Goto Chapter: Top 1 2 3 4 5 6 7 8 9 Ind
 [Top of Book]  [Contents]   [Previous Chapter]   [Next Chapter] 

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 -> B, where A := GeneralizedMorphismCategoryByCospans( AsCapCategory( Source( F ) ) ) and B := GeneralizedMorphismCategoryByCospans( AsCapCategory( Range( F ) ) ).

 [Top of Book]  [Contents]   [Previous Chapter]   [Next Chapter] 
Goto Chapter: Top 1 2 3 4 5 6 7 8 9 Ind

generated by GAPDoc2HTML