GAP (GeneralizedMorphismsForCAP) - Chapter 3: Generalized Morphism Category by Spans

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3 Generalized Morphism Category by Spans
3.1 GAP Categories
3.1-1 IsGeneralizedMorphismCategoryBySpansObject
3.1-2 IsGeneralizedMorphismBySpan
3.2 Properties
3.2-1 HasIdentityAsReversedArrow
3.3 Attributes
3.3-1 UnderlyingHonestObject
3.3-2 Arrow
3.3-3 ReversedArrow
3.3-4 NormalizedSpanTuple
3.3-5 PseudoInverse
3.3-6 GeneralizedInverseBySpan
3.3-7 IdempotentDefinedBySubobjectBySpan
3.3-8 IdempotentDefinedByFactorobjectBySpan
3.3-9 NormalizedSpan
3.4 Operations
3.4-1 GeneralizedMorphismFromFactorToSubobjectBySpan
3.5 Constructors
3.5-1 GeneralizedMorphismBySpan
3.5-2 GeneralizedMorphismBySpan
3.5-3 GeneralizedMorphismBySpanWithRangeAid
3.5-4 AsGeneralizedMorphismBySpan
3.5-5 GeneralizedMorphismCategoryBySpans
3.5-6 GeneralizedMorphismBySpansObject

### 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 of 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 of 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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