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3 Generalized Morphism Category by Spans
 3.1 GAP Categories
 3.2 Properties
 3.3 Attributes
 3.4 Operations
 3.5 Constructors

3 Generalized Morphism Category by Spans

3.1 GAP Categories

3.1-1 IsGeneralizedMorphismCategoryBySpansObject
‣ IsGeneralizedMorphismCategoryBySpansObject( object )( filter )

Returns: true or false

The GAP category of objects in the generalized morphism category by spans.

3.1-2 IsGeneralizedMorphismBySpan
‣ IsGeneralizedMorphismBySpan( object )( filter )

Returns: true or false

The GAP category of morphisms in the generalized morphism category by spans.

3.2 Properties

3.2-1 HasIdentityAsReversedArrow
‣ HasIdentityAsReversedArrow( alpha )( property )

Returns: true or false

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

3.3 Attributes

3.3-1 UnderlyingHonestObject
‣ UnderlyingHonestObject( a )( attribute )

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

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

3.3-2 Arrow
‣ Arrow( alpha )( attribute )

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

The argument is a generalized morphism \(\alpha\) by a span \(a \leftarrow b \rightarrow c\). The output is its arrow \(b \rightarrow c\).

3.3-3 ReversedArrow
‣ ReversedArrow( alpha )( attribute )

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

The argument is a generalized morphism \(\alpha\) by a span \(a \leftarrow b \rightarrow c\). The output is its reversed arrow \(a \leftarrow b\).

3.3-4 NormalizedSpanTuple
‣ NormalizedSpanTuple( alpha )( attribute )

Returns: a pair of morphisms in \(\mathbf{A}\).

The argument is a generalized morphism \(\alpha: a \rightarrow b\) by a span. The output is its normalized span pair \((a \leftarrow d, d \rightarrow b)\).

3.3-5 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 a span. The output is its pseudo inverse \(b \rightarrow a\).

3.3-6 GeneralizedInverseBySpan
‣ GeneralizedInverseBySpan( 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 span.

3.3-7 IdempotentDefinedBySubobjectBySpan
‣ IdempotentDefinedBySubobjectBySpan( 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 span defined by \(\alpha\).

3.3-8 IdempotentDefinedByFactorobjectBySpan
‣ IdempotentDefinedByFactorobjectBySpan( 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 span defined by \(\alpha\).

3.3-9 NormalizedSpan
‣ NormalizedSpan( alpha )( attribute )

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

The argument is a generalized morphism \(\alpha: a \rightarrow b\) by a span. The output is its normalization by span.

3.4 Operations

3.4-1 GeneralizedMorphismFromFactorToSubobjectBySpan
‣ GeneralizedMorphismFromFactorToSubobjectBySpan( 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 span from the factorobject to the subobject.

3.5 Constructors

3.5-1 GeneralizedMorphismBySpan
‣ GeneralizedMorphismBySpan( alpha, beta )( operation )

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

The arguments are morphisms \(\alpha: a \leftarrow c\) and \(\beta: c \rightarrow b\) in \(\mathbf{A}\). The output is a generalized morphism by span with arrow \(\beta\) and reversed arrow \(\alpha\).

3.5-2 GeneralizedMorphismBySpan
‣ GeneralizedMorphismBySpan( 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 span defined by the composition the given three arrows regarded as generalized morphisms.

3.5-3 GeneralizedMorphismBySpanWithRangeAid
‣ GeneralizedMorphismBySpanWithRangeAid( alpha, beta )( operation )

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

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

3.5-4 AsGeneralizedMorphismBySpan
‣ AsGeneralizedMorphismBySpan( 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 span defined by \(\alpha\).

3.5-5 GeneralizedMorphismCategoryBySpans
‣ GeneralizedMorphismCategoryBySpans( A )( attribute )

Returns: a category

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

3.5-6 GeneralizedMorphismBySpansObject
‣ GeneralizedMorphismBySpansObject( 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 spans whose underlying honest object is \(a\).

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