### 34 Commutative diagrams and abstract categories

#### 34.1

##### 34.1-1 HomomorphismChainToCommutativeDiagram
 ‣ 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.

##### 34.1-2 NormalSeriesToQuotientDiagram
 ‣ 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.

##### 34.1-3 NerveOfCommutativeDiagram
 ‣ NerveOfCommutativeDiagram( D ) ( function )

Inputs a commutative diagram D and returns the commutative diagram ND consisting of all possible composites of the arrows in D.

##### 34.1-4 GroupHomologyOfCommutativeDiagram
 ‣ 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.

##### 34.1-5 PersistentHomologyOfCommutativeDiagramOfPGroups
 ‣ 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.

#### 34.2

##### 34.2-1 CategoricalEnrichment
 ‣ 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.

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##### 34.2-2 IdentityArrow
 ‣ IdentityArrow( X ) ( function )

Inputs an object X in some category, and returns the identity arrow on the object X.

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##### 34.2-3 InitialArrow
 ‣ InitialArrow( X ) ( function )

Inputs an object X in some category, and returns the arrow from the initial object in the category to X.

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##### 34.2-4 TerminalArrow
 ‣ TerminalArrow( X ) ( function )

Inputs an object X in some category, and returns the arrow from X to the terminal object in the category.

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##### 34.2-5 HasInitialObject
 ‣ HasInitialObject( Name ) ( function )

Inputs the name of a category and returns true or false depending on whether the category has an initial object.

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##### 34.2-6 HasTerminalObject
 ‣ HasTerminalObject( Name ) ( function )

Inputs the name of a category and returns true or false depending on whether the category has a terminal object.

##### 34.2-7 Source
 ‣ Source( f ) ( function )

Inputs an arrow f in some category, and returns its source.

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##### 34.2-8 Target
 ‣ Target( f ) ( function )

Inputs an arrow f in some category, and returns its target.

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##### 34.2-9 CategoryName
 ‣ CategoryName( X ) ( function )

Inputs an object or arrow X in some category, and returns the name of the category.

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 ‣ 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.

##### 34.2-11 Object
 ‣ Object( X ) ( function )

Inputs an object X in some category, and returns the GAP structure Y such that X=CategoricalEnrichment(Y,CategoryName(X)).

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##### 34.2-12 Mapping
 ‣ Mapping( X ) ( function )

Inputs an arrow f in some category, and returns the GAP structure Y such that f=CategoricalEnrichment(Y,CategoryName(X)).

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##### 34.2-13 IsCategoryObject
 ‣ IsCategoryObject( X ) ( function )

Inputs X and returns true if X is an object in some category.

##### 34.2-14 IsCategoryArrow
 ‣ IsCategoryArrow( X ) ( function )

Inputs X and returns true if X is an arrow in some category.

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