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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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