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Rewrite. 

I ended up re-implementing most of the functionality of this library
while trying to implement a categorical model of a programming
language I was working on, so I went ahead and copied most of it
over here.

The new version is still missing some features, such as
linear functions, monadic bind, the Unc typeclass, and haddock.
It also makes a few different design decisions, which come
with their own trade-offs.
master
James T. Martin 2024-01-04 15:11:41 -08:00
parent 8a1fad57df
commit 69f0312c8d
Signed by: james
GPG Key ID: D6FB2F9892F9B225
43 changed files with 864 additions and 1376 deletions
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!/monoids-in-the-category-of-endofunctors.cabal

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You should have received a copy of the GNU General Public License
along with this program. If not, see <https://www.gnu.org/licenses/>.
Also add information on how to contact you by electronic and paper mail.
If the program does terminal interaction, make it output a short
notice like this when it starts in an interactive mode:
<program> Copyright (C) <year> <name of author>
This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
This is free software, and you are welcome to redistribute it
under certain conditions; type `show c' for details.
The hypothetical commands `show w' and `show c' should show the appropriate
parts of the General Public License. Of course, your program's commands
might be different; for a GUI interface, you would use an "about box".
You should also get your employer (if you work as a programmer) or school,
if any, to sign a "copyright disclaimer" for the program, if necessary.
For more information on this, and how to apply and follow the GNU GPL, see
<https://www.gnu.org/licenses/>.
The GNU General Public License does not permit incorporating your program
into proprietary programs. If your program is a subroutine library, you
may consider it more useful to permit linking proprietary applications with
the library. If this is what you want to do, use the GNU Lesser General
Public License instead of this License. But first, please read
<https://www.gnu.org/licenses/why-not-lgpl.html>.

5
LICENSE.txt Normal file
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@ -0,0 +1,5 @@
Copyright (C) 2022 by James Martin <james@jtmar.me>
Permission to use, copy, modify, and/or distribute this software for any purpose with or without fee is hereby granted.
THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.

12
README.md
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@ -1,18 +1,14 @@
# Monoids in the Category of Endofunctors
This is a **toy** library for studying the field of [Abstract Nonsense](https://en.wikipedia.org/wiki/Abstract_nonsense) through Haskell.
Writing this sort of code makes for a fun puzzle, but I want to be *absolutely clear*
that I would *never* do anything even remotely close to this in a serious codebase.
If you want to see code more representative of me as a programming practitioner,
please refer to pretty much any project other than this one.
Writing this sort of code makes for a fun puzzle,
but you probably shouldn't use this for any serious work.
This library currently includes:
* Category theory (`Category`)
* Recursion schemes (`Category.Functor.Foldable`)
* Category theory
* Recursion schemes (`Functor.Algebra`)
* Dependent types, type-level programming, and codata (`Data`)
* Dependent quantifiers (implemented with the help of a typeclass; `Quantifier`)
For further information, build and read the Haddock.
## Content warning
This library is an abuse of GHC Haskell and an abuse of common sense.
Do not attempt to view this library if you are faint of heart.

2
Setup.hs
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@ -1,2 +0,0 @@
import Distribution.Simple
main = defaultMain

3
hie.yaml
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@ -1,3 +0,0 @@
cradle:
cabal:
component: "lib:haskell-language-server"

141
monoids-in-the-category-of-endofunctors.cabal
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@ -1,100 +1,51 @@
cabal-version: 2.2
name: monoids-in-the-category-of-endofunctors
version: 0.1.0.0
description: Please see the README on GitHub at <https://github.com/jamestmartin/monoids-in-the-categoy-of-endofunctors#readme>
homepage: https://github.com/jamestmartin/monoids-in-the-categoy-of-endofunctors#readme
bug-reports: https://github.com/jamestmartin/monoids-in-the-categoy-of-endofunctors/issues
author: James Martin
maintainer: james@jtmar.me
copyright: Copyright: (C) 2020 James Martin
license: GPL-3.0-or-later
license-file: LICENSE
build-type: Simple
cabal-version: 3.4
name: monoids-in-the-category-of-endofunctors
version: 0.1.0.0
license: 0BSD
license-file: LICENSE.txt
author: James T. Martin
maintainer: james@jtm.dev
build-type: Simple
extra-source-files:
README.md
source-repository head
type: git
location: https://github.com/jamestmartin/monoids-in-the-categoy-of-endofunctors
common common
default-language: GHC2021
default-extensions: BlockArguments, LambdaCase, NoImplicitPrelude, NoStarIsType, RoleAnnotations, TypeFamilies, DefaultSignatures, DataKinds, QuantifiedConstraints
ghc-options: -Wextra
library
exposed-modules:
Category.Base
Category.Constraint
Category.Enriched
Category.Functor
Category.Functor.Foldable
Category.Groupoid
Category.Monoid
Category.Monoidal
Category.Product
Category.Semigroup
Data.Dict
Data.Fin
Data.Identity
Data.Nat
Data.Proxy
Data.Vec
Quantifier
other-modules:
hs-source-dirs:
src
default-extensions:
ApplicativeDo
BangPatterns
BinaryLiterals
BlockArguments
ConstraintKinds
DataKinds
DefaultSignatures
DeriveDataTypeable
DeriveFoldable
DeriveFunctor
DeriveGeneric
DeriveLift
DeriveTraversable
DerivingStrategies
EmptyCase
EmptyDataDeriving
ExistentialQuantification
ExplicitForAll
FlexibleContexts
FlexibleInstances
FunctionalDependencies
GADTs
GeneralizedNewtypeDeriving
HexFloatLiterals
ImportQualifiedPost
InstanceSigs
KindSignatures
LambdaCase
LinearTypes
MultiParamTypeClasses
MultiWayIf
NamedFieldPuns
NamedWildCards
NoImplicitPrelude
NumericUnderscores
OverloadedStrings
PartialTypeSignatures
PatternSynonyms
PolyKinds
PostfixOperators
QuantifiedConstraints
RankNTypes
ScopedTypeVariables
StandaloneDeriving
StandaloneKindSignatures
TupleSections
TypeApplications
TypeFamilyDependencies
TypeInType
TypeOperators
TypeSynonymInstances
ViewPatterns
ghc-options: -Weverything -Wno-missing-export-lists -Wno-missing-import-lists -Wno-missing-safe-haskell-mode -Wno-safe -Wno-unsafe
build-depends:
base >=4.14 && <5
, ghc-prim >=0.7 && <0.8
default-language: Haskell2010
import: common
build-depends: base ^>=4.17.0.0, data-fix ^>= 0.3.2
hs-source-dirs: src
exposed-modules:
Category.Constraint
Category.Kleisli
Data.Dict
Data.Fin
Data.Identity
Data.Nat
Data.Proxy
Data.Vec
Functor.Algebra
Functor.Associative
Functor.Base
Functor.Bifunctor
Functor.Compose
Functor.Distributive
Functor.Exponent
Functor.Identity
Functor.Product
Functor
Object.Monoid
Object.Semigroup
Quantifier
Relation.Base
Relation.Category
Relation.Opposite
Relation.Product
Relation.Reflexive
Relation.Symmetric
Relation.Transitive
Relation
other-modules:

55
src/Category/Base.hs
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@ -1,55 +0,0 @@
-- The default type signatures for `idL` and `idR` trigger
-- redundant constraints because `NiceCat` requires `Category`
-- but it's forced to require `Category` because that's where it's defined.
-- This seems like it might be a bug.
{-# OPTIONS_GHC -Wno-redundant-constraints #-}
module Category.Base
( Category, Obj, idL, idR, (.)
, NiceCat, id
, Yoneda (Op), Op, getOp
) where
import Data.Kind (Constraint, FUN, Type)
-- | Objects are uniquely identified by their identity arrow.
type Obj :: (i -> i -> j) -> i -> j
type Obj morph a = morph a a
type Category :: forall i. (i -> i -> Type) -> Constraint
class Category morph where
idL :: morph a b -> Obj morph a
default idL :: NiceCat morph => morph a b -> Obj morph a
idL _ = id
idR :: morph a b -> Obj morph b
default idR :: NiceCat morph => morph a b -> Obj morph b
idR _ = id
-- | Associative composition of morphisms.
(.) :: morph b c -> morph a b -> morph a c
type NiceCat :: forall i. (i -> i -> Type) -> Constraint
class Category morph => NiceCat morph where
id :: Obj morph a
instance forall m. Category (FUN m) where
f . g = \x -> f (g x)
instance forall m. NiceCat (FUN m) where
id = \x -> x
type Yoneda :: (i -> i -> Type) -> i -> i -> Type
newtype Yoneda morph a b = Op { getOp :: morph b a }
type Op :: (i -> i -> Type) -> i -> i -> Type
type family Op morph where
Op (Yoneda morph) = morph
Op morph = Yoneda morph
instance Category morph => Category (Yoneda morph) where
idL (Op f) = Op (idR f)
idR (Op f) = Op (idL f)
Op f . Op g = Op (g . f)
instance NiceCat morph => NiceCat (Yoneda morph) where
id = Op id

68
src/Category/Constraint.hs
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@ -1,15 +1,18 @@
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE UndecidableSuperClasses #-}
{-# LANGUAGE UndecidableInstances #-}
module Category.Constraint
( (:-) (Sub), (\\)
, ProdC
) where
( (:-) (Sub), (\\)
, ProdC
) where
import Category.Base
import Category.Functor
import Category.Monoidal
import Category.Product
import Data.Dict
--import Functor.Associative
import Functor.Base
--import Functor.Bifunctor
--import Functor.Identity
--import Functor.Product
import Relation
import Data.Kind (Constraint, Type)
type (:-) :: Constraint -> Constraint -> Type
@ -18,48 +21,41 @@ data (:-) c d = Sub (c => Dict d)
(\\) :: a => (b => c) -> (a :- b) -> c
r \\ Sub Dict = r
instance Category (:-) where
f . g = Sub (Dict \\ f \\ g)
instance NiceCat (:-) where
instance Reflexive (:-)
instance Wide (:-) where
id = Sub Dict
instance Functor (Nat (->) (:-)) (Yoneda (:-)) (:-) where
map (Op (Sub f)) = Nat_ \(Sub g) -> Sub case f of Dict -> case g of Dict -> Dict
instance Transitive (:-) where
f . g = Sub (Dict \\ f \\ g)
instance Functor (->) (:-) ((:-) a) where
instance Functor ((:-) a) where
type Cod ((:-) a) = (->)
type Dom ((:-) a) = (:-)
map = (.)
instance Functor (->) (:-) Dict where
instance Functor Dict where
type Cod Dict = (->)
type Dom Dict = (:-)
map f = \Dict -> case f of Sub Dict -> Dict
type UncurryC :: (a -> b -> Constraint) -> (a, b) -> Constraint
class f (Pi1 ab) (Pi2 ab) => UncurryC f ab
instance f (Pi1 ab) (Pi2 ab) => UncurryC f ab
type instance Uncurry = UncurryC
instance Unc (:-) where
uncurry _ = Sub Dict
ununcurry _ = Sub Dict
-- FIXME
--type instance Uncurry = UncurryC
class (c, d) => ProdC c d
instance (c, d) => ProdC c d
-- Note that, to my understanding,
-- it is impossible to define disjunction in the category of constraints,
-- and as far as I know this is the only way in which entailment is monoidal
-- (up to isomorphism), although I haven't seriously thought about it at all.
instance Functor (Nat (:-) (:-)) (:-) ProdC where
map (Sub f) = Nat_ (Sub case f of Dict -> Dict)
instance Functor ProdC where
type Cod ProdC = Nat (:-) (:-)
type Dom ProdC = (:-)
map (Sub f) = Nat \_ -> Sub case f of Dict -> Dict
instance Functor (:-) (:-) (ProdC a) where
instance Functor (ProdC a) where
type Cod (ProdC a) = (:-)
type Dom (ProdC a) = (:-)
map (Sub f) = Sub case f of Dict -> Dict
instance TensorProduct (:-) ProdC where
type Unit (:-) ProdC = ()
prodIL _ = Sub Dict
prodIR _ = Sub Dict
prodEL _ = Sub Dict
prodER _ = Sub Dict
prodAL _ _ _ = Sub Dict
prodAR _ _ _ = Sub Dict
-- TODO: Re-prove that ProdC is a tensor product in the new rewrite.
-- It shouldn't be particularly difficult, but it sounds tedious right now.

21
src/Category/Enriched.hs
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@ -1,21 +0,0 @@
module Category.Enriched
( Enriched, eid, ecomp
) where
import Category.Base
import Category.Monoidal
import Data.Kind (Constraint, FUN, Type)
import GHC.Types (Multiplicity (One))
type Enriched :: (j -> j -> Type) -> (j -> j -> j) -> (i -> i -> j) -> Constraint
class TensorProduct over prod => Enriched over prod morph where
eid :: proxy prod -> over (Unit over prod) (Obj morph a)
ecomp :: over (prod (morph b c) (morph a b)) (morph a c)
instance NiceCat morph => Enriched (->) (,) morph where
eid _ = \() -> id
ecomp = \(f, g) -> f . g
instance Enriched (FUN 'One) (,) (FUN 'One) where
eid _ = \() -> id
ecomp = \(f, g) -> \x -> f (g x)

98
src/Category/Functor.hs
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@ -1,98 +0,0 @@
{-# LANGUAGE UndecidableInstances #-}
module Category.Functor
( Functor, map
, Endo, Endofunctor, endomap
, Contravariant, contramap
, Nat (Nat), runNat, pattern Nat_, natId
, Const (Const), getConst
) where
import Category.Base
import Data.Either (Either (Left, Right))
import Data.Kind (Constraint, FUN, Type)
import Data.Maybe (Maybe (Nothing, Just))
import Data.Proxy
type Functor :: (j -> j -> Type) -> (i -> i -> Type) -> (i -> j) -> Constraint
class (Category dest, Category src) => Functor dest src f where
map :: src a b -> dest (f a) (f b)
type Endo f a = f a a
type Endofunctor :: (i -> i -> Type) -> (i -> i) -> Constraint
class Endo Functor morph f => Endofunctor morph f where
instance Endo Functor morph f => Endofunctor morph f
endomap :: Endofunctor morph f => morph a b -> morph (f a) (f b)
endomap = map
class Contravariant dest src f where
contramap :: src b a -> dest (f a) (f b)
instance {-# INCOHERENT #-} Functor dest (Yoneda src) f => Contravariant dest src f where
contramap f = map (Op f)
instance {-# INCOHERENT #-} Functor dest src f => Contravariant dest (Yoneda src) f where
contramap (Op f) = map f
type Nat :: (j -> j -> Type) -> (i -> i -> Type) -> (i -> j) -> (i -> j) -> Type
data Nat dest src f g = (Functor dest src f, Functor dest src g) => Nat { runNat :: !(forall a. Obj src a -> dest (f a) (g a)) }
type Nat_' :: (j -> j -> Type) -> (i -> i -> Type) -> (i -> j) -> (i -> j) -> Type
data Nat_' dest src f g = (Functor dest src f, Functor dest src g) => Nat_' !(forall a. dest (f a) (g a))
nat_' :: NiceCat src => Nat dest src f g -> Nat_' dest src f g
nat_' (Nat f) = Nat_' (f id)
pattern Nat_ :: forall dest src f g. NiceCat src => (Functor dest src f, Functor dest src g) => (forall a. dest (f a) (g a)) -> Nat dest src f g
{-# COMPLETE Nat_ #-}
pattern Nat_ f <- (nat_' -> Nat_' f)
where Nat_ f = Nat \_ -> f
instance Category (Nat dest src) where
idL (Nat _) = Nat map
idR (Nat _) = Nat map
Nat f . Nat g = Nat \x -> (f x . g x)
natId :: Functor dest src f => Obj (Nat dest src) f
natId = Nat map
instance (forall f. Functor dest src f) => NiceCat (Nat dest src) where
id = natId
instance Functor (->) (FUN m) (FUN m a) where
map f = \g -> f . g
instance Functor (Nat (->) (FUN m)) (Yoneda (FUN m)) (FUN m) where
map (Op f) = Nat_ \g -> g . f
instance Functor (Nat (FUN m) (FUN m)) (FUN m) (,) where
map f = Nat_ \(x, y) -> (f x, y)
instance Functor (FUN m) (FUN m) ((,) a) where
map f = \(x, y) -> (x, f y)
instance Functor (Nat (FUN m) (FUN m)) (FUN m) Either where
map f = Nat_ \case
Left y -> Left (f y)
Right x -> Right x
instance Functor (FUN m) (FUN m) (Either a) where
map f = \case
Left y -> Left y
Right x -> Right (f x)
instance Functor (FUN m) (FUN m) Maybe where
map f = \case
Nothing -> Nothing
Just x -> Just (f x)
instance {-# INCOHERENT #-} Category src => Functor (FUN m) src Proxy where
map _ = \Proxy -> Proxy
type Const :: Type -> i -> Type
newtype Const a b = Const { getConst :: a }
instance Functor (Nat (->) (->)) (->) Const where
map f = Nat_ \(Const x) -> Const (f x)
instance {-# INCOHERENT #-} Category src => Functor (->) src (Const a) where
map _ = \(Const x) -> Const x

22
src/Category/Functor/Foldable.hs
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@ -1,22 +0,0 @@
module Category.Functor.Foldable where
import Category.Base
import Category.Functor
type family Base (t :: i) :: i -> i
class Endofunctor morph (Base t) => Recursive morph t where
project :: morph t (Base t t)
class Endofunctor morph (Base t) => Corecursive morph t where
embed :: morph (Base t t) t
type Algebra morph f a = morph (f a) a
cata :: Recursive morph t => Algebra morph (Base t) a -> morph t a
cata alg = alg . map (cata alg) . project
type Coalgebra morph f a = morph a (f a)
ana :: Corecursive morph t => Coalgebra morph (Base t) a -> morph a t
ana coalg = embed . map (ana coalg) . coalg

8
src/Category/Groupoid.hs
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@ -1,8 +0,0 @@
module Category.Groupoid (Groupoid, inv) where
import Category.Base
import Data.Kind (Constraint, Type)
type Groupoid :: (i -> i -> Type) -> Constraint
class Category morph => Groupoid morph where
inv :: morph a b -> morph b a

23
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{-# LANGUAGE UndecidableInstances #-}
module Category.Kleisli where
import Data.Functor.Identity (Identity (Identity))
import Functor.Base
import Functor.Compose
import Object.Monoid
import Object.Semigroup
import Relation
import Data.Data (Proxy (Proxy))
import Data.Kind (Type)
data Kleisli m x y = Kleisli { runKleisli :: !(Cod m x (m y)) }
data Cokleisli w x y = Cokleisli { runCokleisli :: !(Cod w (w x) y) }
instance (Monoid Compose m, Cod m ~ (->)) => Reflexive (Kleisli m)
instance (Monoid Compose m, Cod m ~ (->)) => Wide (Kleisli m) where
id = Kleisli (runNat (empty (Proxy @Compose)) id . Identity)
instance (Semigroup (Compose :: (Type -> Type) -> (Type -> Type) -> Type -> Type) m, Functor m, Dom m ~ (->), Cod m ~ (->)) => Transitive (Kleisli m) where
Kleisli g . Kleisli f = Kleisli (runNat append id . Compose . map g . f)

29
src/Category/Monoid.hs
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@ -1,29 +0,0 @@
{-# LANGUAGE UndecidableInstances #-}
module Category.Monoid
( Monoid, empty
, (>>=)
) where
import Category.Base
import Category.Functor
import Category.Monoidal
import Category.Semigroup
import Data.Kind (Constraint, Type)
import Data.Maybe (Maybe (Just))
import Data.Proxy
type Monoid :: (i -> i -> Type) -> (i -> i -> i) -> i -> Constraint
class Semigroup morph prod m => Monoid morph prod m where
empty :: proxy morph -> proxy' prod -> morph (Unit morph prod) m
instance Monoid (Nat (->) (->)) Compose Maybe where
empty _ _ = Nat_ \(Identity x) -> Just x
-- | A monad is a monoid object in the monoidal category of endofunctors and natural transformations between them!
class Monoid (Nat (->) (->)) Compose m => Monad m
instance Monoid (Nat (->) (->)) Compose m => Monad m
(>>=) :: forall f a b. Monoid (Nat (->) (->)) Compose f => f a -> (a -> f b) -> f b
m >>= f = runNat (append @_ @(Nat (->) (->))) id (lemma (Compose (map f m)))
where lemma :: forall c. (Functor (->) (->) f => c) -> c
lemma x = case empty (Proxy @(Nat (->) (->))) (Proxy @Compose) :: Nat (->) (->) Identity f of Nat _ -> x

118
src/Category/Monoidal.hs
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@ -1,118 +0,0 @@
module Category.Monoidal
( TensorProduct, Unit, unitObj, prodObj, prodIL, prodIR, prodEL, prodER, prodAL, prodAR
, prodIL_, prodIR_, prodEL_, prodER_, prodAL_, prodAR_
, Compose (Compose), getCompose
, Identity (Identity), getIdentity
) where
import Category.Base
import Category.Functor
import Category.Product
import Data.Either (Either (Left, Right))
import Data.Kind (Constraint, FUN, Type)
import Data.Void (Void)
-- | A category is monoidal if it has a product and a unit for that product.
-- A category can have multiple tensor products and be monoidal in multiple ways,
-- including the category of types itself,
-- so instead of using a @Monoidal@ typeclass, we use a @TensorProduct@ typeclass.
type TensorProduct :: (i -> i -> Type) -> (i -> i -> i) -> Constraint
class Bifunctor morph morph morph prod => TensorProduct (morph :: i -> i -> Type) prod where
type Unit morph prod :: i
-- | The unit is an object.
unitObj :: proxy prod -> Obj morph (Unit morph prod)
default unitObj :: NiceCat morph => proxy prod -> Obj morph (Unit morph prod)
unitObj _ = id
-- | Given two objects, their product is also an object.
prodObj :: Obj morph a -> Obj morph b -> Obj morph (prod a b)
default prodObj :: NiceCat morph => Obj morph a -> Obj morph b -> Obj morph (prod a b)
prodObj _ _ = id
-- | Introduce a product with the value injected into the left side.
prodIL :: Obj morph a -> morph a (prod a (Unit morph prod))
-- | Introduce a product with the value injected into the right side.
prodIR :: Obj morph a -> morph a (prod (Unit morph prod) a)
-- | Eliminate a product with the value projected from the left side.
prodEL :: Obj morph a -> morph (prod a (Unit morph prod)) a
-- | Eliminate a product with the value projected from the right side.
prodER :: Obj morph a -> morph (prod (Unit morph prod) a) a
-- | Reassociate a product, nesting it to the left.
prodAL :: Obj morph a -> Obj morph b -> Obj morph c -> morph (prod a (prod b c)) (prod (prod a b) c)
-- | Reassociate a product, nesting it to the right.
prodAR :: Obj morph a -> Obj morph b -> Obj morph c -> morph (prod (prod a b) c) (prod a (prod b c))
prodIL_ :: (NiceCat morph, TensorProduct morph prod) => morph a (prod a (Unit morph prod))
prodIL_ = prodIL id
prodIR_ :: (NiceCat morph, TensorProduct morph prod) => morph a (prod (Unit morph prod) a)
prodIR_ = prodIR id
prodEL_ :: (NiceCat morph, TensorProduct morph prod) => morph (prod a (Unit morph prod)) a
prodEL_ = prodEL id
prodER_ :: (NiceCat morph, TensorProduct morph prod) => morph (prod (Unit morph prod) a) a
prodER_ = prodER id
prodAL_ :: (NiceCat morph, TensorProduct morph prod) => morph (prod a (prod b c)) (prod (prod a b) c)
prodAL_ = prodAL id id id
prodAR_ :: (NiceCat morph, TensorProduct morph prod) => morph (prod (prod a b) c) (prod a (prod b c))
prodAR_ = prodAR id id id
instance TensorProduct (FUN m) (,) where
type Unit (FUN m) (,) = ()
prodIL _ = \x -> (x, ())
prodIR _ = \x -> ((), x)
prodEL _ = \(x, ()) -> x
prodER _ = \((), x) -> x
prodAL _ _ _ = \(x, (y, z)) -> ((x, y), z)
prodAR _ _ _ = \((x, y), z) -> (x, (y, z))
absurd :: Void %1-> a
absurd = \case{}
instance TensorProduct (FUN m) Either where
type Unit (FUN m) Either = Void
prodIL _ = Left
prodIR _ = Right
prodEL _ = \case
Left x -> x
Right x -> absurd x
prodER _ = \case
Left x -> absurd x
Right x -> x
prodAL _ _ _ = \case
Left x -> Left (Left x)
Right (Left x) -> Left (Right x)
Right (Right x) -> Right x
prodAR _ _ _ = \case
Left (Left x) -> Left x
Left (Right x) -> Right (Left x)
Right x -> Right (Right x)
data Compose f g x = (Functor (->) (->) f, Functor (->) (->) g) => Compose { getCompose :: !(f (g x)) }
newtype Identity x = Identity { getIdentity :: x }
instance Functor (FUN m) (FUN m) Identity where
map f = \(Identity x) -> Identity (f x)
instance Functor (Nat (Nat (->) (->)) (Nat (->) (->))) (Nat (->) (->)) Compose where
map (Nat f) = Nat \(Nat _) -> Nat_ \(Compose x) -> Compose (f id x)
instance Functor (Nat (->) (->)) (Nat (->) (->)) (Compose f) where
map (Nat f) = Nat_ \(Compose x) -> Compose (map (f id) x)
instance Functor (->) (->) (Compose (f :: Type -> Type) g) where
map f = \(Compose x) -> Compose (map @_ @_ @_ @(->) (map f) x)
instance TensorProduct (Nat (->) (->)) Compose where
type Unit (Nat (->) (->)) Compose = Identity
unitObj _ = natId
prodObj _ _ = natId
prodIL (Nat _) = Nat_ \x -> Compose (map Identity x)
prodIR (Nat _) = Nat_ \x -> Compose (Identity x)
prodEL (Nat _) = Nat_ \(Compose x) -> map getIdentity x
prodER (Nat _) = Nat_ \(Compose (Identity x)) -> x
prodAL (Nat _) (Nat _) (Nat _) = Nat_ \(Compose x) -> Compose (Compose (map getCompose x))
prodAR (Nat _) (Nat _) (Nat _) = Nat_ \(Compose (Compose x)) -> Compose (map Compose x)

104
src/Category/Product.hs
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@ -1,104 +0,0 @@
{-# LANGUAGE UndecidableInstances, UndecidableSuperClasses #-}
module Category.Product
( Pi1, Pi2, pairEta
, Product (Product)
, Uncurry, Uncurry' (Uncurry), getUncurry, UncurryN (UncurryN), getUncurryN
, Unc, uncurry, ununcurry
, Bifunctor, bimap_, bimap, first, second
, Profunctor, dimap
) where
import Category.Base
import Category.Functor
import Data.Dict
import Data.Kind (Constraint, FUN, Type)
import Data.Proxy
import Unsafe.Coerce (unsafeCoerce)
-- | The first projection of the type-level tuple.
type Pi1 :: (i, j) -> i
type family Pi1 xy where
Pi1 '(x, _) = x
-- | The second projection of the type-level tuple.
type Pi2 :: (i, j) -> j
type family Pi2 xy where
Pi2 '(_, y) = y
-- | Eta expansion for the pair type *on the type level*. **This does not hold on the value level.**
--
-- This is not provable by GHC's constraint solver, but it is safe to assume.
-- Maybe. Hopefully.
pairEta :: proxy x -> Dict (x ~ '(Pi1 x, Pi2 x))
pairEta = unsafeCoerce (Dict :: Dict ())
-- | The product category of two categories `c` and `d` is the category
-- whose objects are pairs of objects from `c` and `d` and whose arrows
-- are pairs of arrows from `c` and `d`.
type Product :: (i -> i -> Type) -> (j -> j -> Type) -> (i, j) -> (i, j) -> Type
data Product c d a b = Product !(c (Pi1 a) (Pi1 b)) !(d (Pi2 a) (Pi2 b))
instance (Category c, Category d) => Category (Product c d) where
idL (Product f g) = Product (idL f) (idL g)
idR (Product f g) = Product (idR f) (idR g)
Product f1 g1 . Product f2 g2 = Product (f1 . f2) (g1 . g2)
instance (NiceCat c, NiceCat d) => NiceCat (Product c d) where
id = Product id id
type Uncurry :: (a -> b -> c) -> (a, b) -> c
type family Uncurry
type Uncurry' :: (a -> b -> Type) -> (a, b) -> Type
newtype Uncurry' f ab = Uncurry { getUncurry :: f (Pi1 ab) (Pi2 ab) }
type instance Uncurry = Uncurry'
type UncurryN :: (a -> b -> c -> Type) -> (a, b) -> c -> Type
newtype UncurryN f ab x = UncurryN { getUncurryN :: f (Pi1 ab) (Pi2 ab) x }
type instance Uncurry = UncurryN
instance (Category c, Functor (->) c (f a b)) => Functor (->) c (UncurryN f '(a, b)) where
map f (UncurryN x) = UncurryN (map f x)
type Unc :: (c -> c -> Type) -> Constraint
class Category cat => Unc cat where
uncurry :: Obj cat (f a b) -> cat (f a b) (Uncurry f '(a, b))
ununcurry :: Obj cat (f a b) -> cat (Uncurry f '(a, b)) (f a b)
instance Unc (FUN m) where
uncurry _ = Uncurry
ununcurry _ (Uncurry x) = x
instance Unc (Nat (->) (->)) where
uncurry (Nat _) = Nat_ UncurryN
ununcurry (Nat _) = Nat_ \(UncurryN x) -> x
-- | A bifunctor is a functor whose domain is the product category.
type Bifunctor :: (k -> k -> Type) -> (i -> i -> Type) -> (j -> j -> Type) -> (i -> j -> k) -> Constraint
class (Unc cod, Category dom1, Category dom2, Functor cod (Product dom1 dom2) (Uncurry f)) => Bifunctor cod dom1 dom2 f where
bimap_ :: forall a b. Obj dom1 a -> Obj dom2 b -> Obj cod (f a b)
bimap :: forall cod dom1 dom2 f a b c d. Bifunctor cod dom1 dom2 f => dom1 a c -> dom2 b d -> cod (f a b) (f c d)
bimap f g = ununcurry (bimap_ (idR f) (idR g)) . map (Product f g) . uncurry (bimap_ (idL f) (idL g))
first :: forall cod dom f a b c. (NiceCat dom, Bifunctor cod dom dom f) => dom a b -> cod (f a c) (f b c)
first f = bimap f (id :: dom c c)
second :: forall cod dom f a b c. (NiceCat dom, Bifunctor cod dom dom f) => dom b c -> cod (f a b) (f a c)
second g = bimap (id :: dom a a) g
instance (Unc cod, Functor (Nat cod dom2) dom1 f, forall x. Functor cod dom2 (f x), uncurry ~ Uncurry) => Functor cod (Product dom1 dom2) (uncurry f) where
{-# INLINABLE map #-}
map :: forall a b. Product dom1 dom2 a b -> cod (uncurry f a) (uncurry f b)
map (Product f g) = lemma (uncurry (map (idR g)) . runNat (map f) (idR g) . map g . ununcurry (map (idL g)))
where lemma :: ((a ~ '(Pi1 a, Pi2 a), b ~ '(Pi1 b, Pi2 b)) => c) -> c
lemma x = case pairEta (Proxy @a) of Dict -> case pairEta (Proxy @b) of Dict -> x
instance (Unc cod, Category dom1, Functor (Nat cod dom2) dom1 f, forall x. Functor cod dom2 (f x)) => Bifunctor cod dom1 dom2 f where
bimap_ a b = runNat (map a) b . map b
type Profunctor :: (k -> k -> Type) -> (i -> i -> Type) -> (j -> j -> Type) -> (i -> j -> k) -> Constraint
class Bifunctor cod (Yoneda dom1) dom2 f => Profunctor cod dom1 dom2 f
instance Bifunctor cod (Yoneda dom1) dom2 f => Profunctor cod dom1 dom2 f
dimap :: Profunctor cod dom1 dom2 f => dom1 c a -> dom2 b d -> cod (f a b) (f c d)
dimap f g = bimap (Op f) g

18
src/Category/Semigroup.hs
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@ -1,18 +0,0 @@
module Category.Semigroup (Semigroup, append) where
import Category.Functor
import Category.Monoidal
import Data.Kind (Constraint, Type)
import Data.Maybe (Maybe (Nothing, Just))
type Semigroup :: (i -> i -> Type) -> (i -> i -> i) -> i -> Constraint
class TensorProduct morph prod => Semigroup morph prod s where
append :: morph (prod s s) s
instance Semigroup (Nat (->) (->)) Compose Maybe where
append = Nat_ \(Compose x') -> case x' of
Nothing -> Nothing
Just Nothing -> Nothing
Just (Just x) -> Just x
-- TODO: More semigroup/monoid instances. `Either` has two, can I accomodate both?

48
src/Data/Fin.hs
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@ -4,11 +4,13 @@ module Data.Fin
, fin2nat
) where
import Category.Functor
import Category.Functor.Foldable
import Data.Functor.Const (Const (Const))
import Data.Identity
import Data.Kind (Type)
import Data.Nat
import Functor.Algebra
import Functor.Base
import Relation
data Fin :: N -> Type where
FZ :: Fin ('S n)
@ -16,26 +18,42 @@ data Fin :: N -> Type where
data FinF r :: N -> Type where
FZF :: FinF r ('S n)
FSF :: r n -> FinF r ('S n)
type instance Base Fin = FinF
instance Functor (Nat (->) (:~:)) (Nat (->) (:~:)) FinF where
map :: forall f g. Nat (->) (:~:) f g -> Nat (->) (:~:) (FinF f) (FinF g)
map (Nat_ f) = Nat_ \case
instance Functor Fin where
type Cod Fin = (->)
type Dom Fin = (:~:)
map Refl = \case
FZ -> FZ
FS n -> FS (map Refl n)
instance Functor (FinF r) where
type Cod (FinF r) = (->)
type Dom (FinF r) = (:~:)
map Refl = id
instance Functor FinF where
type Cod FinF = Nat (->) (:~:)
type Dom FinF = Nat (->) (:~:)
map (Nat f) = Nat \_ -> \case
FZF -> FZF
(FSF r) -> FSF (f r)
FSF n -> FSF (f id n)
instance Recursive (Nat (->) (:~:)) Fin where
project = Nat_ \case
instance Recursive Fin where
type Base Fin = FinF
project = Nat \_ -> \case
FZ -> FZF
(FS r) -> FSF r
instance Corecursive (Nat (->) (:~:)) Fin where
embed = Nat_ \case
FS n -> FSF n
embed = Nat \_ -> \case
FZF -> FZ
(FSF r) -> FS r
FSF n -> FS n
instance Functor (Const a :: N -> Type) where
type Cod (Const a) = (->)
type Dom (Const a) = (:~:)
map Refl (Const x) = Const x
fin2nat :: Nat (->) (:~:) Fin (Const N)
fin2nat = cata (Nat_ alg)
fin2nat = fold (Nat \_ -> alg)
where alg :: FinF (Const N) n -> Const N n
alg FZF = Const Z
alg (FSF (Const n)) = Const (S n)

24
src/Data/Identity.hs
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@ -1,32 +1,18 @@
module Data.Identity ((:~:) (Refl)) where
import Category.Base
import Category.Functor
import Category.Groupoid
import Relation
import Data.Kind (Type)
type (:~:) :: i -> i -> Type
data (:~:) :: i -> i -> Type where
Refl :: a :~: a
instance Category (:~:) where
instance Transitive (:~:) where
Refl . Refl = Refl
instance NiceCat (:~:) where
instance Reflexive (:~:)
instance Wide (:~:) where
id = Refl
instance Groupoid (:~:) where
instance Symmetric (:~:) where
inv Refl = Refl
-- TODO: There are lots, lots more valid functor instances.
instance {-# INCOHERENT #-} Functor (->) (:~:) f where
map Refl = id
instance {-# INCOHERENT #-} Functor (->) (Yoneda (:~:)) f where
map (Op Refl) = id
instance {-# INCOHERENT #-} Functor (Nat (->) (:~:)) (:~:) f where
map Refl = id
instance {-# INCOHERENT #-} Functor (Nat (->) (:~:)) (Yoneda (:~:)) f where
map (Op Refl) = id

26
src/Data/Nat.hs
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@ -6,10 +6,8 @@ module Data.Nat
, injective, rightZero
) where
import Category.Functor
import Category.Functor.Foldable
import Functor.Algebra
import Data.Dict
import Data.Identity
import Data.Kind (Type)
import Data.Maybe (Maybe (Nothing, Just))
import Quantifier
@ -46,33 +44,15 @@ data NTyF r n where
ZTyF :: NTyF r 'Z
STyF :: r n -> NTyF r ('S n)
instance Functor (Nat (->) (:~:)) (Nat (->) (:~:)) NTyF where
map (Nat_ f) = Nat_ \case
ZTyF -> ZTyF
(STyF r) -> STyF (f r)
instance Recursive N where
type Base N = Maybe
type instance Base N = Maybe
instance Recursive (->) N where
project Z = Nothing
project (S n) = Just n
instance Corecursive (->) N where
embed Nothing = Z
embed (Just n) = S n
type instance Base (Ty N) = NTyF
instance Recursive (Nat (->) (:~:)) (Ty N) where
project = Nat_ \case
ZTy -> ZTyF
(STy r) -> STyF r
instance Corecursive (Nat (->) (:~:)) (Ty N) where
embed = Nat_ \case
ZTyF -> ZTy
(STyF r) -> STy r
-- | Type-level addition.
type family (:+) (m :: N) (n :: N) :: N where
'Z :+ n = n

67
src/Data/Vec.hs
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@ -4,14 +4,13 @@ module Data.Vec
, index
) where
import Category.Base
import Category.Functor
import Category.Functor.Foldable
import Data.Fin
import Data.Identity
import Data.Kind (Type)
import Data.Nat
import Prelude (($))
import Functor.Algebra
import Functor.Base
import Relation
type Vec :: Type -> N -> Type
data Vec a :: N -> Type where
@ -21,30 +20,66 @@ type VecF :: Type -> (N -> Type) -> N -> Type
data VecF a r :: N -> Type where
VZF :: VecF a r 'Z
VSF :: a -> r n -> VecF a r ('S n)
type instance Base (Vec a) = VecF a
instance Functor (Nat (->) (:~:)) (->) Vec where
map f = Nat_ \case
instance Functor (Vec a) where
type Cod (Vec a) = (->)
type Dom (Vec a) = (:~:)
map Refl = \case
VZ -> VZ
(VS x r) -> VS (f x) (runNat (map @_ @_ @(Nat (->) (:~:)) f) id r)
VS x xs -> VS x (map Refl xs)
instance Functor (Nat (->) (:~:)) (Nat (->) (:~:)) (VecF a) where
map (Nat_ f) = Nat_ \case
instance Functor Vec where
type Dom Vec = (->)
type Cod Vec = Nat (->) (:~:)
map f = Nat \_ -> \case
VZ -> VZ
(VS x r) -> VS (f x) (runNat (map f) id r)
instance Functor (VecF a r) where
type Cod (VecF a r) = (->)
type Dom (VecF a r) = (:~:)
map Refl = \case
VZF -> VZF
(VSF x r) -> VSF x (f r)
VSF x xs -> VSF x xs
instance Functor (Nat (Nat (->) (:~:)) (Nat (->) (:~:))) (->) VecF where
map f = Nat_ $ Nat_ \case
instance Functor (VecF a) where
type Cod (VecF a) = Nat (->) (:~:)
type Dom (VecF a) = Nat (->) (:~:)
map (Nat f) = Nat \_ -> \case
VZF -> VZF
(VSF x r) -> VSF x (f id r)
instance Functor VecF where
type Dom VecF = (->)
type Cod VecF = Nat (Nat (->) (:~:)) (Nat (->) (:~:))
map f = Nat \_ -> Nat \_ -> \case
VZF -> VZF
(VSF x r) -> VSF (f x) r
instance Recursive (Vec a) where
type Base (Vec a) = VecF a
project = Nat \_ -> \case
VZ -> VZF
VS x xs -> VSF x xs
embed = Nat \_ -> \case
VZF -> VZ
VSF x xs -> VS x xs
type Ixr :: (N -> Type) -> Type -> N -> Type
newtype Ixr ty r a = Ixr { getIxr :: ty a -> r }
instance Functor (Ixr ty r) where
type Cod (Ixr ty r) = (->)
type Dom (Ixr ty r) = (:~:)
map Refl = id
indexer :: Nat (->) (:~:) Fin (Ixr (Vec a) a)
indexer = cata $ Nat_ \case
FZF -> Ixr \case VS x _ -> x
(FSF (Ixr r)) -> Ixr \case VS _ xs -> r xs
indexer = fold alg
where
alg :: Nat (->) (:~:) (FinF (Ixr (Vec a) a)) (Ixr (Vec a) a)
alg = Nat \_ -> \case
FZF -> Ixr \case VS x _ -> x
(FSF (Ixr r)) -> Ixr \case VS _ xs -> r xs
index :: Fin n -> Vec a n -> a
index = getIxr . runNat indexer id

20
src/Functor.hs Normal file
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@ -0,0 +1,20 @@
module Functor
( module Functor.Associative
, module Functor.Base
, module Functor.Bifunctor
, module Functor.Compose
, module Functor.Exponent
, module Functor.Identity
, module Functor.Product
, module Data.Functor.Identity
) where
import Functor.Associative
import Functor.Base
import Functor.Bifunctor
import Functor.Compose
import Functor.Exponent
import Functor.Identity
import Functor.Product
import Data.Functor.Identity (Identity (Identity), runIdentity)

59
src/Functor/Algebra.hs Normal file
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@ -0,0 +1,59 @@
module Functor.Algebra where
import Functor.Base
import Relation
import Data.Coerce (Coercible, coerce)
import Data.Fix (Fix (Fix), unFix)
import Data.Kind (Constraint, Type)
import GHC.Generics
type Algebra :: (k -> k) -> k -> Type
type Algebra f a = Cod f (f a) a
type Coalgebra :: (k -> k) -> k -> Type
type Coalgebra f a = Cod f a (f a)
type Recursive :: k -> Constraint
class Endofunctor (Base t) => Recursive (t :: k) where
type Base t :: k -> k
embed :: Algebra (Base t) t
project :: Coalgebra (Base t) t
newtype Mu f = Mu { unMu :: forall a. Algebra f a -> a }
data Nu f = forall a. Nu !(Coalgebra f a) !a
instance (Endofunctor f, Cod f ~ (->)) => Recursive (Fix f) where
type Base (Fix f) = f
embed = Fix
project = unFix
instance (Endofunctor f, Cod f ~ (->)) => Recursive (Mu f) where
type Base (Mu f) = f
embed x = Mu \alg -> alg (map (fold alg) x)
project (Mu fold) = fold (map embed)
instance (Endofunctor f, Cod f ~ (->)) => Recursive (Nu f) where
type Base (Nu f) = f
embed = Nu (map project)
project (Nu coalg seed) = map (Nu coalg) (coalg seed)
gproject :: forall t. (Recursive t, Generic t, Generic (Base t t), Coercible (Rep t) (Rep (Base t t)))
=> t -> Base t t
gproject = to . (coerce :: Rep t () -> Rep (Base t t) ()) . from
gembed :: forall t. (Recursive t, Generic t, Generic (Base t t), Coercible (Rep (Base t t)) (Rep t))
=> Base t t -> t
gembed = to . (coerce :: Rep (Base t t) () -> Rep t ()) . from
fold :: Recursive t => Algebra (Base t) a -> Cod (Base t) t a
fold alg = h where h = alg . map h . project
unfold :: Recursive t => Coalgebra (Base t) a -> Cod (Base t) a t
unfold coalg = h where h = embed . map h . coalg
refold :: Endofunctor f => Cod f (f b) b -> Cod f a (f a) -> Cod f a b
refold alg coalg = h where h = alg . map h . coalg
refix :: (Recursive t, Recursive u, Base t ~ Base u, Endofunctor (Base t), Cod (Base t) ~ (->)) => t -> u
refix = refold embed project

31
src/Functor/Associative.hs Normal file
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@ -0,0 +1,31 @@
module Functor.Associative where
import Functor.Base
import Functor.Bifunctor
import Functor.Compose
import Relation
import Data.Either (Either (Left, Right))
import Data.Kind (Constraint)
type Associative :: (k -> k -> k) -> Constraint
class Bifunctor f => Associative f where
assocL :: Object (Cod1 f) x -> Object (Cod1 f) y -> Object (Cod1 f) z -> Cod1 f (f x (f y z)) (f (f x y) z)
assocR :: Object (Cod1 f) x -> Object (Cod1 f) y -> Object (Cod1 f) z -> Cod1 f (f (f x y) z) (f x (f y z))
instance Associative (,) where
assocL _ _ _ ~(x, ~(y, z)) = ((x, y), z)
assocR _ _ _ ~(~(x, y), z) = (x, (y, z))
instance Associative Either where
assocL _ _ _ (Left x) = Left (Left x)
assocL _ _ _ (Right (Left y)) = Left (Right y)
assocL _ _ _ (Right (Right z)) = Right z
assocR _ _ _ (Left (Left x)) = Left x
assocR _ _ _ (Left (Right y)) = Right (Left y)
assocR _ _ _ (Right z) = Right (Right z)
instance Associative Compose where
assocL (Nat _) (Nat _) (Nat _) = Nat \_ (Compose x) -> Compose (Compose (map getCompose x))
assocR (Nat _) (Nat _) (Nat _) = Nat \_ (Compose (Compose x)) -> Compose (map Compose x)

86
src/Functor/Base.hs Normal file
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@ -0,0 +1,86 @@
module Functor.Base where
import Relation
import Data.Functor.Const (Const (Const), getConst)
import Data.Functor.Identity (Identity (Identity), runIdentity)
import Data.Either (Either (Left, Right))
import Data.Kind (Constraint, Type)
import Data.Maybe (Maybe (Just, Nothing))
type Functor :: (j -> k) -> Constraint
class (Category (Cod f), Category (Dom f)) => Functor (f :: j -> k) where
type Cod f :: Relation k
type Dom f :: Relation j
map :: Dom f x y -> Cod f (f x) (f y)
type Endofunctor :: (k -> k) -> Constraint
type Endofunctor f = (Functor f, Cod f ~ Dom f)
-- This can't be placed in a separate Category.Functor file
-- because the Functor instances for built-in types require Nat.
-- | A natural transformation between functors.
type Nat :: (k -> k -> Type) -> (j -> j -> Type) -> (j -> k) -> (j -> k) -> Type
data Nat dest src f g = (Functor f, Functor g, Dom f ~ src, Dom g ~ src, Cod f ~ dest, Cod g ~ dest) => Nat { runNat :: !(forall a. Object src a -> dest (f a) (g a)) }
instance Reflexive (Nat dest src) where
idL (Nat _) = Nat map
idR (Nat _) = Nat map
instance Transitive (Nat dest src) where
Nat f . Nat g = Nat \x -> f x . g x
instance Functor Maybe where
type Cod Maybe = (->)
type Dom Maybe = (->)
map f = \case
Just x -> Just (f x)
Nothing -> Nothing
instance Functor (Either a) where
type Cod (Either a) = (->)
type Dom (Either a) = (->)
map f = \case
Left y -> Left y
Right x -> Right (f x)
instance Functor ((,) a) where
type Cod ((,) a) = (->)
type Dom ((,) a) = (->)
map f = \(x, y) -> (x, f y)
instance Functor ((->) a) where
type Cod ((->) a) = (->)
type Dom ((->) a) = (->)
map = (.)
instance Functor Either where
type Cod Either = Nat (->) (->)
type Dom Either = (->)
map g = Nat \_ -> \case
Left y -> Left (g y)
Right x -> Right x
instance Functor (,) where
type Cod (,) = Nat (->) (->)
type Dom (,) = (->)
map g = Nat \_ (y, x) -> (g y, x)
instance Functor (->) where
type Cod (->) = Nat (->) (->)
type Dom (->) = Opposite (->)
map (Opposite f) = Nat \_ g -> (g . f)
instance Functor Identity where
type Cod Identity = (->)
type Dom Identity = (->)
map f = Identity . f . runIdentity
instance Functor (Const a :: Type -> Type) where
type Cod (Const a) = (->)
type Dom (Const a) = (->)
map _ = Const . getConst
instance Functor (Const :: Type -> Type -> Type) where
type Cod Const = Nat (->) (->)
type Dom Const = (->)
map f = Nat \_ -> Const . f . getConst

100
src/Functor/Bifunctor.hs Normal file
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@ -0,0 +1,100 @@
{-# LANGUAGE PartialTypeSignatures #-}
{-# LANGUAGE UndecidableInstances #-}
{-# OPTIONS_GHC -Wno-partial-type-signatures #-}
module Functor.Bifunctor where
import Functor.Base
import Functor.Compose
import Relation
import Data.Kind (Constraint, Type)
import Data.Either (Either (Left, Right))
type Uncurry :: (a -> b -> c) -> (a, b) -> c
type family Uncurry
type Uncurry' :: (a -> b -> Type) -> (a, b) -> Type
newtype Uncurry' f ab = Uncurry { getUncurry :: f (Pi1 ab) (Pi2 ab) }
type instance Uncurry = Uncurry'
type UncurryN :: (a -> b -> c -> Type) -> (a, b) -> c -> Type
newtype UncurryN f ab x = UncurryN { getUncurryN :: f (Pi1 ab) (Pi2 ab) x }
type instance Uncurry = UncurryN
instance Functor (Uncurry' (,)) where
type Cod (Uncurry' (,)) = (->)
type Dom (Uncurry' (,)) = ProductRel (->) (->)
map (Product g f) = \(Uncurry (y, x)) -> Uncurry (g y, f x)
instance Functor (Uncurry' Either) where
type Cod (Uncurry' Either) = (->)
type Dom (Uncurry' Either) = ProductRel (->) (->)
map (Product g f) = \(Uncurry sum) -> Uncurry case sum of
Left y -> Left (g y)
Right x -> Right (f x)
instance Functor (Uncurry' (->)) where
type Cod (Uncurry' (->)) = (->)
type Dom (Uncurry' (->)) = ProductRel (Opposite (->)) (->)
map (Product (Opposite f) h) = \(Uncurry g) -> Uncurry (h . g . f)
instance (Functor (Pi1 fg), Functor (Pi2 fg), Dom (Pi2 fg) ~ (->)) => Functor (UncurryN (Compose :: (Type -> Type) -> (Type -> Type) -> Type -> Type) fg) where
type Cod (UncurryN Compose fg) = (->)
type Dom (UncurryN Compose fg) = (->)
map f (UncurryN (Compose x)) = UncurryN (Compose (map (map f) x))
instance Functor (UncurryN (Compose :: (Type -> Type) -> (Type -> Type) -> Type -> Type)) where
type Cod (UncurryN Compose) = Nat (->) (->)
type Dom (UncurryN Compose) = ProductRel (Nat (->) (->)) (Nat (->) (->))
map (Product (Nat g) (Nat f)) = Nat \o (UncurryN (Compose x)) -> UncurryN (Compose (g (map o) (map (f o) x)))
type family ProductPi1 x where
ProductPi1 (ProductRel f g) = f
type family ProductPi2 x where
ProductPi2 (ProductRel f g) = g
type Bifunctor :: (i -> j -> k) -> Constraint
class (Category (Dom1 f), Category (Dom2 f), Category (Cod1 f)) => Bifunctor (f :: i -> j -> k) where
type Dom1 f :: Relation i
type Dom2 f :: Relation j
type Cod1 f :: Relation k
bimap :: Dom1 f w y -> Dom2 f x z -> Cod1 f (f w x) (f y z)
-- The category must be wide, or else the `Object (Dom2 f) z` we have to pass around
-- actually makes this function an instance of bimap with x ~ z.
first :: Wide (Dom2 f) => Dom1 f x y -> Cod1 f (f x z) (f y z)
first g = bimap g id
second :: Wide (Dom1 f) => Dom2 f y z -> Cod1 f (f x y) (f x z)
second f = bimap id f
-- We can't make a generic instance for Bifunctor because we can't constrain a type family,
-- nor do we have existentially quantified type variables, so there's no way to state the
-- `Category r` constraint to make this function (or for that matter, the instance of Functor) work.
defaultBimap :: forall r s f w x y z. (Functor (Uncurry' f), Dom (Uncurry' f) ~ ProductRel r s, Cod (Uncurry' f) ~ (->), Category r, Category s) => r w y -> s x z -> Cod (Uncurry' f) (f w x) (f y z)
defaultBimap g f = getUncurry . (map (Product g f) :: Uncurry' f '(w, x) -> Uncurry' f '(y, z)) . Uncurry
instance Bifunctor (,) where
type Dom1 (,) = (->)
type Dom2 (,) = (->)
type Cod1 (,) = (->)
bimap = defaultBimap
instance Bifunctor Either where
type Dom1 Either = (->)
type Dom2 Either = (->)
type Cod1 Either = (->)
bimap = defaultBimap
instance Bifunctor (->) where
type Dom1 (->) = Opposite (->)
type Dom2 (->) = (->)
type Cod1 (->) = (->)
bimap g f = getUncurry . (map (Product g f) :: Uncurry' (->) '(_, _) -> Uncurry' (->) '(_, _)) . Uncurry
instance Bifunctor (Compose :: (Type -> Type) -> (Type -> Type) -> Type -> Type) where
type Dom1 Compose = Nat (->) (->)
type Dom2 Compose = Nat (->) (->)
type Cod1 Compose = Nat (->) (->)
bimap (Nat g) (Nat f) = Nat \_ (Compose x) -> Compose (map (f id) (g id x))
dimap :: (Bifunctor f, Dom1 f ~ Opposite r) => r x w -> Dom2 f z y -> Cod1 f (f w z) (f x y)
dimap f = bimap (Opposite f)

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{-# LANGUAGE UndecidableInstances #-}
module Functor.Compose where
import Functor.Base
import Relation
data Compose g f x = (Functor g, Functor f, Cod g ~ (->), Dom g ~ Cod f) => Compose { getCompose :: !(g (f x)) }
instance Category (Dom f) => Functor (Compose g f) where
type Cod (Compose g f) = (->)
type Dom (Compose g f) = Dom f
map f (Compose x) = Compose (map (map f) x)

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module Functor.Distributive where
import Functor.Base
import Relation
class Endofunctor f => Distributive f where
distribute :: (Endofunctor g, Cod f ~ Cod g) => Cod f (f (g a)) (g (f a))
collect :: (Endofunctor g, Cod f ~ Cod g) => Cod f a (g b) -> Cod f (f a) (g (f b))
collect f = distribute . map f

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module Functor.Exponent where
import Functor.Bifunctor
import Functor.Identity
import Functor.Product
import Relation
class Bifunctor f => InternalHom (f :: k -> k -> k) where
type Unit' f :: k
compose :: Object (Cod1 f) x -> Object (Cod1 f) y -> Object (Cod1 f) z -> Cod1 f (f y z) (f (f x y) (f x z))
identity :: Object (Cod1 f) x -> Cod1 f (Unit' f) (f x x)
abstract :: Cod1 f x (f (Unit' f) x)
apply :: Cod1 f (f (Unit' f) x) x
class (InternalHom hom, TensorProduct prod, Cod1 prod ~ Cod1 hom, Unit prod ~ Unit' hom) => MonoidalClosed hom prod where
curry :: Cod1 hom (hom (prod x y) z) (hom x (hom y z))
uncurry :: Cod1 hom (hom x (hom y z)) (hom (prod x y) z)
class (MonoidalClosed hom prod) => Exponential hom prod where
eval :: Cod1 hom (prod x (hom x y)) y
instance InternalHom (->) where
type Unit' (->) = ()
compose _ _ _ = (.)
identity _ _ = id
abstract = \x _ -> x
apply = \f -> f ()
instance MonoidalClosed (->) (,) where
curry = \f x y -> f (x, y)
uncurry = \f (x, y) -> f x y
instance Exponential (->) (,) where
eval = \(x, f) -> f x

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{-# LANGUAGE UndecidableInstances #-}
module Functor.Identity where
import Functor.Base
import Functor.Bifunctor
import Functor.Compose
import Relation
import Data.Either (Either (Left, Right))
import Data.Functor.Identity (Identity (Identity, runIdentity))
import Data.Kind (Constraint)
import Data.Void (Void)
type LeftIdentity :: (k -> k -> k) -> Constraint
class Bifunctor f => LeftIdentity (f :: k -> k -> k) where
type UnitL f :: k
unitLI :: Object (Cod1 f) x -> Cod1 f x (f (UnitL f) x)
unitLE :: Object (Cod1 f) x -> Cod1 f (f (UnitL f) x) x
unitLI' :: (LeftIdentity f, Wide (Cod1 f)) => Cod1 f x (f (UnitL f) x)
unitLI' = unitLI id
unitLE' :: (LeftIdentity f, Wide (Cod1 f)) => Cod1 f (f (UnitL f) x) x
unitLE' = unitLE id
type RightIdentity :: (k -> k -> k) -> Constraint
class Bifunctor f => RightIdentity (f :: k -> k -> k) where
type UnitR f :: k
unitRI :: Object (Cod1 f) x -> Cod1 f x (f x (UnitR f))
unitRE :: Object (Cod1 f) x -> Cod1 f (f x (UnitR f)) x
-- TODO: unitRI', unitRE'
type LeftRightIdentity :: (k -> k -> k) -> Constraint
class (LeftIdentity f, RightIdentity f, UnitL f ~ UnitR f) => LeftRightIdentity (f :: k -> k -> k) where
type Unit f :: k
instance (LeftIdentity f, RightIdentity f, UnitL f ~ UnitR f) => LeftRightIdentity f where
type Unit f = UnitL f
instance LeftIdentity (,) where
type UnitL (,) = ()
unitLI _ x = ((), x)
unitLE _ (_, x) = x
instance LeftIdentity Either where
type UnitL Either = Void
unitLI _ x = Right x
unitLE _ (Right x) = x
unitLE _ (Left x) = case x of {}
instance LeftIdentity Compose where
type UnitL Compose = Identity
unitLI (Nat _) = Nat \_ -> Compose . Identity
unitLE (Nat _) = Nat \_ -> runIdentity . getCompose
instance RightIdentity (,) where
type UnitR (,) = ()
unitRI _ x = (x, ())
unitRE _ (x, _) = x
instance RightIdentity Either where
type UnitR Either = Void
unitRI _ x = Left x
unitRE _ (Right x) = case x of {}
unitRE _ (Left x) = x
instance RightIdentity Compose where
type UnitR Compose = Identity
unitRI (Nat _) = Nat \_ -> Compose . map Identity
unitRE (Nat _) = Nat \_ -> map runIdentity . getCompose

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module Functor.Product where
import Functor.Associative
import Functor.Bifunctor
import Functor.Identity
import Relation
import Data.Either (Either (Left, Right))
import Data.Kind (Constraint)
import Data.Void (Void)
-- | A category with a tensor product is a monoidal category.
type TensorProduct :: (k -> k -> k) -> Constraint
type TensorProduct f = (Associative f, LeftRightIdentity f)
class HasTerminal (r :: Relation k) where
type Terminal r :: k
drop :: Object r x -> r x (Terminal r)
class HasInitial (r :: Relation k) where
type Initial r :: k
absurd :: Object r x -> r (Initial r) x
type Product :: (k -> k -> k) -> Constraint
class (TensorProduct f, HasTerminal (Cod1 f), Terminal (Cod1 f) ~ Unit f) => Product f where
projectL :: Object (Cod1 f) x -> Object (Cod1 f) y -> Cod1 f (f x y) x
projectR :: Object (Cod1 f) x -> Object (Cod1 f) y -> Cod1 f (f x y) y
type Coproduct :: (k -> k -> k) -> Constraint
class (TensorProduct f, HasInitial (Cod1 f), Initial (Cod1 f) ~ Unit f) => Coproduct f where
injectL :: Object (Cod1 f) x -> Object (Cod1 f) y -> Cod1 f x (f x y)
injectR :: Object (Cod1 f) x -> Object (Cod1 f) y -> Cod1 f y (f x y)
instance HasTerminal (->) where
type Terminal (->) = ()
drop _ = \_ -> ()
instance HasInitial (->) where
type Initial (->) = Void
absurd _ = \case{}
instance Product (,) where
projectL _ _ = \(x, _) -> x
projectR _ _ = \(_, y) -> y
instance Coproduct Either where
injectL _ _ = Left
injectR _ _ = Right

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module Object.Monoid where
import Functor.Base
import Functor.Bifunctor
import Functor.Compose
import Functor.Identity
import Object.Semigroup
import Data.Data (Proxy (Proxy))
import Data.Kind (Constraint, Type)
type Monoid :: (k -> k -> k) -> k -> Constraint
class Semigroup prod m => Monoid prod m where
empty :: proxy prod -> Dom1 prod (Unit prod) m
type Comonoid :: (k -> k -> k) -> k -> Constraint
class Cosemigroup coprod w => Comonoid coprod w where
destroy :: proxy coprod -> Dom1 coprod w (Unit coprod)
empty' :: (Monoid (,) m) => m
empty' = empty (Proxy :: Proxy (,)) ()
instance Comonoid (,) a where
destroy _ = \_ -> ()
-- | A monad is a monoid in the category of endofunctors.
type Monad :: (Type -> Type) -> Constraint
type Monad m = (Endofunctor m, Monoid Compose m)
-- | A comonad is a comonoid in the category of endofunctors.
type Comonad :: (Type -> Type) -> Constraint
type Comonad m = (Endofunctor m, Comonoid Compose m)
-- TODO: Re-implement bind and cobind in rewrite.

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module Object.Semigroup where
import Functor.Bifunctor
import Functor.Product
import Data.Kind (Constraint)
type Semigroup :: (k -> k -> k) -> k -> Constraint
class TensorProduct prod => Semigroup prod s where
append :: Cod1 prod (prod s s) s
type Cosemigroup :: (k -> k -> k) -> k -> Constraint
class TensorProduct prod => Cosemigroup prod s where
split :: Cod1 prod s (prod s s)
(<>) :: Semigroup (,) s => s -> s -> s
x <> y = append (x, y)
instance Cosemigroup (,) a where
split = \x -> (x, x)

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module Relation
( module Relation.Base
, module Relation.Category
, module Relation.Opposite
, module Relation.Product
, module Relation.Reflexive
, module Relation.Symmetric
, module Relation.Transitive
) where
import Relation.Base
import Relation.Category
import Relation.Opposite
import Relation.Product
import Relation.Reflexive
import Relation.Symmetric
import Relation.Transitive

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module Relation.Base where
import Data.Kind (Type)
type Relation :: Type -> Type
type Relation k = k -> k -> Type
type Object :: Relation i -> i -> Type
type Object r x = r x x

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module Relation.Category (Category) where
import Relation.Base
import Relation.Reflexive
import Relation.Transitive
import Data.Kind (Constraint)
type Category :: Relation k -> Constraint
type Category r = (Reflexive r, Transitive r)

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module Relation.Opposite where
import Relation.Base
import Relation.Reflexive
import Relation.Transitive
type Opposite :: Relation k -> Relation k
newtype Opposite r x y = Opposite { getOpposite :: r y x }
instance Reflexive r => Reflexive (Opposite r) where
idL (Opposite f) = Opposite (idR f)
idR (Opposite f) = Opposite (idL f)
instance Wide r => Wide (Opposite r) where
id = Opposite id
instance Transitive r => Transitive (Opposite r) where
Opposite f . Opposite g = Opposite (g . f)
type Op :: Relation k -> Relation k
type family Op r where
Op (Opposite r) = r
Op r = Opposite r

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module Relation.Product where
import Relation.Base
import Relation.Reflexive
import Relation.Transitive
type Pi1 :: (i, j) -> i
type family Pi1 xy where
Pi1 '(x, _) = x
type Pi2 :: (i, j) -> j
type family Pi2 xy where
Pi2 '(_, y) = y
-- | A product of categories.
type ProductRel :: Relation j -> Relation k -> Relation (j, k)
data ProductRel r s x y = Product (r (Pi1 x) (Pi1 y)) (s (Pi2 x) (Pi2 y))
instance (Reflexive r, Reflexive s) => Reflexive (ProductRel r s) where
idL (Product f g) = Product (idL f) (idL g)
idR (Product f g) = Product (idR f) (idR g)
instance (Wide r, Wide s) => Wide (ProductRel r s) where
id = Product id id
instance (Transitive r, Transitive s) => Transitive (ProductRel r s) where
Product g1 g2 . Product f1 f2 = Product (g1 . f1) (g2 . f2)

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module Relation.Reflexive (Reflexive, Wide, idL, idR, id) where
import Relation.Base
import Data.Kind (Constraint)
type Reflexive :: Relation k -> Constraint
class Reflexive r where
idL :: r x y -> Object r x
default idL :: Wide r => r x y -> Object r x
idL _ = id
idR :: r x y -> Object r y
default idR :: Wide r => r x y -> Object r y
idR _ = id
type Wide :: Relation k -> Constraint
class Reflexive r => Wide r where
id :: r x x
instance Reflexive (->)
instance Wide (->) where { id = \x -> x }

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module Relation.Symmetric (Symmetric, inv) where
import Relation.Base
import Data.Kind (Constraint)
type Symmetric :: Relation k -> Constraint
class Symmetric r where
inv :: r x y -> r y x

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module Relation.Transitive (Transitive, (.)) where
import Relation.Base
import Data.Kind (Constraint)
type Transitive :: Relation k -> Constraint
class Transitive r where
(.) :: r y z -> r x y -> r x z
instance Transitive (->) where
g . f = \x -> g (f x)