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12 Chain Maps
12.1 ChainMaps: Categories and Representations
12.1-1 IsHomalgChainMap
12.1-2 IsHomalgChainSelfMap
12.1-3 IsChainMapOfFinitelyPresentedObjectsRep
12.1-4 IsCochainMapOfFinitelyPresentedObjectsRep
12.2 Chain Maps: Constructors
12.2-1 HomalgChainMap
12.3 Chain Maps: Properties
12.3-1 IsMorphism
12.3-2 IsGeneralizedMorphism
12.3-3 IsGeneralizedEpimorphism
12.3-4 IsGeneralizedMonomorphism
12.3-5 IsGeneralizedIsomorphism
12.3-6 IsIdentityMorphism
12.3-7 IsMonomorphism
12.3-8 IsEpimorphism
12.3-9 IsSplitMonomorphism
12.3-10 IsSplitEpimorphism
12.3-11 IsIsomorphism
12.3-12 IsAutomorphism
12.3-13 IsGradedMorphism
12.3-14 IsQuasiIsomorphism
12.4 Chain Maps: Attributes
12.4-1 Source
12.4-2 Range
12.5 Chain Maps: Operations and Functions
12.5-1 ByASmallerPresentation

### 12 Chain Maps

#### 12.1 ChainMaps: Categories and Representations

##### 12.1-1 IsHomalgChainMap

`> IsHomalgChainMap`( cm ) | ( category )

Returns: `true` or `false`

The GAP category of homalg (co)chain maps.

(It is a subcategory of the GAP category `IsHomalgMorphism`.)

##### 12.1-2 IsHomalgChainSelfMap

`> IsHomalgChainSelfMap`( cm ) | ( category )

Returns: `true` or `false`

The GAP category of homalg (co)chain self-maps.

(It is a subcategory of the GAP categories `IsHomalgChainMap` and `IsHomalgEndomorphism`.)

##### 12.1-3 IsChainMapOfFinitelyPresentedObjectsRep

`> IsChainMapOfFinitelyPresentedObjectsRep`( c ) | ( representation )

Returns: `true` or `false`

The GAP representation of chain maps of finitley generated homalg modules.

(It is a representation of the GAP category `IsHomalgChainMap` (12.1-1), which is a subrepresentation of the GAP representation `IsMorphismOfFinitelyGeneratedModulesRep`.)

##### 12.1-4 IsCochainMapOfFinitelyPresentedObjectsRep

`> IsCochainMapOfFinitelyPresentedObjectsRep`( c ) | ( representation )

Returns: `true` or `false`

The GAP representation of cochain maps of finitley generated homalg modules.

(It is a representation of the GAP category `IsHomalgChainMap` (12.1-1), which is a subrepresentation of the GAP representation `IsMorphismOfFinitelyGeneratedModulesRep`.)

#### 12.2 Chain Maps: Constructors

##### 12.2-1 HomalgChainMap

`> HomalgChainMap`( phi[, C][, D][, d] ) | ( function )

Returns: a homalg chain map

The constructor creates a (co)chain map given a source homalg (co)chain complex C, a target homalg (co)chain complex D (--> 9.2), and a homalg map phi (--> 10.2)at (co)homological degree d. The returned (co)chain map will cautiously be indicated using parenthesis: "chain map". To verify if the result is indeed a (co)chain map use `IsMorphism` (12.3-1). If source and target are identical objects, and only then, the (co)chain map is created as a (co)chain selfmap.

The following examples shows a chain map that induces the zero map on homology, but is itself not zero in the derived category:

    gap> ZZ := HomalgRingOfIntegers( );;
    gap> M := 1 * ZZ;
    <The free left module of rank 1 on a free generator>
    gap> Display( M );
    Z^(1 x 1)
    gap> N := HomalgMatrix( "[3]", 1, 1, ZZ );;
    gap> N := LeftPresentation( N );
    <A cyclic left module presented by 1 relation for a cyclic generator>
    gap> Display( N );
    Z/< 3 >
    gap> a := HomalgMap( HomalgMatrix( "[2]", 1, 1, ZZ ), M, M );
    <An endomorphism of a left module>
    gap> c := HomalgMap( HomalgMatrix( "[2]", 1, 1, ZZ ), M, N );
    <A homomorphism of left modules>
    gap> b := HomalgMap( HomalgMatrix( "[1]", 1, 1, ZZ ), M, M );
    <An endomorphism of a left module>
    gap> d := HomalgMap( HomalgMatrix( "[1]", 1, 1, ZZ ), M, N );
    <A homomorphism of left modules>
    gap> C1 := HomalgComplex( a );
    <A non-zero acyclic complex containing a single morphism of left modules at de\
    grees [ 0 .. 1 ]>
    gap> C2 := HomalgComplex( c );
    <A non-zero acyclic complex containing a single morphism of left modules at de\
    grees [ 0 .. 1 ]>
    gap> cm := HomalgChainMap( d, C1, C2 );
    <A "chain map" containing a single left morphism at degree 0>
    gap> Add( cm, b );
    gap> IsMorphism( cm );
    true
    gap> cm;
    <A chain map containing 2 morphisms of left modules at degrees [ 0 .. 1 ]>
    gap> hcm := DefectOfExactness( cm );
    <A chain map of graded objects containing
    2 morphisms of left modules at degrees [ 0 .. 1 ]>
    gap> IsZero( hcm );
    true
    gap> IsZero( Source( hcm ) );
    false
    gap> IsZero( Range( hcm ) );
    false

#### 12.3 Chain Maps: Properties

##### 12.3-1 IsMorphism

`> IsMorphism`( cm ) | ( property )

Returns: `true` or `false`

Check if cm is a well-defined chain map, i.e. independent of all involved presentations.

##### 12.3-2 IsGeneralizedMorphism

`> IsGeneralizedMorphism`( cm ) | ( property )

Returns: `true` or `false`

Check if cm is a generalized morphism.

##### 12.3-3 IsGeneralizedEpimorphism

`> IsGeneralizedEpimorphism`( cm ) | ( property )

Returns: `true` or `false`

Check if cm is a generalized epimorphism.

##### 12.3-4 IsGeneralizedMonomorphism

`> IsGeneralizedMonomorphism`( cm ) | ( property )

Returns: `true` or `false`

Check if cm is a generalized monomorphism.

##### 12.3-5 IsGeneralizedIsomorphism

`> IsGeneralizedIsomorphism`( cm ) | ( property )

Returns: `true` or `false`

Check if cm is a generalized isomorphism.

##### 12.3-6 IsIdentityMorphism

`> IsIdentityMorphism`( cm ) | ( property )

Returns: `true` or `false`

Check if the homalg chain map cm is the identity chain map.

##### 12.3-7 IsMonomorphism

`> IsMonomorphism`( cm ) | ( property )

Returns: `true` or `false`

Check if the homalg chain map cm is a monomorphism.

##### 12.3-8 IsEpimorphism

`> IsEpimorphism`( cm ) | ( property )

Returns: `true` or `false`

Check if the homalg chain map cm is an epimorphism.

##### 12.3-9 IsSplitMonomorphism

`> IsSplitMonomorphism`( cm ) | ( property )

Returns: `true` or `false`

Check if the homalg chain map cm is a split monomorphism.

##### 12.3-10 IsSplitEpimorphism

`> IsSplitEpimorphism`( cm ) | ( property )

Returns: `true` or `false`

Check if the homalg chain map cm is a split epimorphism.

##### 12.3-11 IsIsomorphism

`> IsIsomorphism`( cm ) | ( property )

Returns: `true` or `false`

Check if the homalg chain map cm is an isomorphism.

##### 12.3-12 IsAutomorphism

`> IsAutomorphism`( cm ) | ( property )

Returns: `true` or `false`

Check if the homalg chain map cm is an automorphism.

##### 12.3-13 IsGradedMorphism

`> IsGradedMorphism`( cm ) | ( property )

Returns: `true` or `false`

Check if the source and target complex of the homalg chain map cm are graded objects, i.e. if all their morphisms vanish.

##### 12.3-14 IsQuasiIsomorphism

`> IsQuasiIsomorphism`( cm ) | ( property )

Returns: `true` or `false`

Check if the homalg chain map cm is a quasi-isomorphism.

#### 12.4 Chain Maps: Attributes

##### 12.4-1 Source

`> Source`( cm ) | ( attribute )

Returns: a homalg complex

The source of the homalg chain map cm.

##### 12.4-2 Range

`> Range`( cm ) | ( attribute )

Returns: a homalg complex

The target (range) of the homalg chain map cm.

#### 12.5 Chain Maps: Operations and Functions

##### 12.5-1 ByASmallerPresentation

`> ByASmallerPresentation`( cm ) | ( method )

Returns: a homalg complex

See `ByASmallerPresentation` (11.5-2) on complexes.

    InstallMethod( ByASmallerPresentation,
            "for homalg chain maps",
            [ IsHomalgChainMap ],

      function( cm )

        ByASmallerPresentation( Source( cm ) );
        ByASmallerPresentation( Range( cm ) );

        List( MorphismsOfChainMap( cm ), DecideZero );

        return cm;

    end );

This method performs side effects on its argument cm and returns it.

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