COMMUTATIVE DIAGRAMS
‣ HomomorphismChainToCommutativeDiagram( H ) | ( function ) |
Inputs a list H=[h_1,h_2,...,h_n] of mappings such that the composite h_1h_2...h_n is defined. It returns the list of composable homomorphism as a commutative diagram.
Examples:
‣ NormalSeriesToQuotientDiagram( L ) | ( function ) |
‣ NormalSeriesToQuotientDiagram( L, M ) | ( function ) |
Inputs an increasing (or decreasing) list L=[L_1,L_2,...,L_n] of normal subgroups of a group G with G=L_n. It returns the chain of quotient homomorphisms G/L_i → G/L_i+1 as a commutative diagram.
Optionally a subseries M of L can be entered as a second variable. Then the resulting diagram of quotient groups has two rows of horizontal arrows and one row of vertical arrows.
Examples:
‣ NerveOfCommutativeDiagram( D ) | ( function ) |
Inputs a commutative diagram D and returns the commutative diagram ND consisting of all possible composites of the arrows in D.
Examples:
‣ GroupHomologyOfCommutativeDiagram( D, n ) | ( function ) |
‣ GroupHomologyOfCommutativeDiagram( D, n, prime ) | ( function ) |
‣ GroupHomologyOfCommutativeDiagram( D, n, prime, Resolution_Algorithm ) | ( function ) |
Inputs a commutative diagram D of p-groups and positive integer n. It returns the commutative diagram of vector spaces obtained by applying mod p homology.
Non-prime power groups can also be handled if a prime p is entered as the third argument. Integral homology can be obtained by setting p=0. For p=0 the result is a diagram of groups.
A particular resolution algorithm, such as ResolutionNilpotentGroup, can be entered as a fourth argument. For positive p the default is ResolutionPrimePowerGroup. For p=0 the default is ResolutionFiniteGroup.
Examples:
‣ PersistentHomologyOfCommutativeDiagramOfPGroups( D, n ) | ( function ) |
Inputs a commutative diagram D of finite p-groups and a positive integer n. It returns a list containing, for each homomorphism in the nerve of D, a triple [k,l,m] where k is the dimension of the source of the induced mod p homology map in degree n, l is the dimension of the image, and m is the dimension of the cokernel.
Examples:
ABSTRACT CATEGORIES
‣ CategoricalEnrichment( X, Name ) | ( function ) |
Inputs a structure X such as a group or group homomorphism, together with the name of some existing category such as Name:=Category_of_Groups or Category_of_Abelian_Groups. It returns, as appropriate, an object or arrow in the named category.
Examples: 1
‣ IdentityArrow( X ) | ( function ) |
Inputs an object X in some category, and returns the identity arrow on the object X.
Examples: 1
‣ InitialArrow( X ) | ( function ) |
Inputs an object X in some category, and returns the arrow from the initial object in the category to X.
Examples: 1
‣ TerminalArrow( X ) | ( function ) |
Inputs an object X in some category, and returns the arrow from X to the terminal object in the category.
Examples: 1
‣ HasInitialObject( Name ) | ( function ) |
Inputs the name of a category and returns true or false depending on whether the category has an initial object.
Examples: 1
‣ HasTerminalObject( Name ) | ( function ) |
Inputs the name of a category and returns true or false depending on whether the category has a terminal object.
Examples:
‣ Source( f ) | ( function ) |
Inputs an arrow f in some category, and returns its source.
Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11
‣ Target( f ) | ( function ) |
Inputs an arrow f in some category, and returns its target.
Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8
‣ CategoryName( X ) | ( function ) |
Inputs an object or arrow X in some category, and returns the name of the category.
Examples: 1
‣ CompositionEqualityAdditionMinus | ( global variable ) |
Composition of suitable arrows f,g is given by f*g when the source of f equals the target of g. (Warning: this differes to the standard GAP convention.)
Equality is tested using f=g.
In an additive category the sum and difference of suitable arrows is given by f+g and f-g.
Examples:
‣ Object( X ) | ( function ) |
Inputs an object X in some category, and returns the GAP structure Y such that X=CategoricalEnrichment(Y,CategoryName(X)).
‣ Mapping( X ) | ( function ) |
Inputs an arrow f in some category, and returns the GAP structure Y such that f=CategoricalEnrichment(Y,CategoryName(X)).
‣ IsCategoryObject( X ) | ( function ) |
Inputs X and returns true if X is an object in some category.
Examples:
‣ IsCategoryArrow( X ) | ( function ) |
Inputs X and returns true if X is an arrow in some category.
Examples:
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