CommaCategory.CommaCategory

A comma category is a category where objects are morphisms of another category C and morphisms are commutative squares in C.

For example, see Examples.ExampleCommaCategory.*

Documentation

data CommaObject o1 o2 m3

A CommaObject in the CommaCategory (T|S) is a triplet <e,d,f> where f : T(e) -> S(d).

See Categories for the working mathematician, Saunders Mac Lane, P.46.

Constructors

CommaObject

Fields

• indexSrc :: o1

  e

• indexTgt :: o2

  d

• arrow :: m3

  f

Instances

(Eq o1, Eq o2, Eq m3) => Eq (CommaObject o1 o2 m3)

Defined in CommaCategory.CommaCategory

Methods

(==) :: CommaObject o1 o2 m3 -> CommaObject o1 o2 m3 -> Bool

(/=) :: CommaObject o1 o2 m3 -> CommaObject o1 o2 m3 -> Bool

(Show o1, Show o2, Show m3) => Show (CommaObject o1 o2 m3)

Defined in CommaCategory.CommaCategory

Methods

showsPrec :: Int -> CommaObject o1 o2 m3 -> ShowS

show :: CommaObject o1 o2 m3 -> String

showList :: [CommaObject o1 o2 m3] -> ShowS

(PrettyPrintable o1, PrettyPrintable o2, PrettyPrintable m3) => PrettyPrintable (CommaObject o1 o2 m3)

Defined in CommaCategory.CommaCategory

Methods

pprint :: CommaObject o1 o2 m3 -> String

(Morphism m1 o1, Morphism m2 o2, Eq o1, Eq o2, Eq m3) => Morphism (CommaMorphism o1 o2 m1 m2 m3) (CommaObject o1 o2 m3)

Defined in CommaCategory.CommaCategory

Methods

(@) :: CommaMorphism o1 o2 m1 m2 m3 -> CommaMorphism o1 o2 m1 m2 m3 -> CommaMorphism o1 o2 m1 m2 m3

source :: CommaMorphism o1 o2 m1 m2 m3 -> CommaObject o1 o2 m3

target :: CommaMorphism o1 o2 m1 m2 m3 -> CommaObject o1 o2 m3

(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2, FiniteCategory c3 m3 o3, Morphism m3 o3, Eq m3) => GeneratedFiniteCategory (CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (CommaMorphism o1 o2 m1 m2 m3) (CommaObject o1 o2 m3)

Defined in CommaCategory.CommaCategory

Methods

genAr :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaObject o1 o2 m3 -> CommaObject o1 o2 m3 -> [CommaMorphism o1 o2 m1 m2 m3]

decompose :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaMorphism o1 o2 m1 m2 m3 -> [CommaMorphism o1 o2 m1 m2 m3]

genArrows :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> [CommaMorphism o1 o2 m1 m2 m3]

(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2, FiniteCategory c3 m3 o3, Morphism m3 o3, Eq m3) => FiniteCategory (CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (CommaMorphism o1 o2 m1 m2 m3) (CommaObject o1 o2 m3)

Defined in CommaCategory.CommaCategory

Methods

ob :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> [CommaObject o1 o2 m3]

identity :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaObject o1 o2 m3 -> CommaMorphism o1 o2 m1 m2 m3

ar :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaObject o1 o2 m3 -> CommaObject o1 o2 m3 -> [CommaMorphism o1 o2 m1 m2 m3]

arrows :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> [CommaMorphism o1 o2 m1 m2 m3]

data CommaMorphism o1 o2 m1 m2 m3

A CommaMorphism in the CommaCategory (T|S) is a couple <k,h>.

See Categories for the working mathematician, Saunders Mac Lane, P.46.

Constructors

CommaMorphism

Fields

• srcCM :: CommaObject o1 o2 m3

  The source CommaObject

• tgtCM :: CommaObject o1 o2 m3

  The target CommaObject

• indexAr1 :: m1

  k

• indexAr2 :: m2

  h

Instances

(Eq o1, Eq o2, Eq m3, Eq m1, Eq m2) => Eq (CommaMorphism o1 o2 m1 m2 m3)

Defined in CommaCategory.CommaCategory

Methods

(==) :: CommaMorphism o1 o2 m1 m2 m3 -> CommaMorphism o1 o2 m1 m2 m3 -> Bool

(/=) :: CommaMorphism o1 o2 m1 m2 m3 -> CommaMorphism o1 o2 m1 m2 m3 -> Bool

(Show o1, Show o2, Show m3, Show m1, Show m2) => Show (CommaMorphism o1 o2 m1 m2 m3)

Defined in CommaCategory.CommaCategory

Methods

showsPrec :: Int -> CommaMorphism o1 o2 m1 m2 m3 -> ShowS

show :: CommaMorphism o1 o2 m1 m2 m3 -> String

showList :: [CommaMorphism o1 o2 m1 m2 m3] -> ShowS

(PrettyPrintable m1, PrettyPrintable m2) => PrettyPrintable (CommaMorphism o1 o2 m1 m2 m3)

Defined in CommaCategory.CommaCategory

Methods

pprint :: CommaMorphism o1 o2 m1 m2 m3 -> String

(Morphism m1 o1, Morphism m2 o2, Eq o1, Eq o2, Eq m3) => Morphism (CommaMorphism o1 o2 m1 m2 m3) (CommaObject o1 o2 m3)

Defined in CommaCategory.CommaCategory

Methods

(@) :: CommaMorphism o1 o2 m1 m2 m3 -> CommaMorphism o1 o2 m1 m2 m3 -> CommaMorphism o1 o2 m1 m2 m3

source :: CommaMorphism o1 o2 m1 m2 m3 -> CommaObject o1 o2 m3

target :: CommaMorphism o1 o2 m1 m2 m3 -> CommaObject o1 o2 m3

(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2, FiniteCategory c3 m3 o3, Morphism m3 o3, Eq m3) => GeneratedFiniteCategory (CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (CommaMorphism o1 o2 m1 m2 m3) (CommaObject o1 o2 m3)

Defined in CommaCategory.CommaCategory

Methods

genAr :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaObject o1 o2 m3 -> CommaObject o1 o2 m3 -> [CommaMorphism o1 o2 m1 m2 m3]

decompose :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaMorphism o1 o2 m1 m2 m3 -> [CommaMorphism o1 o2 m1 m2 m3]

genArrows :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> [CommaMorphism o1 o2 m1 m2 m3]

(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2, FiniteCategory c3 m3 o3, Morphism m3 o3, Eq m3) => FiniteCategory (CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (CommaMorphism o1 o2 m1 m2 m3) (CommaObject o1 o2 m3)

Defined in CommaCategory.CommaCategory

Methods

ob :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> [CommaObject o1 o2 m3]

identity :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaObject o1 o2 m3 -> CommaMorphism o1 o2 m1 m2 m3

ar :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaObject o1 o2 m3 -> CommaObject o1 o2 m3 -> [CommaMorphism o1 o2 m1 m2 m3]

arrows :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> [CommaMorphism o1 o2 m1 m2 m3]

data CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3

A CommaCategory is a couple (T|S) with T and S two diagrams with the same target.

See Categories for the working mathematician, Saunders Mac Lane, P.46.

Constructors

CommaCategory

Fields

• leftDiag :: Diagram c1 m1 o1 c3 m3 o3

  T

• rightDiag :: Diagram c2 m2 o2 c3 m3 o3

  S

Instances

(Eq c1, Eq c3, Eq o1, Eq o3, Eq m1, Eq m3, Eq c2, Eq o2, Eq m2) => Eq (CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3)

Defined in CommaCategory.CommaCategory

Methods

(==) :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Bool

(/=) :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Bool

(Show c1, Show c3, Show o1, Show o3, Show m1, Show m3, Show c2, Show o2, Show m2) => Show (CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3)

Defined in CommaCategory.CommaCategory

Methods

showsPrec :: Int -> CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> ShowS

show :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> String

showList :: [CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3] -> ShowS

(PrettyPrintable c1, PrettyPrintable m1, PrettyPrintable o1, PrettyPrintable c2, PrettyPrintable m2, PrettyPrintable o2, PrettyPrintable c3, PrettyPrintable m3, PrettyPrintable o3, FiniteCategory c1 m1 o1, FiniteCategory c2 m2 o2, FiniteCategory c3 m3 o3, Morphism m1 o1, Morphism m2 o2, Morphism m3 o3, Eq m1, Eq o1, Eq m2, Eq o2) => PrettyPrintable (CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3)

Defined in CommaCategory.CommaCategory

Methods

pprint :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> String

(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2, FiniteCategory c3 m3 o3, Morphism m3 o3, Eq m3) => GeneratedFiniteCategory (CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (CommaMorphism o1 o2 m1 m2 m3) (CommaObject o1 o2 m3)

Defined in CommaCategory.CommaCategory

Methods

genAr :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaObject o1 o2 m3 -> CommaObject o1 o2 m3 -> [CommaMorphism o1 o2 m1 m2 m3]

decompose :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaMorphism o1 o2 m1 m2 m3 -> [CommaMorphism o1 o2 m1 m2 m3]

genArrows :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> [CommaMorphism o1 o2 m1 m2 m3]

(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2, FiniteCategory c3 m3 o3, Morphism m3 o3, Eq m3) => FiniteCategory (CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (CommaMorphism o1 o2 m1 m2 m3) (CommaObject o1 o2 m3)

Defined in CommaCategory.CommaCategory

Methods

ob :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> [CommaObject o1 o2 m3]

identity :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaObject o1 o2 m3 -> CommaMorphism o1 o2 m1 m2 m3

ar :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> CommaObject o1 o2 m3 -> CommaObject o1 o2 m3 -> [CommaMorphism o1 o2 m1 m2 m3]

arrows :: CommaCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> [CommaMorphism o1 o2 m1 m2 m3]

mkSliceCategory :: (FiniteCategory c m o, Morphism m o, Eq o) => c -> o -> Maybe (CommaCategory c m o One One One c m o)

Constructs the slice category of a category C over an object o.

Returns Nothing if the object is not in the category.

mkCosliceCategory :: (FiniteCategory c m o, Morphism m o, Eq o) => c -> o -> Maybe (CommaCategory One One One c m o c m o)

Constructs the coslice category of a category C under an object o.

Returns Nothing if the object is not in the category.

mkArrowCategory :: (FiniteCategory c m o, Morphism m o, Eq o) => c -> CommaCategory c m o c m o c m o

Constructs the arrow category of a category C.