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### 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.

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