GAP (HAP commands) - Chapter 34: Commutative diagrams and abstract categories

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34 Commutative diagrams and abstract categories
34.1
34.1-1 HomomorphismChainToCommutativeDiagram
34.1-2 NormalSeriesToQuotientDiagram
34.1-3 NerveOfCommutativeDiagram
34.1-4 GroupHomologyOfCommutativeDiagram
34.1-5 PersistentHomologyOfCommutativeDiagramOfPGroups
34.2
34.2-1 CategoricalEnrichment
34.2-2 IdentityArrow
34.2-3 InitialArrow
34.2-4 TerminalArrow
34.2-5 HasInitialObject
34.2-6 HasTerminalObject
34.2-7 Source
34.2-8 Target
34.2-9 CategoryName
34.2-10 CompositionEqualityAdditionMinus
34.2-11 Object
34.2-12 Mapping
34.2-13 IsCategoryObject
34.2-14 IsCategoryArrow

### 34 Commutative diagrams and abstract categories

COMMUTATIVE DIAGRAMS

#### 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.

Examples:

##### 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 \(G/L_i \rightarrow G/L_{i+1}\) 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:

##### 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 \(D\).

Examples:

##### 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.

Examples:

##### 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.

Examples:

ABSTRACT CATEGORIES

#### 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.

Examples: 1

##### 34.2-2 IdentityArrow

`‣ IdentityArrow`( X ) | ( function )

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

Examples: 1

##### 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 \(X\).

Examples: 1

##### 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.

Examples: 1

##### 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.

Examples: 1

##### 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.

Examples:

##### 34.2-7 Source

`‣ 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 , 12

##### 34.2-8 Target

`‣ Target`( f ) | ( function )

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

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8

##### 34.2-9 CategoryName

`‣ CategoryName`( X ) | ( function )

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

Examples: 1

##### 34.2-10 CompositionEqualityAdditionMinus

`‣ 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:

##### 34.2-11 Object

`‣ Object`( X ) | ( function )

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

Examples: 1 , 2

##### 34.2-12 Mapping

`‣ Mapping`( X ) | ( function )

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

Examples: 1 , 2 , 3 , 4 , 5

##### 34.2-13 IsCategoryObject

`‣ IsCategoryObject`( X ) | ( function )

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

Examples:

##### 34.2-14 IsCategoryArrow

`‣ IsCategoryArrow`( X ) | ( function )

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

Examples:

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