Copyright | Guillaume Sabbagh 2022 |
---|---|
License | GPL-3 |
Maintainer | guillaumesabbagh@protonmail.com |
Stability | experimental |
Portability | portable |
Safe Haskell | Safe-Inferred |
Language | Haskell2010 |
Math.Categories.FunctorCategory
Description
A FunctorCategory
D^C (also written [C,D]) where C is a FiniteCategory
and D is a Category
has Diagram
s F : C -> D
as objects and NaturalTransformation
s between them as morphisms. NaturalTransformation
s compose vertically in this category. See the operator (<=@<=
) for horizontal composition.
A Diagram
is a heterogeneous functor, meaning that the source category might be different from the target category. See FinCat
for a homogeneous ones.
Diagram
s are objects in a FunctorCategory
, they therefore can not be composed with the usual operator (@
). See (<-@<-
) for composing Diagram
s.
Beware that source
and target
are not defined on Diagram
because it is not a Morphism
, use src
and tgt
instead. Also note that a Diagram
does not need to contain the mapping of all morphisms from the source category, it may only contain a mapping for the generating morphisms of the source category.
You can also do left and right whiskering with the operators (<=@<-
) and (<-@<=
).
A FunctorCategory
is a FiniteCategory
if the source and target category are finite, but it is only a Category
if the target category is not finite.
All operators defined in this module respect the following convention: a "->" arrow represent a functor and a "=>" represent a natural transformation. For example (<-@<=
) allows to compose a natural transformation (the "<=" arrow) with a functor (the "<-" arrow), note that composition is always read from right to left.
Synopsis
- data Diagram c1 m1 o1 c2 m2 o2 = Diagram {}
- data DiagramError c1 m1 o1 c2 m2 o2
- checkFiniteDiagram :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2) => Diagram c1 m1 o1 c2 m2 o2 -> Maybe (DiagramError c1 m1 o1 c2 m2 o2)
- checkDiagram :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, Category c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2) => Diagram c1 m1 o1 c2 m2 o2 -> Maybe (DiagramError c1 m1 o1 c2 m2 o2)
- diagram :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2) => c1 -> c2 -> Map o1 o2 -> Map m1 m2 -> Either (DiagramError c1 m1 o1 c2 m2 o2) (Diagram c1 m1 o1 c2 m2 o2)
- finiteDiagram :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2) => c1 -> c2 -> Map o1 o2 -> Map m1 m2 -> Either (DiagramError c1 m1 o1 c2 m2 o2) (Diagram c1 m1 o1 c2 m2 o2)
- (->$) :: Eq o1 => Diagram c1 m1 o1 c2 m2 o2 -> o1 -> o2
- (->£) :: (Category c1 m1 o1, Morphism m1 o1, Morphism m2 o2, Eq m1) => Diagram c1 m1 o1 c2 m2 o2 -> m1 -> m2
- (<-@<-) :: (Category c1 m1 o1, Morphism m1 o1, Eq m1, Category c2 m2 o2, Morphism m2 o2, Eq o2, Eq m2, Morphism m3 o3) => Diagram c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c3 m3 o3
- selectObject :: (Category c m o, Morphism m o, Eq o) => c -> o -> Diagram One One One c m o
- constantDiagram :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Category c2 m2 o2, Morphism m2 o2) => c1 -> c2 -> o2 -> Diagram c1 m1 o1 c2 m2 o2
- discreteDiagram :: (Category c m o, Morphism m o, Eq o) => c -> [o] -> Diagram (DiscreteCategory Int) (DiscreteMorphism Int) Int c m o
- parallelDiagram :: (Category c m o, Morphism m o, Eq o) => c -> m -> m -> Diagram Parallel ParallelAr ParallelOb c m o
- insertionFunctor1 :: (Category c m o, Morphism m o, Eq o) => FullSubcategory c m o -> Diagram (FullSubcategory c m o) m o c m o
- insertionFunctor2 :: (Category c m o, Morphism m o, Eq o) => InheritedFullSubcategory c m o -> Diagram (InheritedFullSubcategory c m o) m o c m o
- completeDiagram :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, Category c2 m2 o2, Morphism m2 o2) => Diagram c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2
- pickRandomDiagram :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, RandomGen g) => c1 -> c2 -> g -> (Diagram c1 m1 o1 c2 m2 o2, g)
- inverseDiagram :: (FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2, FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1) => Diagram c1 m1 o1 c2 m2 o2 -> Either (DiagramError c2 m2 o2 c1 m1 o1) (Diagram c2 m2 o2 c1 m1 o1)
- unsafeInverseDiagram :: Diagram c1 m1 o1 c2 m2 o2 -> Diagram c2 m2 o2 c1 m1 o1
- data NaturalTransformation c1 m1 o1 c2 m2 o2
- components :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> Map o1 m2
- data NaturalTransformationError c1 m1 o1 c2 m2 o2
- checkNaturalTransformation :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, Morphism m2 o2, Eq c2, Eq m2, Eq o2) => NaturalTransformation c1 m1 o1 c2 m2 o2 -> Maybe (NaturalTransformationError c1 m1 o1 c2 m2 o2)
- naturalTransformation :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, Morphism m2 o2, Eq c2, Eq m2, Eq o2) => Diagram c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Map o1 m2 -> Either (NaturalTransformationError c1 m1 o1 c2 m2 o2) (NaturalTransformation c1 m1 o1 c2 m2 o2)
- unsafeNaturalTransformation :: Diagram c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Map o1 m2 -> NaturalTransformation c1 m1 o1 c2 m2 o2
- (=>$) :: Eq o1 => NaturalTransformation c1 m1 o1 c2 m2 o2 -> o1 -> m2
- (<=@<=) :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => NaturalTransformation c2 m2 o2 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c3 m3 o3
- horizontalComposition :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => NaturalTransformation c2 m2 o2 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c3 m3 o3
- (<=@<-) :: (Category c1 m1 o1, Morphism m1 o1, Eq m1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => NaturalTransformation c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c3 m3 o3
- leftWhiskering :: (Category c1 m1 o1, Morphism m1 o1, Eq m1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => NaturalTransformation c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c3 m3 o3
- (<-@<=) :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, Category c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, Morphism m3 o3) => Diagram c2 m2 o2 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c3 m3 o3
- rightWhiskering :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, Category c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, Morphism m3 o3) => Diagram c2 m2 o2 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c3 m3 o3
- data FunctorCategory c1 m1 o1 c2 m2 o2 = FunctorCategory c1 c2
- data PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 = PrecomposedFunctorCategory (Diagram c1 m1 o1 c2 m2 o2) c3
- data PostcomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 = PostcomposedFunctorCategory (Diagram c2 m2 o2 c3 m3 o3) c1
Diagram
data Diagram c1 m1 o1 c2 m2 o2 Source #
A Diagram
is a functor from a FiniteCategory
to a Category
.
A Diagram
can have a source category and a target category with different types. It must obey the following rules :
funct ->$ (source f) == source (funct ->£ f)
funct ->$ (target f) == target (funct ->£ f)
funct ->£ (f @ g) = (funct ->£ f) @ (funct ->£ g)
funct ->£ (identity a) = identity (funct ->$ a)
Diagram
is not private because we can't always check functoriality if the target category is infinite.
However it is recommanded to use the smart constructors diagram
or finiteDiagram
which check the structure of the Diagram
at construction. See also the useful function completeDiagram
.
You can omit the mapping of generated morphisms of the source category.
Constructors
Diagram | |
Instances
(Category c m o, Morphism m o, Eq m, Eq o) => Morphism (FinFunctor c m o) c Source # | |
Defined in Math.Categories.FinCat Methods (@) :: FinFunctor c m o -> FinFunctor c m o -> FinFunctor c m o Source # (@?) :: FinFunctor c m o -> FinFunctor c m o -> Maybe (FinFunctor c m o) Source # source :: FinFunctor c m o -> c Source # target :: FinFunctor c m o -> c Source # | |
(FiniteCategory c m o, Morphism m o, Eq c, Eq m, Eq o) => Category (FinCat c m o) (FinFunctor c m o) c Source # | |
Defined in Math.Categories.FinCat Methods identity :: FinCat c m o -> c -> FinFunctor c m o Source # ar :: FinCat c m o -> c -> c -> Set (FinFunctor c m o) Source # genAr :: FinCat c m o -> c -> c -> Set (FinFunctor c m o) Source # decompose :: FinCat c m o -> FinFunctor c m o -> [FinFunctor c m o] Source # | |
(FiniteCategory c m o, Morphism m o, Eq c, Eq m, Eq o, FiniteCategory cIndex mIndex oIndex, Morphism mIndex oIndex, Eq oIndex, Eq mIndex) => CocompleteCategory (FinCat c m o) (FinFunctor c m o) c (FinCat (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (FinFunctor (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) cIndex mIndex oIndex Source # | Note that computing a |
Defined in Math.FiniteCategories.ColimitCategory Methods colimit :: Diagram cIndex mIndex oIndex (FinCat c m o) (FinFunctor c m o) c -> Cocone cIndex mIndex oIndex (FinCat (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (FinFunctor (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) Source # coprojectBase :: Diagram cIndex mIndex oIndex (FinCat c m o) (FinFunctor c m o) c -> Diagram (FinCat c m o) (FinFunctor c m o) c (FinCat (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (FinFunctor (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) (CGMorphism (Colimit oIndex o) (Colimit oIndex m)) (Colimit oIndex o)) (CompositionGraph (Colimit oIndex o) (Colimit oIndex m)) Source # | |
(FiniteCategory c m o, Morphism m o, Eq c, Eq m, Eq o, FiniteCategory cIndex mIndex oIndex, Morphism mIndex oIndex, Eq cIndex, Eq mIndex, Eq oIndex) => CompleteCategory (FinCat c m o) (FinFunctor c m o) c (FinCat (LimitCategory cIndex mIndex oIndex c m o) (Limit oIndex m) (Limit oIndex o)) (FinFunctor (LimitCategory cIndex mIndex oIndex c m o) (Limit oIndex m) (Limit oIndex o)) (LimitCategory cIndex mIndex oIndex c m o) cIndex mIndex oIndex Source # | |
Defined in Math.FiniteCategories.LimitCategory Methods limit :: Diagram cIndex mIndex oIndex (FinCat c m o) (FinFunctor c m o) c -> Cone cIndex mIndex oIndex (FinCat (LimitCategory cIndex mIndex oIndex c m o) (Limit oIndex m) (Limit oIndex o)) (FinFunctor (LimitCategory cIndex mIndex oIndex c m o) (Limit oIndex m) (Limit oIndex o)) (LimitCategory cIndex mIndex oIndex c m o) Source # projectBase :: Diagram cIndex mIndex oIndex (FinCat c m o) (FinFunctor c m o) c -> Diagram (FinCat c m o) (FinFunctor c m o) c (FinCat (LimitCategory cIndex mIndex oIndex c m o) (Limit oIndex m) (Limit oIndex o)) (FinFunctor (LimitCategory cIndex mIndex oIndex c m o) (Limit oIndex m) (Limit oIndex o)) (LimitCategory cIndex mIndex oIndex c m o) Source # | |
(Eq n, Eq e) => HasCoequalizers (FinCat (CompositionGraph n e) (CGMorphism n e) n) (FinFunctor (CompositionGraph n e) (CGMorphism n e) n) (CompositionGraph n e) Source # | |
Defined in Math.FiniteCategories.ColimitCategory Methods coequalize :: Diagram Parallel ParallelAr ParallelOb (FinCat (CompositionGraph n e) (CGMorphism n e) n) (FinFunctor (CompositionGraph n e) (CGMorphism n e) n) (CompositionGraph n e) -> Cocone Parallel ParallelAr ParallelOb (FinCat (CompositionGraph n e) (CGMorphism n e) n) (FinFunctor (CompositionGraph n e) (CGMorphism n e) n) (CompositionGraph n e) Source # | |
(Eq e, Eq n, Eq oIndex) => HasCoproducts (FinCat (CompositionGraph n e) (CGMorphism n e) n) (FinFunctor (CompositionGraph n e) (CGMorphism n e) n) (CompositionGraph n e) (FinCat (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) (CGMorphism (Colimit oIndex n) (Colimit oIndex e)) (Colimit oIndex n)) (FinFunctor (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) (CGMorphism (Colimit oIndex n) (Colimit oIndex e)) (Colimit oIndex n)) (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) oIndex Source # | |
Defined in Math.FiniteCategories.ColimitCategory Methods coproduct :: Diagram (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinCat (CompositionGraph n e) (CGMorphism n e) n) (FinFunctor (CompositionGraph n e) (CGMorphism n e) n) (CompositionGraph n e) -> Cocone (DiscreteCategory oIndex) (DiscreteMorphism oIndex) oIndex (FinCat (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) (CGMorphism (Colimit oIndex n) (Colimit oIndex e)) (Colimit oIndex n)) (FinFunctor (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) (CGMorphism (Colimit oIndex n) (Colimit oIndex e)) (Colimit oIndex n)) (CompositionGraph (Colimit oIndex n) (Colimit oIndex e)) Source # | |
(PrettyPrint c1, PrettyPrint c2, PrettyPrint o1, PrettyPrint o2, PrettyPrint m1, PrettyPrint m2, Eq o1, Eq o2, Eq m1, Eq m2) => PrettyPrint (Diagram c1 m1 o1 c2 m2 o2) Source # | |
(Simplifiable c1, Simplifiable c2, Simplifiable o1, Simplifiable o2, Simplifiable m1, Simplifiable m2, Eq o1, Eq m1) => Simplifiable (Diagram c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory | |
Generic (Diagram c1 m1 o1 c2 m2 o2) Source # | |
(Show c1, Show c2, Show o1, Show o2, Show m1, Show m2) => Show (Diagram c1 m1 o1 c2 m2 o2) Source # | |
(Eq c1, Eq m1, Eq o1, Eq c2, Eq m2, Eq o2, FiniteCategory c1 m1 o1, Morphism m1 o1) => Eq (Diagram c1 m1 o1 c2 m2 o2) Source # | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, Morphism m2 o2) => Morphism (NaturalTransformation c1 m1 o1 c2 m2 o2) (Diagram c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory Methods (@) :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 Source # (@?) :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> Maybe (NaturalTransformation c1 m1 o1 c2 m2 o2) Source # source :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 Source # target :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 Source # | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, Category c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2) => Category (FunctorCategory c1 m1 o1 c2 m2 o2) (NaturalTransformation c1 m1 o1 c2 m2 o2) (Diagram c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory Methods identity :: FunctorCategory c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 Source # ar :: FunctorCategory c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Set (NaturalTransformation c1 m1 o1 c2 m2 o2) Source # genAr :: FunctorCategory c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Set (NaturalTransformation c1 m1 o1 c2 m2 o2) Source # decompose :: FunctorCategory c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> [NaturalTransformation c1 m1 o1 c2 m2 o2] Source # | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2) => FiniteCategory (FunctorCategory c1 m1 o1 c2 m2 o2) (NaturalTransformation c1 m1 o1 c2 m2 o2) (Diagram c1 m1 o1 c2 m2 o2) Source # | A |
Defined in Math.Categories.FunctorCategory | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, Category c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, Category c3 m3 o3, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => Category (PostcomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (NaturalTransformation c1 m1 o1 c3 m3 o3) (Diagram c1 m1 o1 c3 m3 o3) Source # | |
Defined in Math.Categories.FunctorCategory Methods identity :: PostcomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c3 m3 o3 Source # ar :: PostcomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Set (NaturalTransformation c1 m1 o1 c3 m3 o3) Source # genAr :: PostcomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Set (NaturalTransformation c1 m1 o1 c3 m3 o3) Source # decompose :: PostcomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c3 m3 o3 -> [NaturalTransformation c1 m1 o1 c3 m3 o3] Source # | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, Category c3 m3 o3, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => Category (PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (NaturalTransformation c1 m1 o1 c3 m3 o3) (Diagram c1 m1 o1 c3 m3 o3) Source # | |
Defined in Math.Categories.FunctorCategory Methods identity :: PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c3 m3 o3 Source # ar :: PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Set (NaturalTransformation c1 m1 o1 c3 m3 o3) Source # genAr :: PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Set (NaturalTransformation c1 m1 o1 c3 m3 o3) Source # decompose :: PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c3 m3 o3 -> [NaturalTransformation c1 m1 o1 c3 m3 o3] Source # | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, FiniteCategory c3 m3 o3, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => FiniteCategory (PostcomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (NaturalTransformation c1 m1 o1 c3 m3 o3) (Diagram c1 m1 o1 c3 m3 o3) Source # | A |
Defined in Math.Categories.FunctorCategory | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, FiniteCategory c3 m3 o3, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => FiniteCategory (PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (NaturalTransformation c1 m1 o1 c3 m3 o3) (Diagram c1 m1 o1 c3 m3 o3) Source # | A |
Defined in Math.Categories.FunctorCategory | |
type Rep (Diagram c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory type Rep (Diagram c1 m1 o1 c2 m2 o2) = D1 ('MetaData "Diagram" "Math.Categories.FunctorCategory" "FiniteCategories-0.6.0.1-L2v014CZIYe4CTCeXkyH5k" 'False) (C1 ('MetaCons "Diagram" 'PrefixI 'True) ((S1 ('MetaSel ('Just "src") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 c1) :*: S1 ('MetaSel ('Just "tgt") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 c2)) :*: (S1 ('MetaSel ('Just "omap") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Map o1 o2)) :*: S1 ('MetaSel ('Just "mmap") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Map m1 m2))))) |
Check diagram structure
data DiagramError c1 m1 o1 c2 m2 o2 Source #
A datatype to represent a malformation of a Diagram
.
Instances
checkFiniteDiagram :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2) => Diagram c1 m1 o1 c2 m2 o2 -> Maybe (DiagramError c1 m1 o1 c2 m2 o2) Source #
Check wether the properties of a Diagram
are respected where the source and target category are finite.
checkDiagram :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, Category c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2) => Diagram c1 m1 o1 c2 m2 o2 -> Maybe (DiagramError c1 m1 o1 c2 m2 o2) Source #
Check wether the properties of a Diagram
are respected where the source or target category is infinite.
diagram :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2) => c1 -> c2 -> Map o1 o2 -> Map m1 m2 -> Either (DiagramError c1 m1 o1 c2 m2 o2) (Diagram c1 m1 o1 c2 m2 o2) Source #
Smart constructor of a Diagram
. See finiteDiagram
for constructing finite Diagram
s.
finiteDiagram :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2) => c1 -> c2 -> Map o1 o2 -> Map m1 m2 -> Either (DiagramError c1 m1 o1 c2 m2 o2) (Diagram c1 m1 o1 c2 m2 o2) Source #
Smart constructor of a finite Diagram
.
Operators
(->$) :: Eq o1 => Diagram c1 m1 o1 c2 m2 o2 -> o1 -> o2 Source #
Apply a Diagram
on an object of the source category.
(->£) :: (Category c1 m1 o1, Morphism m1 o1, Morphism m2 o2, Eq m1) => Diagram c1 m1 o1 c2 m2 o2 -> m1 -> m2 Source #
Apply a Diagram
on a morphism of the source category.
(<-@<-) :: (Category c1 m1 o1, Morphism m1 o1, Eq m1, Category c2 m2 o2, Morphism m2 o2, Eq o2, Eq m2, Morphism m3 o3) => Diagram c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c3 m3 o3 Source #
Compose two Diagram
s.
Usual diagrams
selectObject :: (Category c m o, Morphism m o, Eq o) => c -> o -> Diagram One One One c m o Source #
Construct a Diagram
selecting an object in a category.
There is no check that the object belongs in the category.
constantDiagram :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Category c2 m2 o2, Morphism m2 o2) => c1 -> c2 -> o2 -> Diagram c1 m1 o1 c2 m2 o2 Source #
Construct a constant Diagram
on an object of the target Category
given an indexing FiniteCategory
.
There is no check that the object belongs in the category.
discreteDiagram :: (Category c m o, Morphism m o, Eq o) => c -> [o] -> Diagram (DiscreteCategory Int) (DiscreteMorphism Int) Int c m o Source #
parallelDiagram :: (Category c m o, Morphism m o, Eq o) => c -> m -> m -> Diagram Parallel ParallelAr ParallelOb c m o Source #
Insertion diagrams for subcategories
insertionFunctor1 :: (Category c m o, Morphism m o, Eq o) => FullSubcategory c m o -> Diagram (FullSubcategory c m o) m o c m o Source #
The insertion functor from the FullSubcategory
to the original category.
insertionFunctor2 :: (Category c m o, Morphism m o, Eq o) => InheritedFullSubcategory c m o -> Diagram (InheritedFullSubcategory c m o) m o c m o Source #
The insertion functor from the InheritedFullSubcategory
to the original category.
Diagram construction helper
completeDiagram :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1, Category c2 m2 o2, Morphism m2 o2) => Diagram c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 Source #
Complete a partial Diagram
by adding mapping on objects from mapping of arrows and mapping on identities.
Does not check the structure of the resulting Diagram
, you can use checkFiniteDiagram
or checkDiagram
to check afterwards.
pickRandomDiagram :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, RandomGen g) => c1 -> c2 -> g -> (Diagram c1 m1 o1 c2 m2 o2, g) Source #
Choose a random diagram in the functor category of an index category and an image category.
Other diagram functions
inverseDiagram :: (FiniteCategory c2 m2 o2, Morphism m2 o2, Eq m2, Eq o2, FiniteCategory c1 m1 o1, Morphism m1 o1, Eq m1, Eq o1) => Diagram c1 m1 o1 c2 m2 o2 -> Either (DiagramError c2 m2 o2 c1 m1 o1) (Diagram c2 m2 o2 c1 m1 o1) Source #
Return the inverse of a finite Diagram
if possible, return a DiagramError
otherwise. Note that this function fails almost all the time if the mapping of morphisms contains generators only (it would only work if all the generators are in the image of the diagram).
unsafeInverseDiagram :: Diagram c1 m1 o1 c2 m2 o2 -> Diagram c2 m2 o2 c1 m1 o1 Source #
Return the inverse of a Diagram
without checking the structure of the returned Diagram
. See inverseDiagram
for the safe version. Note that this function fails almost all the time if the mapping of morphisms contains generators only (it would only work if all the generators are in the image of the diagram).
Natural transformation
data NaturalTransformation c1 m1 o1 c2 m2 o2 Source #
A NaturalTransformation
between two Diagram
s from C to D is a mapping from objects of C to morphisms of D such that naturality is respected. C must be a FiniteCategory
because we need its objects in the mapping of a NaturalTransformation
.
Formally, let F and G be functors, and eta : Ob(C) -> Ar(D). The following properties should be respected :
source F = source G
target F = target G
(eta =>$ target f) @ (F ->£ f) = (G ->£ f) @ (eta =>$ source f)
NaturalTransformation
is private, use the smart constructor naturalTransformation
to instantiate it.
Instances
(PrettyPrint c1, PrettyPrint c2, PrettyPrint o1, PrettyPrint o2, PrettyPrint m1, PrettyPrint m2, Eq o1, Eq o2, Eq m1, Eq m2) => PrettyPrint (NaturalTransformation c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory Methods pprint :: Int -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> String Source # pprintWithIndentations :: Int -> Int -> String -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> String Source # pprintIndent :: Int -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> String Source # | |
(Simplifiable c1, Simplifiable c2, Simplifiable o1, Simplifiable o2, Simplifiable m1, Simplifiable m2, Eq o1, Eq m1) => Simplifiable (NaturalTransformation c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory Methods simplify :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 # | |
Generic (NaturalTransformation c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory Associated Types type Rep (NaturalTransformation c1 m1 o1 c2 m2 o2) :: Type -> Type # Methods from :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> Rep (NaturalTransformation c1 m1 o1 c2 m2 o2) x # to :: Rep (NaturalTransformation c1 m1 o1 c2 m2 o2) x -> NaturalTransformation c1 m1 o1 c2 m2 o2 # | |
(Show c1, Show m1, Show o1, Show c2, Show m2, Show o2) => Show (NaturalTransformation c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory Methods showsPrec :: Int -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> ShowS # show :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> String # showList :: [NaturalTransformation c1 m1 o1 c2 m2 o2] -> ShowS # | |
(Eq c1, Eq m1, Eq o1, Eq c2, Eq m2, Eq o2, FiniteCategory c1 m1 o1, Morphism m1 o1) => Eq (NaturalTransformation c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory Methods (==) :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> Bool # (/=) :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> Bool # | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, Morphism m2 o2) => Morphism (NaturalTransformation c1 m1 o1 c2 m2 o2) (Diagram c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory Methods (@) :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 Source # (@?) :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> Maybe (NaturalTransformation c1 m1 o1 c2 m2 o2) Source # source :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 Source # target :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 Source # | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, Category c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2) => Category (FunctorCategory c1 m1 o1 c2 m2 o2) (NaturalTransformation c1 m1 o1 c2 m2 o2) (Diagram c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory Methods identity :: FunctorCategory c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 Source # ar :: FunctorCategory c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Set (NaturalTransformation c1 m1 o1 c2 m2 o2) Source # genAr :: FunctorCategory c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Set (NaturalTransformation c1 m1 o1 c2 m2 o2) Source # decompose :: FunctorCategory c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> [NaturalTransformation c1 m1 o1 c2 m2 o2] Source # | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2) => FiniteCategory (FunctorCategory c1 m1 o1 c2 m2 o2) (NaturalTransformation c1 m1 o1 c2 m2 o2) (Diagram c1 m1 o1 c2 m2 o2) Source # | A |
Defined in Math.Categories.FunctorCategory | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, Category c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, Category c3 m3 o3, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => Category (PostcomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (NaturalTransformation c1 m1 o1 c3 m3 o3) (Diagram c1 m1 o1 c3 m3 o3) Source # | |
Defined in Math.Categories.FunctorCategory Methods identity :: PostcomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c3 m3 o3 Source # ar :: PostcomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Set (NaturalTransformation c1 m1 o1 c3 m3 o3) Source # genAr :: PostcomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Set (NaturalTransformation c1 m1 o1 c3 m3 o3) Source # decompose :: PostcomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c3 m3 o3 -> [NaturalTransformation c1 m1 o1 c3 m3 o3] Source # | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, Category c3 m3 o3, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => Category (PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (NaturalTransformation c1 m1 o1 c3 m3 o3) (Diagram c1 m1 o1 c3 m3 o3) Source # | |
Defined in Math.Categories.FunctorCategory Methods identity :: PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c3 m3 o3 Source # ar :: PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Set (NaturalTransformation c1 m1 o1 c3 m3 o3) Source # genAr :: PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Set (NaturalTransformation c1 m1 o1 c3 m3 o3) Source # decompose :: PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c3 m3 o3 -> [NaturalTransformation c1 m1 o1 c3 m3 o3] Source # | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, FiniteCategory c3 m3 o3, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => FiniteCategory (PostcomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (NaturalTransformation c1 m1 o1 c3 m3 o3) (Diagram c1 m1 o1 c3 m3 o3) Source # | A |
Defined in Math.Categories.FunctorCategory | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, FiniteCategory c3 m3 o3, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => FiniteCategory (PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (NaturalTransformation c1 m1 o1 c3 m3 o3) (Diagram c1 m1 o1 c3 m3 o3) Source # | A |
Defined in Math.Categories.FunctorCategory | |
type Rep (NaturalTransformation c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory type Rep (NaturalTransformation c1 m1 o1 c2 m2 o2) = D1 ('MetaData "NaturalTransformation" "Math.Categories.FunctorCategory" "FiniteCategories-0.6.0.1-L2v014CZIYe4CTCeXkyH5k" 'False) (C1 ('MetaCons "NaturalTransformation" 'PrefixI 'True) (S1 ('MetaSel ('Just "srcNT") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Diagram c1 m1 o1 c2 m2 o2)) :*: (S1 ('MetaSel ('Just "tgtNT") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Diagram c1 m1 o1 c2 m2 o2)) :*: S1 ('MetaSel ('Just "components") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Map o1 m2))))) |
Getter
components :: NaturalTransformation c1 m1 o1 c2 m2 o2 -> Map o1 m2 Source #
The components
Check structure
data NaturalTransformationError c1 m1 o1 c2 m2 o2 Source #
A datatype to represent a malformation of a NaturalTransformation
.
Instances
checkNaturalTransformation :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, Morphism m2 o2, Eq c2, Eq m2, Eq o2) => NaturalTransformation c1 m1 o1 c2 m2 o2 -> Maybe (NaturalTransformationError c1 m1 o1 c2 m2 o2) Source #
Check wether the structure of a NaturalTransformation
is respected.
naturalTransformation :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, Morphism m2 o2, Eq c2, Eq m2, Eq o2) => Diagram c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Map o1 m2 -> Either (NaturalTransformationError c1 m1 o1 c2 m2 o2) (NaturalTransformation c1 m1 o1 c2 m2 o2) Source #
The smart constructor of NaturalTransformation
. Checks wether the structure is correct.
unsafeNaturalTransformation :: Diagram c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Map o1 m2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 Source #
Unsafe constructor of NaturalTransformation
for performance purposes. It does not check the structure of the NaturalTransformation
.
Use this constructor only if the NaturalTransformation
is necessarily well formed.
Operators
(=>$) :: Eq o1 => NaturalTransformation c1 m1 o1 c2 m2 o2 -> o1 -> m2 Source #
Apply a NaturalTransformation
on an object of the source Diagram
.
(<=@<=) :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => NaturalTransformation c2 m2 o2 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c3 m3 o3 Source #
Compose horizontally NaturalTransformation
s.
horizontalComposition :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => NaturalTransformation c2 m2 o2 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c3 m3 o3 Source #
Alias of (<=@<=
).
(<=@<-) :: (Category c1 m1 o1, Morphism m1 o1, Eq m1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => NaturalTransformation c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c3 m3 o3 Source #
Left whiskering allows to compose a Diagram
with a NaturalTransformation
.
leftWhiskering :: (Category c1 m1 o1, Morphism m1 o1, Eq m1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => NaturalTransformation c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c3 m3 o3 Source #
Alias of (<=@<-
).
(<-@<=) :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, Category c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, Morphism m3 o3) => Diagram c2 m2 o2 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c3 m3 o3 Source #
Right whiskering allows to compose a NaturalTransformation
with a Diagram
.
rightWhiskering :: (FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, Category c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, Morphism m3 o3) => Diagram c2 m2 o2 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c3 m3 o3 Source #
Alias of (<-@<=
).
Functor categories
data FunctorCategory c1 m1 o1 c2 m2 o2 Source #
A FunctorCategory
D^C where C is a FiniteCategory
and D is a Category
has Diagram
s F : C -> D
as objects and NaturalTransformation
s between them as morphisms. NaturalTransformation
s compose vertically in this category.
Constructors
FunctorCategory c1 c2 |
Instances
(PrettyPrint c1, PrettyPrint c2) => PrettyPrint (FunctorCategory c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory Methods pprint :: Int -> FunctorCategory c1 m1 o1 c2 m2 o2 -> String Source # pprintWithIndentations :: Int -> Int -> String -> FunctorCategory c1 m1 o1 c2 m2 o2 -> String Source # pprintIndent :: Int -> FunctorCategory c1 m1 o1 c2 m2 o2 -> String Source # | |
(Simplifiable c1, Simplifiable c2) => Simplifiable (FunctorCategory c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory Methods simplify :: FunctorCategory c1 m1 o1 c2 m2 o2 -> FunctorCategory c1 m1 o1 c2 m2 o2 # | |
Generic (FunctorCategory c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory Associated Types type Rep (FunctorCategory c1 m1 o1 c2 m2 o2) :: Type -> Type # Methods from :: FunctorCategory c1 m1 o1 c2 m2 o2 -> Rep (FunctorCategory c1 m1 o1 c2 m2 o2) x # to :: Rep (FunctorCategory c1 m1 o1 c2 m2 o2) x -> FunctorCategory c1 m1 o1 c2 m2 o2 # | |
(Show c1, Show c2) => Show (FunctorCategory c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory Methods showsPrec :: Int -> FunctorCategory c1 m1 o1 c2 m2 o2 -> ShowS # show :: FunctorCategory c1 m1 o1 c2 m2 o2 -> String # showList :: [FunctorCategory c1 m1 o1 c2 m2 o2] -> ShowS # | |
(Eq c1, Eq c2) => Eq (FunctorCategory c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory Methods (==) :: FunctorCategory c1 m1 o1 c2 m2 o2 -> FunctorCategory c1 m1 o1 c2 m2 o2 -> Bool # (/=) :: FunctorCategory c1 m1 o1 c2 m2 o2 -> FunctorCategory c1 m1 o1 c2 m2 o2 -> Bool # | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, Category c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2) => Category (FunctorCategory c1 m1 o1 c2 m2 o2) (NaturalTransformation c1 m1 o1 c2 m2 o2) (Diagram c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory Methods identity :: FunctorCategory c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 Source # ar :: FunctorCategory c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Set (NaturalTransformation c1 m1 o1 c2 m2 o2) Source # genAr :: FunctorCategory c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Diagram c1 m1 o1 c2 m2 o2 -> Set (NaturalTransformation c1 m1 o1 c2 m2 o2) Source # decompose :: FunctorCategory c1 m1 o1 c2 m2 o2 -> NaturalTransformation c1 m1 o1 c2 m2 o2 -> [NaturalTransformation c1 m1 o1 c2 m2 o2] Source # | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2) => FiniteCategory (FunctorCategory c1 m1 o1 c2 m2 o2) (NaturalTransformation c1 m1 o1 c2 m2 o2) (Diagram c1 m1 o1 c2 m2 o2) Source # | A |
Defined in Math.Categories.FunctorCategory | |
type Rep (FunctorCategory c1 m1 o1 c2 m2 o2) Source # | |
Defined in Math.Categories.FunctorCategory type Rep (FunctorCategory c1 m1 o1 c2 m2 o2) = D1 ('MetaData "FunctorCategory" "Math.Categories.FunctorCategory" "FiniteCategories-0.6.0.1-L2v014CZIYe4CTCeXkyH5k" 'False) (C1 ('MetaCons "FunctorCategory" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 c1) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 c2))) |
data PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 Source #
A FunctorCategory
D^C precomposed by a functor F : B -> C
where B and C are FiniteCategory
and D is a Category
.
It has Diagram
s G o F : B -> D
as objects and NaturalTransformation
s between them as morphisms. NaturalTransformation
s compose vertically in this category.
Constructors
PrecomposedFunctorCategory (Diagram c1 m1 o1 c2 m2 o2) c3 |
Instances
(Eq o1, Eq o2, Eq m1, Eq m2, PrettyPrint c1, PrettyPrint c2, PrettyPrint o1, PrettyPrint o2, PrettyPrint m1, PrettyPrint m2, PrettyPrint c3) => PrettyPrint (PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) Source # | |
Defined in Math.Categories.FunctorCategory Methods pprint :: Int -> PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> String Source # pprintWithIndentations :: Int -> Int -> String -> PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> String Source # pprintIndent :: Int -> PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> String Source # | |
(Eq o1, Eq m1, Simplifiable c1, Simplifiable c2, Simplifiable o1, Simplifiable o2, Simplifiable m1, Simplifiable m2, Simplifiable c3) => Simplifiable (PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) Source # | |
Defined in Math.Categories.FunctorCategory Methods simplify :: PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 # | |
Generic (PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) Source # | |
Defined in Math.Categories.FunctorCategory Associated Types type Rep (PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) :: Type -> Type # Methods from :: PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Rep (PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) x # to :: Rep (PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) x -> PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 # | |
(Show c1, Show c2, Show o1, Show o2, Show m1, Show m2, Show c3) => Show (PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) Source # | |
Defined in Math.Categories.FunctorCategory Methods showsPrec :: Int -> PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> ShowS # show :: PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> String # showList :: [PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3] -> ShowS # | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, Eq c2, Eq m2, Eq o2, Eq c3) => Eq (PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) Source # | |
Defined in Math.Categories.FunctorCategory Methods (==) :: PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Bool # (/=) :: PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Bool # | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, Category c3 m3 o3, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => Category (PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (NaturalTransformation c1 m1 o1 c3 m3 o3) (Diagram c1 m1 o1 c3 m3 o3) Source # | |
Defined in Math.Categories.FunctorCategory Methods identity :: PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c3 m3 o3 Source # ar :: PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Set (NaturalTransformation c1 m1 o1 c3 m3 o3) Source # genAr :: PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Diagram c1 m1 o1 c3 m3 o3 -> Set (NaturalTransformation c1 m1 o1 c3 m3 o3) Source # decompose :: PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 -> NaturalTransformation c1 m1 o1 c3 m3 o3 -> [NaturalTransformation c1 m1 o1 c3 m3 o3] Source # | |
(FiniteCategory c1 m1 o1, Morphism m1 o1, Eq c1, Eq m1, Eq o1, FiniteCategory c2 m2 o2, Morphism m2 o2, Eq c2, Eq m2, Eq o2, FiniteCategory c3 m3 o3, Morphism m3 o3, Eq c3, Eq m3, Eq o3) => FiniteCategory (PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) (NaturalTransformation c1 m1 o1 c3 m3 o3) (Diagram c1 m1 o1 c3 m3 o3) Source # | A |
Defined in Math.Categories.FunctorCategory | |
type Rep (PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) Source # | |
Defined in Math.Categories.FunctorCategory type Rep (PrecomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3) = D1 ('MetaData "PrecomposedFunctorCategory" "Math.Categories.FunctorCategory" "FiniteCategories-0.6.0.1-L2v014CZIYe4CTCeXkyH5k" 'False) (C1 ('MetaCons "PrecomposedFunctorCategory" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Diagram c1 m1 o1 c2 m2 o2)) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 c3))) |
data PostcomposedFunctorCategory c1 m1 o1 c2 m2 o2 c3 m3 o3 Source #
A FunctorCategory
D^C postcomposed by a functor F : D -> E
where C is a FiniteCategory
and D and E are Category
.
It has Diagram
s F o G : C -> E
as objects and NaturalTransformation
s between them as morphisms. NaturalTransformation
s compose vertically in this category.
Constructors
PostcomposedFunctorCategory (Diagram c2 m2 o2 c3 m3 o3) c1 |