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Let A be an additive category. The Adelman category of A is the free abelian category induced by A. An object x of the Adelman category of A consists of a composable pair (\rho: a \rightarrow b, \gamma: b \rightarrow c) in A. We call \rho the relation morphism, and \gamma the co-relation morphism of x.

Given two objects x = (\rho: a \rightarrow b, \gamma: b \rightarrow c) and y = (\rho': a' \rightarrow b', \gamma': b' \rightarrow c'), a morphism \alpha from x to y in the Adelman category of A consists of a morphism \beta: b \rightarrow b', called the morphism datum, that has to fit into some commutative diagram of the form Any such morphism \omega is called a relation witness, any such morphism \psi is called a co-relation witness. Two morphisms between x and y with morphism data \beta and \beta' are congruent iff there exists \sigma_1: b \rightarrow a' and \sigma_2: c \rightarrow b' such that \beta - \beta' = \sigma_1 \cdot \rho' + \gamma \cdot \sigma_2. We call any such pair (\sigma_1, \sigma_2) a witness pair for \beta, \beta' being congruent.

#### 4.1 GAP Categories

 ‣ IsAdelmanCategoryObject( a ) ( filter )

Returns: true or false

The GAP category of objects of an Adelman category. Every object of an Adelman category lies in this GAP category.

 ‣ IsAdelmanCategoryMorphism( alpha ) ( filter )

Returns: true or false

The GAP category of morphisms of an Adelman category. Every morphism of an Adelman category lies in this GAP category.

 ‣ IsAdelmanCategory( C ) ( filter )

Returns: true or false

The GAP category of Adelman categories. Every CAP category which was created as an Adelman category lies in this GAP category.

#### 4.2 Constructors

 ‣ AdelmanCategory( A ) ( attribute )

Returns: a category

The argument is an additive CAP category A. The output is the Adelman category of A.

 ‣ AdelmanCategoryObject( alpha, beta ) ( operation )

Returns: an object

The arguments are two morphisms \alpha: a \rightarrow b, \beta: b \rightarrow c of the same additive category A. The output is an object in the Adelman category of A whose relation morphism is \alpha and whose co-relation morphism is \beta.

 ‣ AdelmanCategoryMorphism( x, alpha, y ) ( operation )

Returns: a morphism in \mathrm{Hom}(x, y)

Let A be an additive category. The arguments are an object x in the Adelman category of A, a morphism \alpha: a \rightarrow b of A, and an object y in the Adelman category of A. The output is a morphism in the Adelman category of A whose morphism datum is given by \alpha.

 ‣ AsAdelmanCategoryObject( a ) ( attribute )

Returns: an object

The argument is an object a of an additive category A. The output is an object in the Adelman category of A whose relation morphism is 0 \rightarrow a and whose co-relation morphism is a \rightarrow 0.

 ‣ AsAdelmanCategoryMorphism( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}( x, y )

The argument is a morphism \alpha: a \rightarrow b of an additive category A. The output is a morphism in the Adelman category of A whose source x is AsAdelmanCategoryObject( a ), whose range y is AsAdelmanCategoryObject( b ), and whose morphism datum is \alpha.

##### 4.2-6 \/
 ‣ \/( a, C ) ( operation )

Returns: an object

This is a convenience method. The first argument is an object a which either lies in an additive category A (which was not created as a Freyd category) or in a Freyd category F of an underlying additive category A. The second argument is an Adelman category C of A. If a lies in A this method returns AsAdelmanCategoryObject( a ). If a lies in F, this method return an object in C whose relation morphism is the same as the relation morphism of a, and whose co-relation morphism is 0.

##### 4.2-7 \/
 ‣ \/( alpha, C ) ( operation )

Returns: a morphism in \mathrm{Hom}( x, y )

This is a convenience method. The first argument is a morphism \alpha which lies in an additive category A. The second argument is an Adelman category C of A. This method returns AsAdelmanCategoryMorphism( alpha ). We set x = \mathrm{ AsAdelmanCategoryObject( Source( \alpha ) ) } and y = \mathrm{ AsAdelmanCategoryObject( Range( \alpha ) ) }.

#### 4.3 Attributes and Properties

##### 4.3-1 UnderlyingCategory
 ‣ UnderlyingCategory( C ) ( attribute )

Returns: a category

The argument is an Adelman category C. The output is its underlying category A with which it was constructed.

##### 4.3-2 RelationMorphism
 ‣ RelationMorphism( x ) ( attribute )

Returns: a morphism in \mathrm{Hom}( a, b )

The argument is an object x in an Adelman category. The output is its relation morphism \rho: a \rightarrow b.

##### 4.3-3 CorelationMorphism
 ‣ CorelationMorphism( x ) ( attribute )

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

The argument is an object x in an Adelman category. The output is its co-relation morphism \gamma: b \rightarrow c.

##### 4.3-4 UnderlyingMorphism
 ‣ UnderlyingMorphism( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}( b, b' )

The argument is a morphism \alpha in an Adelman category. The output is its morphism datum \beta: b \rightarrow b'.

##### 4.3-5 RelationWitness
 ‣ RelationWitness( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}( a, a' )

The argument is a morphism \alpha in an Adelman category. The output is its relation witness \omega: a \rightarrow a'.

##### 4.3-6 CorelationWitness
 ‣ CorelationWitness( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}( c, c' )

The argument is a morphism \alpha in an Adelman category. The output is its co-relation witness \psi: c \rightarrow c'.

##### 4.3-7 WitnessPairForBeingCongruentToZero
 ‣ WitnessPairForBeingCongruentToZero( alpha ) ( attribute )

Returns: a list of morphisms

The argument is a morphism \alpha congruent to zero in an Adelman category. The output is a witness pair.

##### 4.3-8 MereExistenceOfWitnessPairForBeingCongruentToZero
 ‣ MereExistenceOfWitnessPairForBeingCongruentToZero( alpha ) ( attribute )

Returns: a boolean

The argument is a morphism \alpha in an Adelman category. The output is true if \alpha is congruent to zero, else false.

 ‣ IsSequenceAsAdelmanCategoryObject( x ) ( property )

Returns: a boolean

The argument is an object x in an Adelman category. The output is true if the composition of its relation morphism and its co-relation morphism yields zero. Otherwise, the output is false.

#### 4.4 Exactness properties: objects as functors

The Adelman category of A can also be interpreted as the category A-\mathrm{mod}-\mathrm{mod}, i.e., the category of finitely presented functors whose domain is given by A-\mathrm{mod}, and whose codomain is the category of abelian groups. The category A-\mathrm{mod}-\mathrm{mod} embeds into A-\mathrm{Mod}-\mathrm{mod} via extension by filtered colimits. Thus, any object in the Adelman category of A can be interpreted as a functor from A-\mathrm{Mod} (an abelian category) into the category of abelian groups (also an abelian category). Via this interpretation, it makes sense to ask for exactness properties of an object in the Adelman category.

##### 4.4-1 IsExact
 ‣ IsExact( x ) ( property )

Returns: a boolean

The argument is an object x in an Adelman category. The output is true if x corresponds to an exact functor. Otherwise, the output is false.

##### 4.4-2 IsLeftExact
 ‣ IsLeftExact( x ) ( property )

Returns: a boolean

The argument is an object x in an Adelman category. The output is true if x corresponds to a left exact functor. Otherwise, the output is false.

##### 4.4-3 IsRightExact
 ‣ IsRightExact( x ) ( property )

Returns: a boolean

The argument is an object x in an Adelman category. The output is true if x corresponds to a right exact functor. Otherwise, the output is false.

##### 4.4-4 IsMonoPreserving
 ‣ IsMonoPreserving( x ) ( property )

Returns: a boolean

The argument is an object x in an Adelman category. The output is true if x corresponds to a mono preserving functor. Otherwise, the output is false.

##### 4.4-5 IsEpiPreserving
 ‣ IsEpiPreserving( x ) ( property )

Returns: a boolean

The argument is an object x in an Adelman category. The output is true if x corresponds to an epi preserving functor. Otherwise, the output is false.

##### 4.4-6 IsImagePreserving
 ‣ IsImagePreserving( x ) ( property )

Returns: a boolean

The argument is an object x in an Adelman category. The output is true if x corresponds to an image preserving functor. Otherwise, the output is false.

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