cat.Category
open Category
module _ { ' : Level} ( : Category ') {A B obj : Object } where
IsProduct : (π₁ : [ obj , A ]) (π₂ : [ obj , B ]) Set ( ')
IsProduct π₁ π₂
= {X : Object } (x₁ : [ X , A ]) (x₂ : [ X , B ])
∃![ x ] ( [ π₁ x ] x₁ × [ π₂ x ] x₂)
-- Tip from Andrea; Consider this style for efficiency:
-- record IsProduct { ' : Level} ( : Category {} {'})
-- {A B obj : Object } (π₁ : Arrow obj A) (π₂ : Arrow obj B) : Set (') where
-- field
-- isProduct : ∀ {X : .Object} (x₁ : .Arrow X A) (x₂ : .Arrow X B)
-- → ∃![ x ] ( ._⊕_ π₁ x ≡ x₁ × . _⊕_ π₂ x ≡ x₂)
record Product { ' : Level} { : Category '} (A B : Object ) : Set ( ') where
no-eta-equality
field
obj : Object
proj₁ : [ obj , A ]
proj₂ : [ obj , B ]
{{isProduct}} : IsProduct proj₁ proj₂
arrowProduct : {X} (π₁ : [ X , A ]) (π₂ : [ X , B ])
[ X , obj ]
arrowProduct π₁ π₂ = proj₁ (isProduct π₁ π₂)
record HasProducts { ' : Level} ( : Category ') : Set ( ') where
field
product : (A B : Object ) Product { = } A B
open Product
objectProduct : (A B : Object ) Object
objectProduct A B = Product.obj (product A B)
-- The product mentioned in awodey in Def 6.1 is not the regular product of arrows.
-- It's a "parallel" product
parallelProduct : {A A' B B' : Object } [ A , A' ] [ B , B' ]
[ objectProduct A B , objectProduct A' B' ]
parallelProduct {A = A} {A' = A'} {B = B} {B' = B'} a b = arrowProduct (product A' B')
( [ a (product A B) .proj₁ ])
( [ b (product A B) .proj₂ ])