‣ IsCoPreSheafCategory( category ) | ( filter ) |
Returns: true or false
The GAP category of copresheaf categories.
‣ IsCellInCoPreSheafCategory( cell ) | ( filter ) |
Returns: true or false
The GAP category of cells in a copresheaf category.
‣ IsObjectInCoPreSheafCategory( obj ) | ( filter ) |
Returns: true or false
The GAP category of objects in a copresheaf category.
‣ IsMorphismInCoPreSheafCategory( mor ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in a copresheaf category.
‣ Source( coPSh ) | ( attribute ) |
Returns: a CAP category
The source category of the copresheaf category coPSh.
‣ Target( coPSh ) | ( attribute ) |
Returns: a CAP category
The target category of the copresheaf category coPSh.
‣ Source( F ) | ( attribute ) |
Returns: a CAP category
The source of the functor underlying functor object F.
‣ Target( F ) | ( attribute ) |
Returns: a CAP category
The target of the functor underlying the functor object F.
‣ OppositeOfSource( coPSh ) | ( attribute ) |
Returns: a CAP category
The opposite of the source category of the copresheaf category coPSh.
‣ ValuesOfCoPreSheaf( F ) | ( attribute ) |
Returns: a pair of lists
The input is functor F in a copresheaf category coPSh. The output is pair of lists. The first is the list of values of the functor F on all objects of the source category of coPSh. The second is the list of values of the functor F on all *generating* morphisms of the source category of coPSh.
‣ ValuesOnAllObjects( eta ) | ( attribute ) |
Returns: a list
Returns the values of the natural transformation eta in a copresheaf category coPSh on all objects of the source category of coPSh.
The 2-cell underlying the functor object F_or_eta.
‣ UnderlyingCapTwoCategoryCell( F_or_eta ) | ( attribute ) |
Returns: a CAP functor or natural transformation
‣ CoYonedaEmbedding( B ) | ( attribute ) |
Returns: a CAP functor
‣ CoYonedaEmbeddingOfSourceCategory( coPSh ) | ( attribute ) |
Returns: a CAP functor
‣ ApplyObjectInCoPreSheafCategoryToObject( F, obj ) | ( operation ) |
Returns: a CAP object
Apply the presheaf F to the object obj. The shorthand is F(obj).
‣ ApplyObjectInCoPreSheafCategoryToMorphism( F, mor ) | ( operation ) |
Returns: a CAP morphism
Apply the presheaf F to the morphism mor. The shorthand is F(mor).
‣ ApplyMorphismInCoPreSheafCategoryToObject( eta, obj ) | ( operation ) |
Returns: a CAP morphism
Apply the presheaf morphism eta to the object obj. The shorthand is eta(o).
‣ CoPreSheaves( B, D ) | ( operation ) |
‣ CoPreSheaves( B, k ) | ( operation ) |
‣ CoPreSheaves( B ) | ( operation ) |
Returns: a CAP category
Construct the category CoPreSheaves( B, D )= FunctorCategory( B, D )^op of copresheaves from the small category B to the category D as objects and their natural transformations as morphisms.
‣ CreateCoPreSheaf( F ) | ( attribute ) |
‣ CreateCoPreSheaf( B, rec_images_of_objects, rec_images_of_morphisms ) | ( operation ) |
‣ CreateCoPreSheaf( B, images_of_objects, images_of_morphisms ) | ( operation ) |
Returns: an object in a CAP category
Turn the functor F:B \(\to\) D into an object in the category of functors coPSh := CoPreSheaves( B, D ). An alternative input is the source category B and two defining records rec_images_of_objects and rec_images_of_morphisms of F. Another alternative input is the source category B and two defining lists images_of_objects and images_of_morphisms of F. The order of their entries must correspond to that of the vertices and arrows of the underlying quiver.
For the convenience of the user the following input is also valid: If images_of_objects is a list of nonnegative integers, images_of_morphisms is a list of matrices, and \(k:=\) CommutativeRingOfLinearCategory( B ) is a field then the two lists are interpreted as objects and morphisms in a matrix category or a category of rows over \(k\), respectively.
‣ CreateCoPreSheafMorphismByValues( arg1, arg2, arg3, arg4 ) | ( operation ) |
‣ CreateCoPreSheafMorphism( eta ) | ( attribute ) |
‣ CreateCoPreSheafMorphism( U, e, V ) | ( operation ) |
‣ CreateCoPreSheafMorphism( U, e, V ) | ( operation ) |
‣ CreateCoPreSheafMorphism( arg1, arg2, arg3 ) | ( operation ) |
Returns: a morphism in a CAP category
Turn the natrual transformation eta:\(F \to G\) into a morphism U := CreateCoPreSheaf( F ) \(\to\) V := CreateCoPreSheaf( G ) in the category of functors coPSh := CoPreSheaves( B, C ), where B := Source( F ) = Source( G ) and C := Target( F ) = Target( G ).
An alternative input is the triple (U, e, V), where e is a defining record of eta.
Another alternative input is the triple (U, e, V), where e is a defining list of eta.
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