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2 Generalized Morphism Category by Cospans
 2.1 GAP Categories
 2.2 Properties
 2.3 Attributes
 2.4 Operations
 2.5 Constructors
 2.6 Constructors of lifts of exact functors and natrual (iso)morphisms

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 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 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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