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### 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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