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Universal Objects are Isomorphic #80

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Universal Objects are Isomorphic #80

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a636f39
Terminal Objects are Isomorphic
Jake-Gillberg Apr 5, 2020
9dde59a
Update TerminalObject.lidr
Jake-Gillberg Apr 5, 2020
14cb833
CoProducts are Isomorphic
Jake-Gillberg Apr 16, 2020
1543be7
proof of isomorphism using dual category
Jake-Gillberg Apr 17, 2020
b5a2610
add missing import
Jake-Gillberg Apr 17, 2020
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102 changes: 88 additions & 14 deletions src/CoLimits/CoProduct.lidr
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Expand Up @@ -22,38 +22,112 @@ along with this program. If not, see <https://www.gnu.org/licenses/>.
> module CoLimits.CoProduct
>
> import Basic.Category
> import Basic.Isomorphism
>
> %access public export
> %default total
> %auto_implicits off
>
> record CommutingMorphism
> (cat : Category)
> (a : obj cat) (b : obj cat) (carrier : obj cat) (c : obj cat)
> (inl : mor cat a carrier) (inr : mor cat b carrier)
> (f : mor cat a c) (g : mor cat b c)
> (l : obj cat) (r : obj cat) (carrier : obj cat) (c : obj cat)
> (inl : mor cat l carrier) (inr : mor cat r carrier)
> (f : mor cat l c) (g : mor cat r c)
> where
> constructor MkCommutingMorphism
> challenger : mor cat carrier c
> commutativityLeft : compose cat a carrier c inl challenger = f
> commutativityRight : compose cat b carrier c inr challenger = g
> commutativityLeft : compose cat l carrier c inl challenger = f
> commutativityRight : compose cat r carrier c inr challenger = g
>
> record CoProduct
> (cat : Category)
> (a : obj cat) (b : obj cat)
> (l : obj cat) (r : obj cat)
> where
> constructor MkCoProduct
> carrier: obj cat
> inl: mor cat a carrier
> inr: mor cat b carrier
> inl: mor cat l carrier
> inr: mor cat r carrier
> exists:
> (c : obj cat)
> -> (f : mor cat a c)
> -> (g : mor cat b c)
> -> CommutingMorphism cat a b carrier c inl inr f g
> -> (f : mor cat l c)
> -> (g : mor cat r c)
> -> CommutingMorphism cat l r carrier c inl inr f g
> unique:
> (c : obj cat)
> -> (f : mor cat a c)
> -> (g : mor cat b c)
> -> (h : CommutingMorphism cat a b carrier c inl inr f g)
> -> (f : mor cat l c)
> -> (g : mor cat r c)
> -> (h : CommutingMorphism cat l r carrier c inl inr f g)
> -> challenger h = challenger (exists c f g)
>
> coProductMorphism :
> (cat : Category)
> -> (l, r : obj cat)
> -> (a, b : CoProduct cat l r)
> -> CommutingMorphism
> cat
> l r (carrier a) (carrier b)
> (inl a) (inr a)
> (inl b) (inr b)
> coProductMorphism cat l r a b = exists a (carrier b) (inl b) (inr b)
>
> composeCoProductMorphisms :
> (cat : Category)
> -> (l, r : obj cat)
> -> (a, b : CoProduct cat l r)
> -> CommutingMorphism
> cat
> l r (carrier a) (carrier a)
> (inl a) (inr a)
> (inl a) (inr a)
> composeCoProductMorphisms cat l r a b =
> let
> mor = coProductMorphism cat l r a b
> inv = coProductMorphism cat l r b a
> in
> MkCommutingMorphism
> (compose cat (carrier a) (carrier b) (carrier a)
> (challenger mor) (challenger inv))
> (rewrite associativity cat l (carrier a) (carrier b) (carrier a)
> (inl a) (challenger mor) (challenger inv) in
> rewrite commutativityLeft mor in
> rewrite commutativityLeft inv in Refl)
> (rewrite associativity cat r (carrier a) (carrier b) (carrier a)
> (inr a) (challenger mor) (challenger inv) in
> rewrite commutativityRight mor in
> rewrite commutativityRight inv in Refl)
>
> idCommutingMorphism :
> (cat : Category)
> -> (l, r : obj cat)
> -> (a : CoProduct cat l r)
> -> CommutingMorphism
> cat
> l r (carrier a) (carrier a)
> (inl a) (inr a)
> (inl a) (inr a)
> idCommutingMorphism cat l r a = MkCommutingMorphism
> (identity cat (carrier a))
> (rightIdentity cat l (carrier a) (inl a))
> (rightIdentity cat r (carrier a) (inr a))
>
> coProductsAreIsomorphic :
> (cat : Category)
> -> (l, r : obj cat)
> -> (a, b : CoProduct cat l r)
> -> Isomorphic cat (carrier a) (carrier b)
> coProductsAreIsomorphic cat l r a b =
> let
> mor = coProductMorphism cat l r a b
> inv = coProductMorphism cat l r b a
> in
> buildIsomorphic
> (challenger mor)
> (challenger inv)
> (rewrite unique a (carrier a) (inl a) (inr a)
> (composeCoProductMorphisms cat l r a b) in
> rewrite unique a (carrier a) (inl a) (inr a)
> (idCommutingMorphism cat l r a) in Refl)
> (rewrite unique b (carrier b) (inl b) (inr b)
> (composeCoProductMorphisms cat l r b a) in
> rewrite unique b (carrier b) (inl b) (inr b)
> (idCommutingMorphism cat l r b) in Refl)
14 changes: 14 additions & 0 deletions src/Dual/DualCategory.lidr
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Expand Up @@ -120,3 +120,17 @@ along with this program. If not, see <https://www.gnu.org/licenses/>.
> (idFunctor (dualCategory $ dualCategory cat))
> (\_ => Refl)
> (\_, _, _ => Refl))
>
> dualPreservesIsomorphic :
> Isomorphic (dualCategory cat) a b
> -> Isomorphic cat a b
> dualPreservesIsomorphic
> (MkIsomorphic
> morphism
> (MkIsomorphism inverse
> (MkInverseMorphisms lawLeft lawRight))) =
> MkIsomorphic
> inverse
> (MkIsomorphism
> morphism
> (MkInverseMorphisms lawLeft lawRight))
7 changes: 7 additions & 0 deletions src/Limits/Product.lidr
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Expand Up @@ -22,6 +22,7 @@ along with this program. If not, see <https://www.gnu.org/licenses/>.
> module Limits.Product
>
> import Basic.Category
> import Basic.Isomorphism
> import public CoLimits.CoProduct
> import public Dual.DualCategory
>
Expand All @@ -30,3 +31,9 @@ along with this program. If not, see <https://www.gnu.org/licenses/>.
>
> Product : (cat : Category) -> (a : obj cat) -> (b : obj cat) -> Type
> Product cat a b = CoProduct (dualCategory cat) a b
>
> productsAreIsomorphic :
> (a, b : Product cat l r)
> -> Isomorphic cat (carrier a) (carrier b)
> productsAreIsomorphic {cat} {l} {r} a b =
> dualPreservesIsomorphic (coProductsAreIsomorphic (dualCategory cat) l r a b)
7 changes: 7 additions & 0 deletions src/Limits/TerminalObject.lidr
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Expand Up @@ -24,9 +24,16 @@ along with this program. If not, see <https://www.gnu.org/licenses/>.
> import Basic.Category
> import public CoLimits.InitialObject
> import public Dual.DualCategory
> import Basic.Isomorphism
>
> %access public export
> %default total
>
> TerminalObject : Category -> Type
> TerminalObject cat = InitialObject (dualCategory cat)
>
> terminalObjectsAreIsomorphic :
> (a, b : TerminalObject cat)
> -> Isomorphic cat (carrier a) (carrier b)
> terminalObjectsAreIsomorphic {cat} a b =
> dualPreservesIsomorphic (initialObjectsAreIsomorphic (dualCategory cat) a b)