Copyright	Guillaume Sabbagh 2022
License	GPL-3
Maintainer	guillaumesabbagh@protonmail.com
Stability	experimental
Portability	portable
Safe Haskell	Safe-Inferred
Language	Haskell2010

Math.Categories.FinCat

Contents

Homogeneous functor
FinCat
Orphan instances

Description

The FinCat category has as objects finite categories and as morphisms homogeneous functors between them.

Functors must be homogeneous because otherwise we would not be able to compose them with the Morphism typeclass.

The FinCat datatype should not be confused with the FiniteCategory typeclass. The FiniteCategory typeclass describes axioms a structure should follow to be considered a finite category. The FinCat type is itself a Category.

A FinFunctor is a Diagram where the source and target category are the same. The source category of a FinFunctor should be finite.

Synopsis

type FinFunctor c m o = Diagram c m o c m o
data FinCat c m o = FinCat

Homogeneous functor

type FinFunctor c m o = Diagram c m o c m o Source #

A FinFunctor funct between two categories is a map between objects and a map between arrows of the two categories such that :

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)

A FinFunctor is a type of Diagram.

It is meant to be a morphism between categories within FinCat, it is homogeneous, the type of the source category must be the same as the type of the target category.

See Diagram in Math.Categories.FunctorCategory for heterogeneous ones.

FinCat

data FinCat c m o Source #

The FinCat category has as objects finite categories and as morphisms homogeneous functors between them.

Constructors

FinCat

Instances

Instances details

PrettyPrint (FinCat c m o) Source #
Instance details Defined in Math.Categories.FinCat Methods pprint :: Int -> FinCat c m o -> String Source # pprintWithIndentations :: Int -> Int -> String -> FinCat c m o -> String Source # pprintIndent :: Int -> FinCat c m o -> String Source #
Simplifiable (FinCat c m o) Source #
Instance details Defined in Math.Categories.FinCat Methods simplify :: FinCat c m o -> FinCat c m o #
Generic (FinCat c m o) Source #
Instance details Defined in Math.Categories.FinCat Associated Types type Rep (FinCat c m o) :: Type -> Type Methods from :: FinCat c m o -> Rep (FinCat c m o) x to :: Rep (FinCat c m o) x -> FinCat c m o
Show (FinCat c m o) Source #
Instance details Defined in Math.Categories.FinCat Methods showsPrec :: Int -> FinCat c m o -> ShowS show :: FinCat c m o -> String showList :: [FinCat c m o] -> ShowS
Eq (FinCat c m o) Source #
Instance details Defined in Math.Categories.FinCat Methods (==) :: FinCat c m o -> FinCat c m o -> Bool (/=) :: FinCat c m o -> FinCat c m o -> Bool
(FiniteCategory c m o, Morphism m o, Eq c, Eq m, Eq o) => Category (FinCat c m o) (FinFunctor c m o) c Source #
Instance details 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 `ColimitCategory` is MUCH slower than computing a `LimitCategory` as coequalizers of categories are trickier than equalizers of categories. We can only compute colimits of `CompositionGraph`s as coequalizing might create new formal morphisms when gluing two objects. Therefore `colimit` transforms all given categories into `CompositionGraph`s. If you already have a `CompositionGraph`, consider using `colimitOfCompositionGraphs` instead of `colimit`.
Instance details 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 #
Instance details 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 #
Instance details 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 #
Instance details 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 #
type Rep (FinCat c m o) Source #
Instance details Defined in Math.Categories.FinCat type Rep (FinCat c m o) = D1 ('MetaData "FinCat" "Math.Categories.FinCat" "FiniteCategories-0.6.4.0-inplace" 'False) (C1 ('MetaCons "FinCat" 'PrefixI 'False) (U1 :: Type -> Type))

Orphan instances

(Category c m o, Morphism m o, Eq m, Eq o) => Morphism (FinFunctor c m o) c Source #
Instance details 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 #