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

Math.FiniteCategories.ColimitCategory

Contents

Orphan instances

Description

A ColimitCategory is the colimit of a diagram in FinCat. To compute colimits in a usual category, see Math.CocompleteCategory. To compute colimits in a custom FiniteCategory, see colimits in Math.ConeCategory.

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 CompositionGraphs as coequalizing might create new formal morphisms when gluing two objects. Therefore colimit transforms all given categories into CompositionGraphs. If you already have a CompositionGraph, consider using colimitOfCompositionGraphs instead of colimit.

Synopsis

glueObject :: (Eq a, Eq b) => CompositionGraph a b -> a -> a -> FinFunctor (CompositionGraph a b) (CGMorphism a b) a
glueObjects :: (Eq a, Eq b) => CompositionGraph a b -> Set a -> FinFunctor (CompositionGraph a b) (CGMorphism a b) a
glueMorphism :: (Eq a, Eq b) => CompositionGraph a b -> CGMorphism a b -> CGMorphism a b -> FinFunctor (CompositionGraph a b) (CGMorphism a b) a
glueMorphisms :: (Eq a, Eq b) => CompositionGraph a b -> Set (CGMorphism a b) -> FinFunctor (CompositionGraph a b) (CGMorphism a b) a
colimitOfCompositionGraphs :: (FiniteCategory cIndex mIndex oIndex, Morphism mIndex oIndex, Eq cIndex, Eq mIndex, Eq oIndex, Eq n, Eq e) => Diagram cIndex mIndex oIndex (FinCat (CompositionGraph n e) (CGMorphism n e) n) (FinFunctor (CompositionGraph n e) (CGMorphism n e) n) (CompositionGraph n e) -> Cocone cIndex mIndex 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))
coprojectBaseCompositionGraphs :: (FiniteCategory cIndex mIndex oIndex, Morphism mIndex oIndex, Eq mIndex, Eq oIndex, Eq e, Eq n) => Diagram cIndex mIndex oIndex (FinCat (CompositionGraph n e) (CGMorphism n e) n) (FinFunctor (CompositionGraph n e) (CGMorphism n e) n) (CompositionGraph n e) -> Diagram (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))

Documentation

glueObject :: (Eq a, Eq b) => CompositionGraph a b -> a -> a -> FinFunctor (CompositionGraph a b) (CGMorphism a b) a Source #

Glue two objects of a CompositionGraph and return an insertion diagram into the new CompositionGraph.

glueObjects :: (Eq a, Eq b) => CompositionGraph a b -> Set a -> FinFunctor (CompositionGraph a b) (CGMorphism a b) a Source #

Glue objects of a CompositionGraph and return an insertion diagram into the new CompositionGraph.

glueMorphism :: (Eq a, Eq b) => CompositionGraph a b -> CGMorphism a b -> CGMorphism a b -> FinFunctor (CompositionGraph a b) (CGMorphism a b) a Source #

Glue two morphisms with same source and target of a CompositionGraph and return an insertion diagram into the new CompositionGraph. The first argument should not be composite if the second is a generator.

glueMorphisms :: (Eq a, Eq b) => CompositionGraph a b -> Set (CGMorphism a b) -> FinFunctor (CompositionGraph a b) (CGMorphism a b) a Source #

Glue morphisms with same source and target of a CompositionGraph and return an insertion diagram into the new CompositionGraph.

colimitOfCompositionGraphs :: (FiniteCategory cIndex mIndex oIndex, Morphism mIndex oIndex, Eq cIndex, Eq mIndex, Eq oIndex, Eq n, Eq e) => Diagram cIndex mIndex oIndex (FinCat (CompositionGraph n e) (CGMorphism n e) n) (FinFunctor (CompositionGraph n e) (CGMorphism n e) n) (CompositionGraph n e) -> Cocone cIndex mIndex 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 #

Computes the colimit of a Diagram of CompositionGraphs, use this if you already have a Diagram of CompositionGraphs.

coprojectBaseCompositionGraphs :: (FiniteCategory cIndex mIndex oIndex, Morphism mIndex oIndex, Eq mIndex, Eq oIndex, Eq e, Eq n) => Diagram cIndex mIndex oIndex (FinCat (CompositionGraph n e) (CGMorphism n e) n) (FinFunctor (CompositionGraph n e) (CGMorphism n e) n) (CompositionGraph n e) -> Diagram (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)) Source #

Uncoproject a base of CompositionGraphs.

Orphan instances

(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 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 #
(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 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 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 #