GAP (GeneralizedMorphismsForCAP) - Chapter 2: Generalized Morphism Category by Cospans

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2 Generalized Morphism Category by Cospans
2.1 GAP Categories
2.1-1 IsGeneralizedMorphismCategoryByCospansObject
2.1-2 IsGeneralizedMorphismByCospan
2.2 Properties
2.2-1 HasIdentityAsReversedArrow
2.3 Attributes
2.3-1 UnderlyingHonestObject
2.3-2 Arrow
2.3-3 ReversedArrow
2.3-4 NormalizedCospanTuple
2.3-5 PseudoInverse
2.3-6 GeneralizedInverseByCospan
2.3-7 IdempotentDefinedBySubobjectByCospan
2.3-8 IdempotentDefinedByFactorobjectByCospan
2.3-9 NormalizedCospan
2.4 Operations
2.4-1 GeneralizedMorphismFromFactorToSubobjectByCospan
2.5 Constructors
2.5-1 GeneralizedMorphismByCospan
2.5-2 GeneralizedMorphismByCospan
2.5-3 GeneralizedMorphismByCospanWithSourceAid
2.5-4 AsGeneralizedMorphismByCospan
2.5-5 GeneralizedMorphismCategoryByCospans
2.5-6 GeneralizedMorphismByCospansObject
2.6 Constructors of lifts of exact functors and natrual (iso)morphisms
2.6-1 AsGeneralizedMorphismByCospan

### 2 Generalized Morphism Category by Cospans

#### 2.1 GAP Categories

##### 2.1-1 IsGeneralizedMorphismCategoryByCospansObject

`‣ IsGeneralizedMorphismCategoryByCospansObject`( object ) | ( filter )

Returns: `true` or `false`

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

##### 2.1-2 IsGeneralizedMorphismByCospan

`‣ IsGeneralizedMorphismByCospan`( object ) | ( filter )

Returns: `true` or `false`

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

#### 2.2 Properties

##### 2.2-1 HasIdentityAsReversedArrow

`‣ HasIdentityAsReversedArrow`( alpha ) | ( property )

Returns: `true` or `false`

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

#### 2.3 Attributes

##### 2.3-1 UnderlyingHonestObject

`‣ UnderlyingHonestObject`( a ) | ( attribute )

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

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

##### 2.3-2 Arrow

`‣ Arrow`( alpha ) | ( attribute )

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

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

##### 2.3-3 ReversedArrow

`‣ ReversedArrow`( alpha ) | ( attribute )

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

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

##### 2.3-4 NormalizedCospanTuple

`‣ NormalizedCospanTuple`( alpha ) | ( attribute )

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

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

##### 2.3-5 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 cospan. The output is its pseudo inverse \(b \rightarrow a\).

##### 2.3-6 GeneralizedInverseByCospan

`‣ GeneralizedInverseByCospan`( 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 cospan.

##### 2.3-7 IdempotentDefinedBySubobjectByCospan

`‣ IdempotentDefinedBySubobjectByCospan`( 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 cospan defined by \(\alpha\).

##### 2.3-8 IdempotentDefinedByFactorobjectByCospan

`‣ IdempotentDefinedByFactorobjectByCospan`( 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 cospan defined by \(\alpha\).

##### 2.3-9 NormalizedCospan

`‣ NormalizedCospan`( 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 cospan. The output is its normalization by cospan.

#### 2.4 Operations

##### 2.4-1 GeneralizedMorphismFromFactorToSubobjectByCospan

`‣ GeneralizedMorphismFromFactorToSubobjectByCospan`( 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 cospan from the factorobject to the subobject.

#### 2.5 Constructors

##### 2.5-1 GeneralizedMorphismByCospan

`‣ GeneralizedMorphismByCospan`( alpha, beta ) | ( operation )

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

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

##### 2.5-2 GeneralizedMorphismByCospan

`‣ GeneralizedMorphismByCospan`( 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 cospan defined by the composition of the given three arrows regarded as generalized morphisms.

##### 2.5-3 GeneralizedMorphismByCospanWithSourceAid

`‣ GeneralizedMorphismByCospanWithSourceAid`( alpha, beta ) | ( operation )

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

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

##### 2.5-4 AsGeneralizedMorphismByCospan

`‣ AsGeneralizedMorphismByCospan`( 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 cospan defined by \(\alpha\).

##### 2.5-5 GeneralizedMorphismCategoryByCospans

`‣ GeneralizedMorphismCategoryByCospans`( A ) | ( attribute )

Returns: a category

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

##### 2.5-6 GeneralizedMorphismByCospansObject

`‣ GeneralizedMorphismByCospansObject`( 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 cospans whose underlying honest object is \(a\).

#### 2.6 Constructors of lifts of exact functors and natrual (iso)morphisms

##### 2.6-1 AsGeneralizedMorphismByCospan

`‣ AsGeneralizedMorphismByCospan`( F, name ) | ( operation )

Lift the exact functor F to a functor \(A \to B\), where \(A := \) `GeneralizedMorphismCategoryByCospans( AsCapCategory( Source(` F`) ) )` and \(B := \) `GeneralizedMorphismCategoryByCospans( AsCapCategory( Range(` F`) ) )`.

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