Goto Chapter: Top 1 2 3 4 5 6 7 8 9 Ind
 [Top of Book]  [Contents]   [Previous Chapter]   [Next Chapter] 

4 Generalized Morphism Category by Three Arrows
 4.1 GAP Categories
 4.2 Properties
 4.3 Attributes
 4.4 Operations
 4.5 Constructors

4 Generalized Morphism Category by Three Arrows

4.1 GAP Categories

4.1-1 IsGeneralizedMorphismCategoryByThreeArrowsObject
‣ IsGeneralizedMorphismCategoryByThreeArrowsObject( object )( filter )

Returns: true or false

The GAP category of objects in the generalized morphism category by three arrows.

4.1-2 IsGeneralizedMorphismByThreeArrows
‣ IsGeneralizedMorphismByThreeArrows( object )( filter )

Returns: true or false

The GAP category of morphisms in the generalized morphism category by three arrows.

4.2 Properties

4.2-1 HasIdentitiesAsReversedArrows
‣ HasIdentitiesAsReversedArrows( alpha )( property )

Returns: true or false

The argument is a generalized morphism \(\alpha\) by three arrows \(a \leftarrow b \rightarrow c \leftarrow d\). The output is true if \(a \leftarrow b\) and \(c \leftarrow d\) are congruent to identity morphisms, false otherwise.

4.2-2 HasIdentityAsSourceAid
‣ HasIdentityAsSourceAid( alpha )( property )

Returns: true or false

The argument is a generalized morphism \(\alpha\) by three arrows \(a \leftarrow b \rightarrow c \leftarrow d\). The output is true if \(a \leftarrow b\) is congruent to an identity morphism, false otherwise.

4.2-3 HasIdentityAsRangeAid
‣ HasIdentityAsRangeAid( alpha )( property )

Returns: true or false

The argument is a generalized morphism \(\alpha\) by three arrows \(a \leftarrow b \rightarrow c \leftarrow d\). The output is true if \(c \leftarrow d\) is congruent to an identity morphism, false otherwise.

4.3 Attributes

4.3-1 UnderlyingHonestObject
‣ UnderlyingHonestObject( a )( attribute )

Returns: an object in \(\mathbf{A}\)

The argument is an object \(a\) in the generalized morphism category by three arrows. The output is its underlying honest object.

4.3-2 SourceAid
‣ SourceAid( alpha )( attribute )

Returns: a morphism in \(\mathrm{Hom}_{\mathbf{A}}(b,a)\)

The argument is a generalized morphism \(\alpha\) by three arrows \(a \leftarrow b \rightarrow c \leftarrow d\). The output is its source aid \(a \leftarrow b\).

4.3-3 RangeAid
‣ RangeAid( alpha )( attribute )

Returns: a morphism in \(\mathrm{Hom}_{\mathbf{A}}(d,c)\)

The argument is a generalized morphism \(\alpha\) by three arrows \(a \leftarrow b \rightarrow c \leftarrow d\). The output is its range aid \(c \leftarrow d\).

4.3-4 Arrow
‣ Arrow( alpha )( attribute )

Returns: a morphism in \(\mathrm{Hom}_{\mathbf{A}}(b,c)\)

The argument is a generalized morphism \(\alpha\) by three arrows \(a \leftarrow b \rightarrow c \leftarrow d\). The output is its range aid \(b \rightarrow c\).

4.3-5 PseudoInverse
‣ PseudoInverse( alpha )( attribute )

Returns: a morphism in \(\mathrm{Hom}_{\mathbf{G(A)}}(b,a)\)

The argument is a generalized morphism \(\alpha: a \rightarrow b\) by three arrows. The output is its pseudo inverse \(b \rightarrow a\).

4.3-6 GeneralizedInverseByThreeArrows
‣ GeneralizedInverseByThreeArrows( alpha )( attribute )

Returns: a morphism in \(\mathrm{Hom}_{\mathbf{G(A)}}(b,a)\)

The argument is a morphism \(\alpha: a \rightarrow b \in \mathbf{A}\). The output is its generalized inverse \(b \rightarrow a\) by three arrows.

4.3-7 IdempotentDefinedBySubobjectByThreeArrows
‣ IdempotentDefinedBySubobjectByThreeArrows( alpha )( attribute )

Returns: a morphism in \(\mathrm{Hom}_{\mathbf{G(A)}}(b,b)\)

The argument is a subobject \(\alpha: a \hookrightarrow b \in \mathbf{A}\). The output is the idempotent \(b \rightarrow b \in \mathbf{G(A)}\) by three arrows defined by \(\alpha\).

4.3-8 IdempotentDefinedByFactorobjectByThreeArrows
‣ IdempotentDefinedByFactorobjectByThreeArrows( alpha )( attribute )

Returns: a morphism in \(\mathrm{Hom}_{\mathbf{G(A)}}(b,b)\)

The argument is a factorobject \(\alpha: b \twoheadrightarrow a \in \mathbf{A}\). The output is the idempotent \(b \rightarrow b \in \mathbf{G(A)}\) by three arrows defined by \(\alpha\).

4.4 Operations

4.4-1 GeneralizedMorphismFromFactorToSubobjectByThreeArrows
‣ GeneralizedMorphismFromFactorToSubobjectByThreeArrows( beta, alpha )( operation )

Returns: a morphism in \(\mathrm{Hom}_{\mathbf{G(A)}}(c,a)\)

The arguments are a a factorobject \(\beta: b \twoheadrightarrow c\), and a subobject \(\alpha: a \hookrightarrow b\). The output is the generalized morphism by three arrows from the factorobject to the subobject.

4.4-2 CommonCoastriction
‣ CommonCoastriction( L )( operation )

Returns: a list of generalized morphisms

The argument is a list \(L\) of generalized morphisms by three arrows having the same range. The output is a list of generalized morphisms by three arrows which is the comman coastriction of \(L\).

4.5 Constructors

4.5-1 GeneralizedMorphismByThreeArrows
‣ GeneralizedMorphismByThreeArrows( alpha, beta, gamma )( operation )

Returns: a morphism in \(\mathrm{Hom}_{\mathbf{G(A)}}(a,d)\)

The arguments are morphisms \(\alpha: a \leftarrow b\), \(\beta: b \rightarrow c\), and \(\gamma: c \leftarrow d\) in \(\mathbf{A}\). The output is a generalized morphism by three arrows with source aid \(\alpha\), arrow \(\beta\), and range aid \(\gamma\).

4.5-2 GeneralizedMorphismByThreeArrowsWithSourceAid
‣ GeneralizedMorphismByThreeArrowsWithSourceAid( alpha, beta )( operation )

Returns: a morphism in \(\mathrm{Hom}_{\mathbf{G(A)}}(a,c)\)

The arguments are morphisms \(\alpha: a \leftarrow b\), and \(\beta: b \rightarrow c\) in \(\mathbf{A}\). The output is a generalized morphism by three arrows defined by the composition the given two arrows regarded as generalized morphisms.

4.5-3 GeneralizedMorphismByThreeArrowsWithRangeAid
‣ GeneralizedMorphismByThreeArrowsWithRangeAid( beta, gamma )( operation )

Returns: a morphism in \(\mathrm{Hom}_{\mathbf{G(A)}}(b,d)\)

The arguments are morphisms \(\beta: b \rightarrow c\), and \(\gamma: c \leftarrow d\) in \(\mathbf{A}\). The output is a generalized morphism by three arrows defined by the composition the given two arrows regarded as generalized morphisms.

4.5-4 AsGeneralizedMorphismByThreeArrows
‣ AsGeneralizedMorphismByThreeArrows( alpha )( attribute )

Returns: a morphism in \(\mathrm{Hom}_{\mathbf{G(A)}}(a,b)\)

The argument is a morphism \(\alpha: a \rightarrow b\) in \(\mathbf{A}\). The output is the honest generalized morphism by three arrows defined by \(\alpha\).

4.5-5 GeneralizedMorphismCategoryByThreeArrows
‣ GeneralizedMorphismCategoryByThreeArrows( A )( attribute )

Returns: a category

The argument is an abelian category \(\mathbf{A}\). The output is its generalized morphism category \(\mathbf{G(A)}\) by three arrows.

4.5-6 GeneralizedMorphismByThreeArrowsObject
‣ GeneralizedMorphismByThreeArrowsObject( a )( attribute )

Returns: an object in \(\mathbf{G(A)}\)

The argument is an object \(a\) in an abelian category \(\mathbf{A}\). The output is the object in the generalized morphism category by three arrows whose underlying honest object is \(a\).

 [Top of Book]  [Contents]   [Previous Chapter]   [Next Chapter] 
Goto Chapter: Top 1 2 3 4 5 6 7 8 9 Ind

generated by GAPDoc2HTML