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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 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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