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32 Commutative diagrams and abstract categories

### 32 Commutative diagrams and abstract categories

COMMUTATIVE DIAGRAMS

`HomomorphismChainToCommutativeDiagram(H)` 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.

`NormalSeriesToQuotientDiagram(L)` `NormalSeriesToQuotientDiagram(L,M)`

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.

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

`GroupHomologyOfCommutativeDiagram(D,n)` `GroupHomologyOfCommutativeDiagram(D,n,prime)` `GroupHomologyOfCommutativeDiagram(D,n,prime,Resolution_Algorithm)`

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.

`PersistentHomologyOfCommutativeDiagramOfPGroups(D,n)` 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.

ABSTRACT CATEGORIES

`CategoricalEnrichment(X,Name)` 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.
`IdentityArrow(X)` Inputs an object X in some category, and returns the identity arrow on the object X.
`InitialArrow(X)` Inputs an object X in some category, and returns the arrow from the initial object in the category to X.
`TerminalArrow(X)` Inputs an object X in some category, and returns the arrow from X to the terminal object in the category.
`HasInitialObject(Name)` Inputs the name of a category and returns true or false depending on whether the category has an initial object.
`HasTerminalObject(Name)` Inputs the name of a category and returns true or false depending on whether the category has a terminal object.
`Source(f)` Inputs an arrow f in some category, and returns its source.
`Target(f)` Inputs an arrow f in some category, and returns its target.
`CategoryName(X)` Inputs an object or arrow X in some category, and returns the name of the category.
`"*", "=", "+", "-"` 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.
`Object(X)` Inputs an object X in some category, and returns the GAP structure Y such that X=CategoricalEnrichment(Y,CategoryName(X)).
`Mapping(X)` Inputs an arrow f in some category, and returns the GAP structure Y such that f=CategoricalEnrichment(Y,CategoryName(X)).
`IsCategoryObject(X)` Inputs X and returns true if X is an object in some category.
`IsCategoryArrow(X)` Inputs X and returns true if X is an arrow in some category.

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