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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