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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