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.

`‣ 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.

`‣ 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 ) ) }.

`‣ 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.

`‣ 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.

`‣ 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.

`‣ 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'.

`‣ 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'.

`‣ 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'.

`‣ 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.

`‣ 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`

.

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.

`‣ 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`

.

`‣ 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`

.

`‣ 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`

.

`‣ 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`

.

`‣ 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`

.

`‣ 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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