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### 6 Universal Objects

#### 6.1 Kernel

For a given morphism \alpha: A \rightarrow B, a kernel of \alpha consists of three parts:

• an object K,

• a morphism \iota: K \rightarrow A such that \alpha \circ \iota \sim_{K,B} 0,

• a dependent function u mapping each morphism \tau: T \rightarrow A satisfying \alpha \circ \tau \sim_{T,B} 0 to a morphism u(\tau): T \rightarrow K such that \iota \circ u( \tau ) \sim_{T,A} \tau.

The triple ( K, \iota, u ) is called a kernel of \alpha if the morphisms u( \tau ) are uniquely determined up to congruence of morphisms. We denote the object K of such a triple by \mathrm{KernelObject}(\alpha). We say that the morphism u(\tau) is induced by the universal property of the kernel. \\ \mathrm{KernelObject} is a functorial operation. This means: for \mu: A \rightarrow A', \nu: B \rightarrow B', \alpha: A \rightarrow B, \alpha': A' \rightarrow B' such that \nu \circ \alpha \sim_{A,B'} \alpha' \circ \mu, we obtain a morphism \mathrm{KernelObject}( \alpha ) \rightarrow \mathrm{KernelObject}( \alpha' ).

##### 6.1-1 KernelObject
 ‣ KernelObject( alpha ) ( attribute )

Returns: an object

The argument is a morphism \alpha. The output is the kernel K of \alpha.

##### 6.1-2 KernelEmbedding
 ‣ KernelEmbedding( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{KernelObject}(\alpha),A)

The argument is a morphism \alpha: A \rightarrow B. The output is the kernel embedding \iota: \mathrm{KernelObject}(\alpha) \rightarrow A.

##### 6.1-3 KernelEmbeddingWithGivenKernelObject
 ‣ KernelEmbeddingWithGivenKernelObject( alpha, K ) ( operation )

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

The arguments are a morphism \alpha: A \rightarrow B and an object K = \mathrm{KernelObject}(\alpha). The output is the kernel embedding \iota: K \rightarrow A.

##### 6.1-4 KernelLift
 ‣ KernelLift( alpha, tau ) ( operation )

Returns: a morphism in \mathrm{Hom}(T,\mathrm{KernelObject}(\alpha))

The arguments are a morphism \alpha: A \rightarrow B and a test morphism \tau: T \rightarrow A satisfying \alpha \circ \tau \sim_{T,B} 0. The output is the morphism u(\tau): T \rightarrow \mathrm{KernelObject}(\alpha) given by the universal property of the kernel.

##### 6.1-5 KernelLiftWithGivenKernelObject
 ‣ KernelLiftWithGivenKernelObject( alpha, tau, K ) ( operation )

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

The arguments are a morphism \alpha: A \rightarrow B, a test morphism \tau: T \rightarrow A satisfying \alpha \circ \tau \sim_{T,B} 0, and an object K = \mathrm{KernelObject}(\alpha). The output is the morphism u(\tau): T \rightarrow K given by the universal property of the kernel.

 ‣ AddKernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation KernelObject. F: \alpha \mapsto \mathrm{KernelObject}(\alpha).

 ‣ AddKernelEmbedding( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation KernelEmbedding. F: \alpha \mapsto \iota.

 ‣ AddKernelEmbeddingWithGivenKernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation KernelEmbeddingWithGivenKernelObject. F: (\alpha, K) \mapsto \iota.

 ‣ AddKernelLift( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation KernelLift. F: (\alpha, \tau) \mapsto u(\tau).

 ‣ AddKernelLiftWithGivenKernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation KernelLiftWithGivenKernelObject. F: (\alpha, \tau, K) \mapsto u.

##### 6.1-11 KernelObjectFunctorial
 ‣ KernelObjectFunctorial( L ) ( operation )

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

The argument is a list L = [ \alpha: A \rightarrow B, [ \mu: A \rightarrow A', \nu: B \rightarrow B' ], \alpha': A' \rightarrow B' ] of morphisms. The output is the morphism \mathrm{KernelObject}( \alpha ) \rightarrow \mathrm{KernelObject}( \alpha' ) given by the functorality of the kernel.

##### 6.1-12 KernelObjectFunctorial
 ‣ KernelObjectFunctorial( alpha, mu, alpha_prime ) ( operation )

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

The arguments are three morphisms \alpha: A \rightarrow B, \mu: A \rightarrow A', \alpha': A' \rightarrow B'. The output is the morphism \mathrm{KernelObject}( \alpha ) \rightarrow \mathrm{KernelObject}( \alpha' ) given by the functorality of the kernel.

##### 6.1-13 KernelObjectFunctorialWithGivenKernelObjects
 ‣ KernelObjectFunctorialWithGivenKernelObjects( s, alpha, mu, alpha_prime, r ) ( operation )

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

The arguments are an object s = \mathrm{KernelObject}( \alpha ), three morphisms \alpha: A \rightarrow B, \mu: A \rightarrow A', \alpha': A' \rightarrow B', and an object r = \mathrm{KernelObject}( \alpha' ). The output is the morphism \mathrm{KernelObject}( \alpha ) \rightarrow \mathrm{KernelObject}( \alpha' ) given by the functorality of the kernel.

 ‣ AddKernelObjectFunctorialWithGivenKernelObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation KernelObjectFunctorialWithGivenKernelObjects. F: (\mathrm{KernelObject}( \alpha ), \alpha, \mu, \alpha', \mathrm{KernelObject}( \alpha' )) \mapsto (\mathrm{KernelObject}( \alpha ) \rightarrow \mathrm{KernelObject}( \alpha' )).

#### 6.2 Cokernel

For a given morphism \alpha: A \rightarrow B, a cokernel of \alpha consists of three parts:

• an object K,

• a morphism \epsilon: B \rightarrow K such that \epsilon \circ \alpha \sim_{A,K} 0,

• a dependent function u mapping each \tau: B \rightarrow T satisfying \tau \circ \alpha \sim_{A, T} 0 to a morphism u(\tau):K \rightarrow T such that u(\tau) \circ \epsilon \sim_{B,T} \tau.

The triple ( K, \epsilon, u ) is called a cokernel of \alpha if the morphisms u( \tau ) are uniquely determined up to congruence of morphisms. We denote the object K of such a triple by \mathrm{CokernelObject}(\alpha). We say that the morphism u(\tau) is induced by the universal property of the cokernel. \\ \mathrm{CokernelObject} is a functorial operation. This means: for \mu: A \rightarrow A', \nu: B \rightarrow B', \alpha: A \rightarrow B, \alpha': A' \rightarrow B' such that \nu \circ \alpha \sim_{A,B'} \alpha' \circ \mu, we obtain a morphism \mathrm{CokernelObject}( \alpha ) \rightarrow \mathrm{CokernelObject}( \alpha' ).

##### 6.2-1 CokernelObject
 ‣ CokernelObject( alpha ) ( attribute )

Returns: an object

The argument is a morphism \alpha: A \rightarrow B. The output is the cokernel K of \alpha.

##### 6.2-2 CokernelProjection
 ‣ CokernelProjection( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}(B, \mathrm{CokernelObject}( \alpha ))

The argument is a morphism \alpha: A \rightarrow B. The output is the cokernel projection \epsilon: B \rightarrow \mathrm{CokernelObject}( \alpha ).

##### 6.2-3 CokernelProjectionWithGivenCokernelObject
 ‣ CokernelProjectionWithGivenCokernelObject( alpha, K ) ( operation )

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

The arguments are a morphism \alpha: A \rightarrow B and an object K = \mathrm{CokernelObject}(\alpha). The output is the cokernel projection \epsilon: B \rightarrow \mathrm{CokernelObject}( \alpha ).

##### 6.2-4 CokernelColift
 ‣ CokernelColift( alpha, tau ) ( operation )

Returns: a morphism in \mathrm{Hom}(\mathrm{CokernelObject}(\alpha),T)

The arguments are a morphism \alpha: A \rightarrow B and a test morphism \tau: B \rightarrow T satisfying \tau \circ \alpha \sim_{A, T} 0. The output is the morphism u(\tau): \mathrm{CokernelObject}(\alpha) \rightarrow T given by the universal property of the cokernel.

##### 6.2-5 CokernelColiftWithGivenCokernelObject
 ‣ CokernelColiftWithGivenCokernelObject( alpha, tau, K ) ( operation )

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

The arguments are a morphism \alpha: A \rightarrow B, a test morphism \tau: B \rightarrow T satisfying \tau \circ \alpha \sim_{A, T} 0, and an object K = \mathrm{CokernelObject}(\alpha). The output is the morphism u(\tau): K \rightarrow T given by the universal property of the cokernel.

 ‣ AddCokernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation CokernelObject. F: \alpha \mapsto K.

 ‣ AddCokernelProjection( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation CokernelProjection. F: \alpha \mapsto \epsilon.

 ‣ AddCokernelProjectionWithGivenCokernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation CokernelProjection. F: (\alpha, K) \mapsto \epsilon.

 ‣ AddCokernelColift( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation CokernelProjection. F: (\alpha, \tau) \mapsto u(\tau).

 ‣ AddCokernelColiftWithGivenCokernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation CokernelProjection. F: (\alpha, \tau, K) \mapsto u(\tau).

##### 6.2-11 CokernelObjectFunctorial
 ‣ CokernelObjectFunctorial( L ) ( operation )

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

The argument is a list L = [ \alpha: A \rightarrow B, [ \mu:A \rightarrow A', \nu: B \rightarrow B' ], \alpha': A' \rightarrow B' ]. The output is the morphism \mathrm{CokernelObject}( \alpha ) \rightarrow \mathrm{CokernelObject}( \alpha' ) given by the functorality of the cokernel.

##### 6.2-12 CokernelObjectFunctorial
 ‣ CokernelObjectFunctorial( alpha, nu, alpha_prime ) ( operation )

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

The arguments are three morphisms \alpha: A \rightarrow B, \nu: B \rightarrow B', \alpha': A' \rightarrow B'. The output is the morphism \mathrm{CokernelObject}( \alpha ) \rightarrow \mathrm{CokernelObject}( \alpha' ) given by the functorality of the cokernel.

##### 6.2-13 CokernelObjectFunctorialWithGivenCokernelObjects
 ‣ CokernelObjectFunctorialWithGivenCokernelObjects( s, alpha, nu, alpha_prime, r ) ( operation )

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

The arguments are an object s = \mathrm{CokernelObject}( \alpha ), three morphisms \alpha: A \rightarrow B, \nu: B \rightarrow B', \alpha': A' \rightarrow B', and an object r = \mathrm{CokernelObject}( \alpha' ). The output is the morphism \mathrm{CokernelObject}( \alpha ) \rightarrow \mathrm{CokernelObject}( \alpha' ) given by the functorality of the cokernel.

 ‣ AddCokernelObjectFunctorialWithGivenCokernelObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation CokernelObjectFunctorialWithGivenCokernelObjects. F: (\mathrm{CokernelObject}( \alpha ), \alpha, \nu, \alpha', \mathrm{CokernelObject}( \alpha' )) \mapsto (\mathrm{CokernelObject}( \alpha ) \rightarrow \mathrm{CokernelObject}( \alpha' )).

#### 6.3 Zero Object

A zero object consists of three parts:

• an object Z,

• a function u_{\mathrm{in}} mapping each object A to a morphism u_{\mathrm{in}}(A): A \rightarrow Z,

• a function u_{\mathrm{out}} mapping each object A to a morphism u_{\mathrm{out}}(A): Z \rightarrow A.

The triple (Z, u_{\mathrm{in}}, u_{\mathrm{out}}) is called a zero object if the morphisms u_{\mathrm{in}}(A), u_{\mathrm{out}}(A) are uniquely determined up to congruence of morphisms. We denote the object Z of such a triple by \mathrm{ZeroObject}. We say that the morphisms u_{\mathrm{in}}(A) and u_{\mathrm{out}}(A) are induced by the universal property of the zero object.

##### 6.3-1 ZeroObject
 ‣ ZeroObject( C ) ( attribute )

Returns: an object

The argument is a category C. The output is a zero object Z of C.

##### 6.3-2 ZeroObject
 ‣ ZeroObject( c ) ( attribute )

Returns: an object

This is a convenience method. The argument is a cell c. The output is a zero object Z of the category C for which c \in C.

##### 6.3-3 MorphismFromZeroObject
 ‣ MorphismFromZeroObject( A ) ( attribute )

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

This is a convenience method. The argument is an object A. It calls \mathrm{UniversalMorphismFromZeroObject} on A.

##### 6.3-4 MorphismIntoZeroObject
 ‣ MorphismIntoZeroObject( A ) ( attribute )

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

This is a convenience method. The argument is an object A. It calls \mathrm{UniversalMorphismIntoZeroObject} on A.

##### 6.3-5 UniversalMorphismFromZeroObject
 ‣ UniversalMorphismFromZeroObject( A ) ( attribute )

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

The argument is an object A. The output is the universal morphism u_{\mathrm{out}}: \mathrm{ZeroObject} \rightarrow A.

##### 6.3-6 UniversalMorphismFromZeroObjectWithGivenZeroObject
 ‣ UniversalMorphismFromZeroObjectWithGivenZeroObject( A, Z ) ( operation )

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

The arguments are an object A, and a zero object Z = \mathrm{ZeroObject}. The output is the universal morphism u_{\mathrm{out}}: Z \rightarrow A.

##### 6.3-7 UniversalMorphismIntoZeroObject
 ‣ UniversalMorphismIntoZeroObject( A ) ( attribute )

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

The argument is an object A. The output is the universal morphism u_{\mathrm{in}}: A \rightarrow \mathrm{ZeroObject}.

##### 6.3-8 UniversalMorphismIntoZeroObjectWithGivenZeroObject
 ‣ UniversalMorphismIntoZeroObjectWithGivenZeroObject( A, Z ) ( operation )

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

The arguments are an object A, and a zero object Z = \mathrm{ZeroObject}. The output is the universal morphism u_{\mathrm{in}}: A \rightarrow Z.

##### 6.3-9 IsomorphismFromZeroObjectToInitialObject
 ‣ IsomorphismFromZeroObjectToInitialObject( C ) ( attribute )

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

The argument is a category C. The output is the unique isomorphism \mathrm{ZeroObject} \rightarrow \mathrm{InitialObject}.

##### 6.3-10 IsomorphismFromInitialObjectToZeroObject
 ‣ IsomorphismFromInitialObjectToZeroObject( C ) ( attribute )

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

The argument is a category C. The output is the unique isomorphism \mathrm{InitialObject} \rightarrow \mathrm{ZeroObject}.

##### 6.3-11 IsomorphismFromZeroObjectToTerminalObject
 ‣ IsomorphismFromZeroObjectToTerminalObject( C ) ( attribute )

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

The argument is a category C. The output is the unique isomorphism \mathrm{ZeroObject} \rightarrow \mathrm{TerminalObject}.

##### 6.3-12 IsomorphismFromTerminalObjectToZeroObject
 ‣ IsomorphismFromTerminalObjectToZeroObject( C ) ( attribute )

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

The argument is a category C. The output is the unique isomorphism \mathrm{TerminalObject} \rightarrow \mathrm{ZeroObject}.

 ‣ AddZeroObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation ZeroObject. F: () \mapsto \mathrm{ZeroObject}.

 ‣ AddUniversalMorphismIntoZeroObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoZeroObject. F: A \mapsto u_{\mathrm{in}}(A).

 ‣ AddUniversalMorphismIntoZeroObjectWithGivenZeroObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoZeroObjectWithGivenZeroObject. F: (A, Z) \mapsto u_{\mathrm{in}}(A).

 ‣ AddUniversalMorphismFromZeroObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromZeroObject. F: A \mapsto u_{\mathrm{out}}(A).

 ‣ AddUniversalMorphismFromZeroObjectWithGivenZeroObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromZeroObjectWithGivenZeroObject. F: (A,Z) \mapsto u_{\mathrm{out}}(A).

 ‣ AddIsomorphismFromZeroObjectToInitialObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromZeroObjectToInitialObject. F: () \mapsto (\mathrm{ZeroObject} \rightarrow \mathrm{InitialObject}).

 ‣ AddIsomorphismFromInitialObjectToZeroObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromInitialObjectToZeroObject. F: () \mapsto ( \mathrm{InitialObject} \rightarrow \mathrm{ZeroObject}).

 ‣ AddIsomorphismFromZeroObjectToTerminalObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromZeroObjectToTerminalObject. F: () \mapsto (\mathrm{ZeroObject} \rightarrow \mathrm{TerminalObject}).

 ‣ AddIsomorphismFromTerminalObjectToZeroObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromTerminalObjectToZeroObject. F: () \mapsto ( \mathrm{TerminalObject} \rightarrow \mathrm{ZeroObject}).

##### 6.3-22 ZeroObjectFunctorial
 ‣ ZeroObjectFunctorial( C ) ( attribute )

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

The argument is a category C. The output is the unique morphism \mathrm{ZeroObject} \rightarrow \mathrm{ZeroObject}.

 ‣ AddZeroObjectFunctorial( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation ZeroObjectFunctorial. F: () \mapsto (T \rightarrow T).

#### 6.4 Terminal Object

A terminal object consists of two parts:

• an object T,

• a function u mapping each object A to a morphism u( A ): A \rightarrow T.

The pair ( T, u ) is called a terminal object if the morphisms u( A ) are uniquely determined up to congruence of morphisms. We denote the object T of such a pair by \mathrm{TerminalObject}. We say that the morphism u( A ) is induced by the universal property of the terminal object. \\ \mathrm{TerminalObject} is a functorial operation. This just means: There exists a unique morphism T \rightarrow T.

##### 6.4-1 TerminalObject
 ‣ TerminalObject( C ) ( attribute )

Returns: an object

The argument is a category C. The output is a terminal object T of C.

##### 6.4-2 TerminalObject
 ‣ TerminalObject( c ) ( attribute )

Returns: an object

This is a convenience method. The argument is a cell c. The output is a terminal object T of the category C for which c \in C.

##### 6.4-3 UniversalMorphismIntoTerminalObject
 ‣ UniversalMorphismIntoTerminalObject( A ) ( attribute )

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

The argument is an object A. The output is the universal morphism u(A): A \rightarrow \mathrm{TerminalObject}.

##### 6.4-4 UniversalMorphismIntoTerminalObjectWithGivenTerminalObject
 ‣ UniversalMorphismIntoTerminalObjectWithGivenTerminalObject( A, T ) ( operation )

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

The argument are an object A, and an object T = \mathrm{TerminalObject}. The output is the universal morphism u(A): A \rightarrow T.

 ‣ AddTerminalObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation TerminalObject. F: () \mapsto T.

 ‣ AddUniversalMorphismIntoTerminalObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoTerminalObject. F: A \mapsto u(A).

 ‣ AddUniversalMorphismIntoTerminalObjectWithGivenTerminalObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoTerminalObjectWithGivenTerminalObject. F: (A,T) \mapsto u(A).

##### 6.4-8 TerminalObjectFunctorial
 ‣ TerminalObjectFunctorial( C ) ( attribute )

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

The argument is a category C. The output is the unique morphism \mathrm{TerminalObject} \rightarrow \mathrm{TerminalObject}.

 ‣ AddTerminalObjectFunctorial( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation TerminalObjectFunctorial. F: () \mapsto (T \rightarrow T).

#### 6.5 Initial Object

An initial object consists of two parts:

• an object I,

• a function u mapping each object A to a morphism u( A ): I \rightarrow A.

The pair (I,u) is called a initial object if the morphisms u(A) are uniquely determined up to congruence of morphisms. We denote the object I of such a triple by \mathrm{InitialObject}. We say that the morphism u( A ) is induced by the universal property of the initial object. \\ \mathrm{InitialObject} is a functorial operation. This just means: There exists a unique morphisms I \rightarrow I.

##### 6.5-1 InitialObject
 ‣ InitialObject( C ) ( attribute )

Returns: an object

The argument is a category C. The output is an initial object I of C.

##### 6.5-2 InitialObject
 ‣ InitialObject( c ) ( attribute )

Returns: an object

This is a convenience method. The argument is a cell c. The output is an initial object I of the category C for which c \in C.

##### 6.5-3 UniversalMorphismFromInitialObject
 ‣ UniversalMorphismFromInitialObject( A ) ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{InitialObject} \rightarrow A).

The argument is an object A. The output is the universal morphism u(A): \mathrm{InitialObject} \rightarrow A.

##### 6.5-4 UniversalMorphismFromInitialObjectWithGivenInitialObject
 ‣ UniversalMorphismFromInitialObjectWithGivenInitialObject( A, I ) ( operation )

Returns: a morphism in \mathrm{Hom}(\mathrm{InitialObject} \rightarrow A).

The arguments are an object A, and an object I = \mathrm{InitialObject}. The output is the universal morphism u(A): \mathrm{InitialObject} \rightarrow A.

 ‣ AddInitialObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation InitialObject. F: () \mapsto I.

 ‣ AddUniversalMorphismFromInitialObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromInitialObject. F: A \mapsto u(A).

 ‣ AddUniversalMorphismFromInitialObjectWithGivenInitialObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromInitialObjectWithGivenInitialObject. F: (A,I) \mapsto u(A).

##### 6.5-8 InitialObjectFunctorial
 ‣ InitialObjectFunctorial( C ) ( attribute )

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

The argument is a category C. The output is the unique morphism \mathrm{InitialObject} \rightarrow \mathrm{InitialObject}.

 ‣ AddInitialObjectFunctorial( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation InitialObjectFunctorial. F: () \rightarrow ( I \rightarrow I ).

#### 6.6 Direct Sum

For a given list D = (S_1, \dots, S_n) in an Ab-category, a direct sum consists of five parts:

• an object S,

• a list of morphisms \pi = (\pi_i: S \rightarrow S_i)_{i = 1 \dots n},

• a list of morphisms \iota = (\iota_i: S_i \rightarrow S)_{i = 1 \dots n},

• a dependent function u_{\mathrm{in}} mapping every list \tau = ( \tau_i: T \rightarrow S_i )_{i = 1 \dots n} to a morphism u_{\mathrm{in}}(\tau): T \rightarrow S such that \pi_i \circ u_{\mathrm{in}}(\tau) \sim_{T,S_i} \tau_i for all i = 1, \dots, n.

• a dependent function u_{\mathrm{out}} mapping every list \tau = ( \tau_i: S_i \rightarrow T )_{i = 1 \dots n} to a morphism u_{\mathrm{out}}(\tau): S \rightarrow T such that u_{\mathrm{out}}(\tau) \circ \iota_i \sim_{S_i, T} \tau_i for all i = 1, \dots, n,

such that

• \sum_{i=1}^{n} \iota_i \circ \pi_i = \mathrm{id}_S,

• \pi_j \circ \iota_i = \delta_{i,j},

where \delta_{i,j} \in \mathrm{Hom}( S_i, S_j ) is the identity if i=j, and 0 otherwise. The 5-tuple (S, \pi, \iota, u_{\mathrm{in}}, u_{\mathrm{out}}) is called a direct sum of D. We denote the object S of such a 5-tuple by \bigoplus_{i=1}^n S_i. We say that the morphisms u_{\mathrm{in}}(\tau), u_{\mathrm{out}}(\tau) are induced by the universal property of the direct sum. \\ \mathrm{DirectSum} is a functorial operation. This means: For (\mu_i: S_i \rightarrow S'_i)_{i=1\dots n}, we obtain a morphism \bigoplus_{i=1}^n S_i \rightarrow \bigoplus_{i=1}^n S_i'.

##### 6.6-1 DirectSumOp
 ‣ DirectSumOp( D, method_selection_object ) ( operation )

Returns: an object

The argument is a list of objects D = (S_1, \dots, S_n) and an object for method selection. The output is the direct sum \bigoplus_{i=1}^n S_i.

##### 6.6-2 ProjectionInFactorOfDirectSum
 ‣ ProjectionInFactorOfDirectSum( D, k ) ( operation )

Returns: a morphism in \mathrm{Hom}( \bigoplus_{i=1}^n S_i, S_k )

The arguments are a list of objects D = (S_1, \dots, S_n) and an integer k. The output is the k-th projection \pi_k: \bigoplus_{i=1}^n S_i \rightarrow S_k.

##### 6.6-3 ProjectionInFactorOfDirectSumOp
 ‣ ProjectionInFactorOfDirectSumOp( D, k, method_selection_object ) ( operation )

Returns: a morphism in \mathrm{Hom}( \bigoplus_{i=1}^n S_i, S_k )

The arguments are a list of objects D = (S_1, \dots, S_n), an integer k, and an object for method selection. The output is the k-th projection \pi_k: \bigoplus_{i=1}^n S_i \rightarrow S_k.

##### 6.6-4 ProjectionInFactorOfDirectSumWithGivenDirectSum
 ‣ ProjectionInFactorOfDirectSumWithGivenDirectSum( D, k, S ) ( operation )

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

The arguments are a list of objects D = (S_1, \dots, S_n), an integer k, and an object S = \bigoplus_{i=1}^n S_i. The output is the k-th projection \pi_k: S \rightarrow S_k.

##### 6.6-5 InjectionOfCofactorOfDirectSum
 ‣ InjectionOfCofactorOfDirectSum( D, k ) ( operation )

Returns: a morphism in \mathrm{Hom}( S_k, \bigoplus_{i=1}^n S_i )

The arguments are a list of objects D = (S_1, \dots, S_n) and an integer k. The output is the k-th injection \iota_k: S_k \rightarrow \bigoplus_{i=1}^n S_i.

##### 6.6-6 InjectionOfCofactorOfDirectSumOp
 ‣ InjectionOfCofactorOfDirectSumOp( D, k, method_selection_object ) ( operation )

Returns: a morphism in \mathrm{Hom}( S_k, \bigoplus_{i=1}^n S_i )

The arguments are a list of objects D = (S_1, \dots, S_n), an integer k, and an object for method selection. The output is the k-th injection \iota_k: S_k \rightarrow \bigoplus_{i=1}^n S_i.

##### 6.6-7 InjectionOfCofactorOfDirectSumWithGivenDirectSum
 ‣ InjectionOfCofactorOfDirectSumWithGivenDirectSum( D, k, S ) ( operation )

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

The arguments are a list of objects D = (S_1, \dots, S_n), an integer k, and an object S = \bigoplus_{i=1}^n S_i. The output is the k-th injection \iota_k: S_k \rightarrow S.

##### 6.6-8 UniversalMorphismIntoDirectSum
 ‣ UniversalMorphismIntoDirectSum( arg ) ( function )

Returns: a morphism in \mathrm{Hom}(T, \bigoplus_{i=1}^n S_i)

This is a convenience method. There are three different ways to use this method:

• The arguments are a list of objects D = (S_1, \dots, S_n) and a list of morphisms \tau = ( \tau_i: T \rightarrow S_i )_{i = 1 \dots n}.

• The argument is a list of morphisms \tau = ( \tau_i: T \rightarrow S_i )_{i = 1 \dots n}.

• The arguments are morphisms \tau_1: T \rightarrow S_1, \dots, \tau_n: T \rightarrow S_n.

The output is the morphism u_{\mathrm{in}}(\tau): T \rightarrow \bigoplus_{i=1}^n S_i given by the universal property of the direct sum.

##### 6.6-9 UniversalMorphismIntoDirectSumOp
 ‣ UniversalMorphismIntoDirectSumOp( D, tau, method_selection_object ) ( operation )

Returns: a morphism in \mathrm{Hom}(T, \bigoplus_{i=1}^n S_i)

The arguments are a list of objects D = (S_1, \dots, S_n), a list of morphisms \tau = ( \tau_i: T \rightarrow S_i )_{i = 1 \dots n}, and an object for method selection. The output is the morphism u_{\mathrm{in}}(\tau): T \rightarrow \bigoplus_{i=1}^n S_i given by the universal property of the direct sum.

##### 6.6-10 UniversalMorphismIntoDirectSumWithGivenDirectSum
 ‣ UniversalMorphismIntoDirectSumWithGivenDirectSum( D, tau, S ) ( operation )

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

The arguments are a list of objects D = (S_1, \dots, S_n), a list of morphisms \tau = ( \tau_i: T \rightarrow S_i )_{i = 1 \dots n}, and an object S = \bigoplus_{i=1}^n S_i. The output is the morphism u_{\mathrm{in}}(\tau): T \rightarrow S given by the universal property of the direct sum.

##### 6.6-11 UniversalMorphismFromDirectSum
 ‣ UniversalMorphismFromDirectSum( arg ) ( function )

Returns: a morphism in \mathrm{Hom}(\bigoplus_{i=1}^n S_i, T)

This is a convenience method. There are three different ways to use this method:

• The arguments are a list of objects D = (S_1, \dots, S_n) and a list of morphisms \tau = ( \tau_i: S_i \rightarrow T )_{i = 1 \dots n}.

• The argument is a list of morphisms \tau = ( \tau_i: S_i \rightarrow T )_{i = 1 \dots n}.

• The arguments are morphisms S_1 \rightarrow T, \dots, S_n \rightarrow T.

The output is the morphism u_{\mathrm{out}}(\tau): \bigoplus_{i=1}^n S_i \rightarrow T given by the universal property of the direct sum.

##### 6.6-12 UniversalMorphismFromDirectSumOp
 ‣ UniversalMorphismFromDirectSumOp( D, tau, method_selection_object ) ( operation )

Returns: a morphism in \mathrm{Hom}(\bigoplus_{i=1}^n S_i, T)

The arguments are a list of objects D = (S_1, \dots, S_n), a list of morphisms \tau = ( \tau_i: S_i \rightarrow T )_{i = 1 \dots n}, and an object for method selection. The output is the morphism u_{\mathrm{out}}(\tau): \bigoplus_{i=1}^n S_i \rightarrow T given by the universal property of the direct sum.

##### 6.6-13 UniversalMorphismFromDirectSumWithGivenDirectSum
 ‣ UniversalMorphismFromDirectSumWithGivenDirectSum( D, tau, S ) ( operation )

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

The arguments are a list of objects D = (S_1, \dots, S_n), a list of morphisms \tau = ( \tau_i: S_i \rightarrow T )_{i = 1 \dots n}, and an object S = \bigoplus_{i=1}^n S_i. The output is the morphism u_{\mathrm{out}}(\tau): S \rightarrow T given by the universal property of the direct sum.

##### 6.6-14 IsomorphismFromDirectSumToDirectProduct
 ‣ IsomorphismFromDirectSumToDirectProduct( D ) ( operation )

Returns: a morphism in \mathrm{Hom}( \bigoplus_{i=1}^n S_i, \prod_{i=1}^{n}S_i )

The argument is a list of objects D = (S_1, \dots, S_n). The output is the canonical isomorphism \bigoplus_{i=1}^n S_i \rightarrow \prod_{i=1}^{n}S_i.

##### 6.6-15 IsomorphismFromDirectSumToDirectProductOp
 ‣ IsomorphismFromDirectSumToDirectProductOp( D, method_selection_object ) ( operation )

Returns: a morphism in \mathrm{Hom}( \bigoplus_{i=1}^n S_i, \prod_{i=1}^{n}S_i )

The arguments are a list of objects D = (S_1, \dots, S_n) and an object for method selection. The output is the canonical isomorphism \bigoplus_{i=1}^n S_i \rightarrow \prod_{i=1}^{n}S_i.

##### 6.6-16 IsomorphismFromDirectProductToDirectSum
 ‣ IsomorphismFromDirectProductToDirectSum( D ) ( operation )

Returns: a morphism in \mathrm{Hom}( \prod_{i=1}^{n}S_i, \bigoplus_{i=1}^n S_i )

The argument is a list of objects D = (S_1, \dots, S_n). The output is the canonical isomorphism \prod_{i=1}^{n}S_i \rightarrow \bigoplus_{i=1}^n S_i.

##### 6.6-17 IsomorphismFromDirectProductToDirectSumOp
 ‣ IsomorphismFromDirectProductToDirectSumOp( D, method_selection_object ) ( operation )

Returns: a morphism in \mathrm{Hom}( \prod_{i=1}^{n}S_i, \bigoplus_{i=1}^n S_i )

The argument is a list of objects D = (S_1, \dots, S_n) and an object for method selection. The output is the canonical isomorphism \prod_{i=1}^{n}S_i \rightarrow \bigoplus_{i=1}^n S_i.

##### 6.6-18 IsomorphismFromDirectSumToCoproduct
 ‣ IsomorphismFromDirectSumToCoproduct( D ) ( operation )

Returns: a morphism in \mathrm{Hom}( \bigoplus_{i=1}^n S_i, \bigsqcup_{i=1}^{n}S_i )

The argument is a list of objects D = (S_1, \dots, S_n). The output is the canonical isomorphism \bigoplus_{i=1}^n S_i \rightarrow \bigsqcup_{i=1}^{n}S_i.

##### 6.6-19 IsomorphismFromDirectSumToCoproductOp
 ‣ IsomorphismFromDirectSumToCoproductOp( D, method_selection_object ) ( operation )

Returns: a morphism in \mathrm{Hom}( \bigoplus_{i=1}^n S_i, \bigsqcup_{i=1}^{n}S_i )

The argument is a list of objects D = (S_1, \dots, S_n) and an object for method selection. The output is the canonical isomorphism \bigoplus_{i=1}^n S_i \rightarrow \bigsqcup_{i=1}^{n}S_i.

##### 6.6-20 IsomorphismFromCoproductToDirectSum
 ‣ IsomorphismFromCoproductToDirectSum( D ) ( operation )

Returns: a morphism in \mathrm{Hom}( \bigsqcup_{i=1}^{n}S_i, \bigoplus_{i=1}^n S_i )

The argument is a list of objects D = (S_1, \dots, S_n). The output is the canonical isomorphism \bigsqcup_{i=1}^{n}S_i \rightarrow \bigoplus_{i=1}^n S_i.

##### 6.6-21 IsomorphismFromCoproductToDirectSumOp
 ‣ IsomorphismFromCoproductToDirectSumOp( D, method_selection_object ) ( operation )

Returns: a morphism in \mathrm{Hom}( \bigsqcup_{i=1}^{n}S_i, \bigoplus_{i=1}^n S_i )

The argument is a list of objects D = (S_1, \dots, S_n) and an object for method selection. The output is the canonical isomorphism \bigsqcup_{i=1}^{n}S_i \rightarrow \bigoplus_{i=1}^n S_i.

##### 6.6-22 MorphismBetweenDirectSums
 ‣ MorphismBetweenDirectSums( M ) ( operation )
 ‣ MorphismBetweenDirectSums( S, M, T ) ( operation )

Returns: a morphism in \mathrm{Hom}(\bigoplus_{i=1}^{m}A_i, \bigoplus_{j=1}^n B_j)

The argument M = ( ( \phi_{i,j}: A_i \rightarrow B_j )_{j = 1 \dots n} )_{i = 1 \dots m} is a list of lists of morphisms. The output is the morphism \bigoplus_{i=1}^{m}A_i \rightarrow \bigoplus_{j=1}^n B_j defined by the matrix M. The extra arguments S = \bigoplus_{i=1}^{m}A_i and T = \bigoplus_{j=1}^n B_j are source and target of the output, respectively. They must be provided in case M is an empty list or a list of empty lists.

##### 6.6-23 MorphismBetweenDirectSumsOp
 ‣ MorphismBetweenDirectSumsOp( M, m, n, method_selection_morphism ) ( operation )

Returns: a morphism in \mathrm{Hom}(\bigoplus_{i=1}^{m}A_i, \bigoplus_{j=1}^n B_j)

The arguments are a list M = ( \phi_{1,1}, \phi_{1,2}, \dots, \phi_{1,n}, \phi_{2,1}, \dots, \phi_{m,n} ) of morphisms \phi_{i,j}: A_i \rightarrow B_j, an integer m, an integer n, and a method selection morphism. The output is the morphism \bigoplus_{i=1}^{m}A_i \rightarrow \bigoplus_{j=1}^n B_j defined by the list M regarded as a matrix of dimension m \times n.

 ‣ AddProjectionInFactorOfDirectSum( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation ProjectionInFactorOfDirectSum. F: (D,k) \mapsto \pi_{k}.

 ‣ AddProjectionInFactorOfDirectSumWithGivenDirectSum( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation ProjectionInFactorOfDirectSumWithGivenDirectSum. F: (D,k,S) \mapsto \pi_{k}.

 ‣ AddInjectionOfCofactorOfDirectSum( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation InjectionOfCofactorOfDirectSum. F: (D,k) \mapsto \iota_{k}.

 ‣ AddInjectionOfCofactorOfDirectSumWithGivenDirectSum( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation InjectionOfCofactorOfDirectSumWithGivenDirectSum. F: (D,k,S) \mapsto \iota_{k}.

 ‣ AddUniversalMorphismIntoDirectSum( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoDirectSum. F: (D,\tau) \mapsto u_{\mathrm{in}}(\tau).

 ‣ AddUniversalMorphismIntoDirectSumWithGivenDirectSum( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoDirectSumWithGivenDirectSum. F: (D,\tau,S) \mapsto u_{\mathrm{in}}(\tau).

 ‣ AddUniversalMorphismFromDirectSum( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromDirectSum. F: (D,\tau) \mapsto u_{\mathrm{out}}(\tau).

 ‣ AddUniversalMorphismFromDirectSumWithGivenDirectSum( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromDirectSumWithGivenDirectSum. F: (D,\tau,S) \mapsto u_{\mathrm{out}}(\tau).

 ‣ AddIsomorphismFromDirectSumToDirectProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromDirectSumToDirectProduct. F: D \mapsto (\bigoplus_{i=1}^n S_i \rightarrow \prod_{i=1}^{n}S_i).

 ‣ AddIsomorphismFromDirectProductToDirectSum( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromDirectProductToDirectSum. F: D \mapsto ( \prod_{i=1}^{n}S_i \rightarrow \bigoplus_{i=1}^n S_i ).

 ‣ AddIsomorphismFromDirectSumToCoproduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromDirectSumToCoproduct. F: D \mapsto ( \bigoplus_{i=1}^n S_i \rightarrow \bigsqcup_{i=1}^{n}S_i ).

 ‣ AddIsomorphismFromCoproductToDirectSum( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromCoproductToDirectSum. F: D \mapsto ( \bigsqcup_{i=1}^{n}S_i \rightarrow \bigoplus_{i=1}^n S_i ).

 ‣ AddDirectSum( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation DirectSum. F: D \mapsto \bigoplus_{i=1}^n S_i.

##### 6.6-37 DirectSumFunctorial
 ‣ DirectSumFunctorial( L ) ( operation )

Returns: a morphism in \mathrm{Hom}( \bigoplus_{i=1}^n S_i, \bigoplus_{i=1}^n S_i' )

The argument is a list of morphisms L = ( \mu_1: S_1 \rightarrow S_1', \dots, \mu_n: S_n \rightarrow S_n' ). The output is a morphism \bigoplus_{i=1}^n S_i \rightarrow \bigoplus_{i=1}^n S_i' given by the functorality of the direct sum.

##### 6.6-38 DirectSumFunctorialWithGivenDirectSums
 ‣ DirectSumFunctorialWithGivenDirectSums( d_1, L, d_2 ) ( operation )

Returns: a morphism in \mathrm{Hom}( d_1, d_2 )

The arguments are an object d_1 = \bigoplus_{i=1}^n S_i, a list of morphisms L = ( \mu_1: S_1 \rightarrow S_1', \dots, \mu_n: S_n \rightarrow S_n' ), and an object d_2 = \bigoplus_{i=1}^n S_i'. The output is a morphism d_1 \rightarrow d_2 given by the functorality of the direct sum.

 ‣ AddDirectSumFunctorialWithGivenDirectSums( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation DirectSumFunctorialWithGivenDirectSums. F: (\bigoplus_{i=1}^n S_i, ( \mu_1, \dots, \mu_n ), \bigoplus_{i=1}^n S_i') \mapsto (\bigoplus_{i=1}^n S_i \rightarrow \bigoplus_{i=1}^n S_i').

#### 6.7 Coproduct

For a given list of objects D = ( I_1, \dots, I_n ), a coproduct of D consists of three parts:

• an object I,

• a list of morphisms \iota = ( \iota_i: I_i \rightarrow I )_{i = 1 \dots n}

• a dependent function u mapping each list of morphisms \tau = ( \tau_i: I_i \rightarrow T ) to a morphism u( \tau ): I \rightarrow T such that u( \tau ) \circ \iota_i \sim_{I_i, T} \tau_i for all i = 1, \dots, n.

The triple ( I, \iota, u ) is called a coproduct of D if the morphisms u( \tau ) are uniquely determined up to congruence of morphisms. We denote the object I of such a triple by \bigsqcup_{i=1}^n I_i. We say that the morphism u( \tau ) is induced by the universal property of the coproduct. \\ \mathrm{Coproduct} is a functorial operation. This means: For (\mu_i: I_i \rightarrow I'_i)_{i=1\dots n}, we obtain a morphism \bigsqcup_{i=1}^n I_i \rightarrow \bigsqcup_{i=1}^n I_i'.

##### 6.7-1 Coproduct
 ‣ Coproduct( D ) ( attribute )

Returns: an object

The argument is a list of objects D = ( I_1, \dots, I_n ). The output is the coproduct \bigsqcup_{i=1}^n I_i.

##### 6.7-2 Coproduct
 ‣ Coproduct( I1, I2 ) ( operation )

Returns: an object

This is a convenience method. The arguments are two objects I_1, I_2. The output is the coproduct I_1 \bigsqcup I_2.

##### 6.7-3 Coproduct
 ‣ Coproduct( I1, I2 ) ( operation )

Returns: an object

This is a convenience method. The arguments are three objects I_1, I_2, I_3. The output is the coproduct I_1 \bigsqcup I_2 \bigsqcup I_3.

##### 6.7-4 CoproductOp
 ‣ CoproductOp( D, method_selection_object ) ( operation )

Returns: an object

The arguments are a list of objects D = ( I_1, \dots, I_n ) and a method selection object. The output is the coproduct \bigsqcup_{i=1}^n I_i.

##### 6.7-5 InjectionOfCofactorOfCoproduct
 ‣ InjectionOfCofactorOfCoproduct( D, k ) ( operation )

Returns: a morphism in \mathrm{Hom}(I_k, \bigsqcup_{i=1}^n I_i)

The arguments are a list of objects D = ( I_1, \dots, I_n ) and an integer k. The output is the k-th injection \iota_k: I_k \rightarrow \bigsqcup_{i=1}^n I_i.

##### 6.7-6 InjectionOfCofactorOfCoproductOp
 ‣ InjectionOfCofactorOfCoproductOp( D, k, method_selection_object ) ( operation )

Returns: a morphism in \mathrm{Hom}(I_k, \bigsqcup_{i=1}^n I_i)

The arguments are a list of objects D = ( I_1, \dots, I_n ), an integer k, and a method selection object. The output is the k-th injection \iota_k: I_k \rightarrow \bigsqcup_{i=1}^n I_i.

##### 6.7-7 InjectionOfCofactorOfCoproductWithGivenCoproduct
 ‣ InjectionOfCofactorOfCoproductWithGivenCoproduct( D, k, I ) ( operation )

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

The arguments are a list of objects D = ( I_1, \dots, I_n ), an integer k, and an object I = \bigsqcup_{i=1}^n I_i. The output is the k-th injection \iota_k: I_k \rightarrow I.

##### 6.7-8 UniversalMorphismFromCoproduct
 ‣ UniversalMorphismFromCoproduct( arg ) ( function )

Returns: a morphism in \mathrm{Hom}(\bigsqcup_{i=1}^n I_i, T)

This is a convenience method. There are three different ways to use this method.

• The arguments are a list of objects D = ( I_1, \dots, I_n ), a list of morphisms \tau = ( \tau_i: I_i \rightarrow T ).

• The argument is a list of morphisms \tau = ( \tau_i: I_i \rightarrow T ).

• The arguments are morphisms \tau_1: I_1 \rightarrow T, \dots, \tau_n: I_n \rightarrow T

The output is the morphism u( \tau ): \bigsqcup_{i=1}^n I_i \rightarrow T given by the universal property of the coproduct.

##### 6.7-9 UniversalMorphismFromCoproductOp
 ‣ UniversalMorphismFromCoproductOp( D, tau, method_selection_object ) ( operation )

Returns: a morphism in \mathrm{Hom}(\bigsqcup_{i=1}^n I_i, T)

The arguments are a list of objects D = ( I_1, \dots, I_n ), a list of morphisms \tau = ( \tau_i: I_i \rightarrow T ), and a method selection object. The output is the morphism u( \tau ): \bigsqcup_{i=1}^n I_i \rightarrow T given by the universal property of the coproduct.

##### 6.7-10 UniversalMorphismFromCoproductWithGivenCoproduct
 ‣ UniversalMorphismFromCoproductWithGivenCoproduct( D, tau, I ) ( operation )

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

The arguments are a list of objects D = ( I_1, \dots, I_n ), a list of morphisms \tau = ( \tau_i: I_i \rightarrow T ), and an object I = \bigsqcup_{i=1}^n I_i. The output is the morphism u( \tau ): I \rightarrow T given by the universal property of the coproduct.

 ‣ AddCoproduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation Coproduct. F: ( (I_1, \dots, I_n) ) \mapsto I.

 ‣ AddInjectionOfCofactorOfCoproduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation InjectionOfCofactorOfCoproduct. F: ( (I_1, \dots, I_n), i ) \mapsto \iota_i.

 ‣ AddInjectionOfCofactorOfCoproductWithGivenCoproduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation InjectionOfCofactorOfCoproductWithGivenCoproduct. F: ( (I_1, \dots, I_n), i, I ) \mapsto \iota_i.

 ‣ AddUniversalMorphismFromCoproduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromCoproduct. F: ( (I_1, \dots, I_n), \tau ) \mapsto u( \tau ).

 ‣ AddUniversalMorphismFromCoproductWithGivenCoproduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromCoproductWithGivenCoproduct. F: ( (I_1, \dots, I_n), \tau, I ) \mapsto u( \tau ).

##### 6.7-16 CoproductFunctorial
 ‣ CoproductFunctorial( L ) ( operation )

Returns: a morphism in \mathrm{Hom}(\bigsqcup_{i=1}^n I_i, \bigsqcup_{i=1}^n I_i')

The argument is a list L = ( \mu_1: I_1 \rightarrow I_1', \dots, \mu_n: I_n \rightarrow I_n' ). The output is a morphism \bigsqcup_{i=1}^n I_i \rightarrow \bigsqcup_{i=1}^n I_i' given by the functorality of the coproduct.

##### 6.7-17 CoproductFunctorialWithGivenCoproducts
 ‣ CoproductFunctorialWithGivenCoproducts( s, L, r ) ( operation )

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

The arguments are an object s = \bigsqcup_{i=1}^n I_i, a list L = ( \mu_1: I_1 \rightarrow I_1', \dots, \mu_n: I_n \rightarrow I_n' ), and an object r = \bigsqcup_{i=1}^n I_i'. The output is a morphism \bigsqcup_{i=1}^n I_i \rightarrow \bigsqcup_{i=1}^n I_i' given by the functorality of the coproduct.

 ‣ AddCoproductFunctorialWithGivenCoproducts( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation CoproductFunctorialWithGivenCoproducts. F: (\bigsqcup_{i=1}^n I_i, (\mu_1, \dots, \mu_n), \bigsqcup_{i=1}^n I_i') \rightarrow (\bigsqcup_{i=1}^n I_i \rightarrow \bigsqcup_{i=1}^n I_i').

#### 6.8 Direct Product

For a given list of objects D = ( P_1, \dots, P_n ), a direct product of D consists of three parts:

• an object P,

• a list of morphisms \pi = ( \pi_i: P \rightarrow P_i )_{i = 1 \dots n}

• a dependent function u mapping each list of morphisms \tau = ( \tau_i: T \rightarrow P_i )_{i = 1, \dots, n} to a morphism u(\tau): T \rightarrow P such that \pi_i \circ u( \tau ) \sim_{T,P_i} \tau_i for all i = 1, \dots, n.

The triple ( P, \pi, u ) is called a direct product of D if the morphisms u( \tau ) are uniquely determined up to congruence of morphisms. We denote the object P of such a triple by \prod_{i=1}^n P_i. We say that the morphism u( \tau ) is induced by the universal property of the direct product. \\ \mathrm{DirectProduct} is a functorial operation. This means: For (\mu_i: P_i \rightarrow P'_i)_{i=1\dots n}, we obtain a morphism \prod_{i=1}^n P_i \rightarrow \prod_{i=1}^n P_i'.

##### 6.8-1 DirectProductOp
 ‣ DirectProductOp( D ) ( operation )

Returns: an object

The arguments are a list of objects D = ( P_1, \dots, P_n ) and an object for method selection. The output is the direct product \prod_{i=1}^n P_i.

##### 6.8-2 ProjectionInFactorOfDirectProduct
 ‣ ProjectionInFactorOfDirectProduct( D, k ) ( operation )

Returns: a morphism in \mathrm{Hom}(\prod_{i=1}^n P_i, P_k)

The arguments are a list of objects D = ( P_1, \dots, P_n ) and an integer k. The output is the k-th projection \pi_k: \prod_{i=1}^n P_i \rightarrow P_k.

##### 6.8-3 ProjectionInFactorOfDirectProductOp
 ‣ ProjectionInFactorOfDirectProductOp( D, k, method_selection_object ) ( operation )

Returns: a morphism in \mathrm{Hom}(\prod_{i=1}^n P_i, P_k)

The arguments are a list of objects D = ( P_1, \dots, P_n ), an integer k, and an object for method selection. The output is the k-th projection \pi_k: \prod_{i=1}^n P_i \rightarrow P_k.

##### 6.8-4 ProjectionInFactorOfDirectProductWithGivenDirectProduct
 ‣ ProjectionInFactorOfDirectProductWithGivenDirectProduct( D, k, P ) ( operation )

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

The arguments are a list of objects D = ( P_1, \dots, P_n ), an integer k, and an object P = \prod_{i=1}^n P_i. The output is the k-th projection \pi_k: P \rightarrow P_k.

##### 6.8-5 UniversalMorphismIntoDirectProduct
 ‣ UniversalMorphismIntoDirectProduct( arg ) ( function )

Returns: a morphism in \mathrm{Hom}(T, \prod_{i=1}^n P_i)

This is a convenience method. There are three different ways to use this method.

• The arguments are a list of objects D = ( P_1, \dots, P_n ) and a list of morphisms \tau = ( \tau_i: T \rightarrow P_i )_{i = 1, \dots, n}.

• The argument is a list of morphisms \tau = ( \tau_i: T \rightarrow P_i )_{i = 1, \dots, n}.

• The arguments are morphisms \tau_1: T \rightarrow P_1, \dots, \tau_n: T \rightarrow P_n.

The output is the morphism u(\tau): T \rightarrow \prod_{i=1}^n P_i given by the universal property of the direct product.

##### 6.8-6 UniversalMorphismIntoDirectProductOp
 ‣ UniversalMorphismIntoDirectProductOp( D, tau, method_selection_object ) ( operation )

Returns: a morphism in \mathrm{Hom}(T, \prod_{i=1}^n P_i)

The arguments are a list of objects D = ( P_1, \dots, P_n ), a list of morphisms \tau = ( \tau_i: T \rightarrow P_i )_{i = 1, \dots, n}, and an object for method selection. The output is the morphism u(\tau): T \rightarrow \prod_{i=1}^n P_i given by the universal property of the direct product.

##### 6.8-7 UniversalMorphismIntoDirectProductWithGivenDirectProduct
 ‣ UniversalMorphismIntoDirectProductWithGivenDirectProduct( D, tau, P ) ( operation )

Returns: a morphism in \mathrm{Hom}(T, \prod_{i=1}^n P_i)

The arguments are a list of objects D = ( P_1, \dots, P_n ), a list of morphisms \tau = ( \tau_i: T \rightarrow P_i )_{i = 1, \dots, n}, and an object P = \prod_{i=1}^n P_i. The output is the morphism u(\tau): T \rightarrow \prod_{i=1}^n P_i given by the universal property of the direct product.

 ‣ AddDirectProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation DirectProduct. F: ( (P_1, \dots, P_n) ) \mapsto P

 ‣ AddProjectionInFactorOfDirectProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation ProjectionInFactorOfDirectProduct. F: ( (P_1, \dots, P_n),k ) \mapsto \pi_k

 ‣ AddProjectionInFactorOfDirectProductWithGivenDirectProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation ProjectionInFactorOfDirectProductWithGivenDirectProduct. F: ( (P_1, \dots, P_n),k,P ) \mapsto \pi_k

 ‣ AddUniversalMorphismIntoDirectProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoDirectProduct. F: ( (P_1, \dots, P_n), \tau ) \mapsto u( \tau )

 ‣ AddUniversalMorphismIntoDirectProductWithGivenDirectProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoDirectProductWithGivenDirectProduct. F: ( (P_1, \dots, P_n), \tau, P ) \mapsto u( \tau )

##### 6.8-13 DirectProductFunctorial
 ‣ DirectProductFunctorial( L ) ( operation )

Returns: a morphism in \mathrm{Hom}( \prod_{i=1}^n P_i, \prod_{i=1}^n P_i' )

The argument is a list of morphisms L = (\mu_i: P_i \rightarrow P'_i)_{i=1\dots n}. The output is a morphism \prod_{i=1}^n P_i \rightarrow \prod_{i=1}^n P_i' given by the functorality of the direct product.

##### 6.8-14 DirectProductFunctorialWithGivenDirectProducts
 ‣ DirectProductFunctorialWithGivenDirectProducts( s, L, r ) ( operation )

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

The arguments are an object s = \prod_{i=1}^n P_i, a list of morphisms L = (\mu_i: P_i \rightarrow P'_i)_{i=1\dots n}, and an object r = \prod_{i=1}^n P_i'. The output is a morphism \prod_{i=1}^n P_i \rightarrow \prod_{i=1}^n P_i' given by the functorality of the direct product.

 ‣ AddDirectProductFunctorialWithGivenDirectProducts( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation DirectProductFunctorialWithGivenDirectProducts. F: ( \prod_{i=1}^n P_i, (\mu_i: P_i \rightarrow P'_i)_{i=1\dots n}, \prod_{i=1}^n P_i' ) \mapsto (\prod_{i=1}^n P_i \rightarrow \prod_{i=1}^n P_i')

#### 6.9 Fiber Product

For a given list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n}, a fiber product of D consists of three parts:

• an object P,

• a list of morphisms \pi = ( \pi_i: P \rightarrow P_i )_{i = 1 \dots n} such that \beta_i \circ \pi_i \sim_{P, B} \beta_j \circ \pi_j for all pairs i,j.

• a dependent function u mapping each list of morphisms \tau = ( \tau_i: T \rightarrow P_i ) such that \beta_i \circ \tau_i \sim_{T, B} \beta_j \circ \tau_j for all pairs i,j to a morphism u( \tau ): T \rightarrow P such that \pi_i \circ u( \tau ) \sim_{T, P_i} \tau_i for all i = 1, \dots, n.

The triple ( P, \pi, u ) is called a fiber product of D if the morphisms u( \tau ) are uniquely determined up to congruence of morphisms. We denote the object P of such a triple by \mathrm{FiberProduct}(D). We say that the morphism u( \tau ) is induced by the universal property of the fiber product. \\ \mathrm{FiberProduct} is a functorial operation. This means: For a second diagram D' = (\beta_i': P_i' \rightarrow B')_{i = 1 \dots n} and a natural morphism between pullback diagrams (i.e., a collection of morphisms (\mu_i: P_i \rightarrow P'_i)_{i=1\dots n} and \beta: B \rightarrow B' such that \beta_i' \circ \mu_i \sim_{P_i,B'} \beta \circ \beta_i for i = 1, \dots, n) we obtain a morphism \mathrm{FiberProduct}( D ) \rightarrow \mathrm{FiberProduct}( D' ).

##### 6.9-1 IsomorphismFromFiberProductToKernelOfDiagonalDifference
 ‣ IsomorphismFromFiberProductToKernelOfDiagonalDifference( D ) ( operation )

Returns: a morphism in \mathrm{Hom}(\mathrm{FiberProduct}(D), \Delta)

The argument is a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n}. The output is a morphism \mathrm{FiberProduct}(D) \rightarrow \Delta, where \Delta denotes the kernel object equalizing the morphisms \beta_i.

##### 6.9-2 IsomorphismFromFiberProductToKernelOfDiagonalDifferenceOp
 ‣ IsomorphismFromFiberProductToKernelOfDiagonalDifferenceOp( D, method_selection_morphism ) ( operation )

Returns: a morphism in \mathrm{Hom}(\mathrm{FiberProduct}(D), \Delta)

The arguments are a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} and a morphism for method selection. The output is a morphism \mathrm{FiberProduct}(D) \rightarrow \Delta, where \Delta denotes the kernel object equalizing the morphisms \beta_i.

 ‣ AddIsomorphismFromFiberProductToKernelOfDiagonalDifference( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromFiberProductToKernelOfDiagonalDifference. F: ( ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} ) \mapsto \mathrm{FiberProduct}(D) \rightarrow \Delta

##### 6.9-4 IsomorphismFromKernelOfDiagonalDifferenceToFiberProduct
 ‣ IsomorphismFromKernelOfDiagonalDifferenceToFiberProduct( D ) ( operation )

Returns: a morphism in \mathrm{Hom}(\Delta, \mathrm{FiberProduct}(D))

The argument is a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n}. The output is a morphism \Delta \rightarrow \mathrm{FiberProduct}(D), where \Delta denotes the kernel object equalizing the morphisms \beta_i.

##### 6.9-5 IsomorphismFromKernelOfDiagonalDifferenceToFiberProductOp
 ‣ IsomorphismFromKernelOfDiagonalDifferenceToFiberProductOp( D ) ( operation )

Returns: a morphism in \mathrm{Hom}(\Delta, \mathrm{FiberProduct}(D))

The argument is a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} and a morphism for method selection. The output is a morphism \Delta \rightarrow \mathrm{FiberProduct}(D), where \Delta denotes the kernel object equalizing the morphisms \beta_i.

 ‣ AddIsomorphismFromKernelOfDiagonalDifferenceToFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromKernelOfDiagonalDifferenceToFiberProduct. F: ( ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} ) \mapsto \Delta \rightarrow \mathrm{FiberProduct}(D)

##### 6.9-7 DirectSumDiagonalDifference
 ‣ DirectSumDiagonalDifference( D ) ( operation )

Returns: a morphism in \mathrm{Hom}( \bigoplus_{i=1}^n P_i, B )

The argument is a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n}. The output is a morphism \bigoplus_{i=1}^n P_i \rightarrow B such that its kernel equalizes the \beta_i.

##### 6.9-8 DirectSumDiagonalDifferenceOp
 ‣ DirectSumDiagonalDifferenceOp( D, method_selection_morphism ) ( operation )

Returns: a morphism in \mathrm{Hom}( \bigoplus_{i=1}^n P_i, B )

The argument is a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} and a morphism for method selection. The output is a morphism \bigoplus_{i=1}^n P_i \rightarrow B such that its kernel equalizes the \beta_i.

 ‣ AddDirectSumDiagonalDifference( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation DirectSumDiagonalDifference. F: ( D ) \mapsto \mathrm{DirectSumDiagonalDifference}(D)

##### 6.9-10 FiberProductEmbeddingInDirectSum
 ‣ FiberProductEmbeddingInDirectSum( D ) ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{FiberProduct}(D), \bigoplus_{i=1}^n P_i )

The argument is a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n}. The output is the natural embedding \mathrm{FiberProduct}(D) \rightarrow \bigoplus_{i=1}^n P_i.

##### 6.9-11 FiberProductEmbeddingInDirectSumOp
 ‣ FiberProductEmbeddingInDirectSumOp( D, method_selection_morphism ) ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{FiberProduct}(D), \bigoplus_{i=1}^n P_i )

The argument is a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} and a morphism for method selection. The output is the natural embedding \mathrm{FiberProduct}(D) \rightarrow \bigoplus_{i=1}^n P_i.

 ‣ AddFiberProductEmbeddingInDirectSum( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation FiberProductEmbeddingInDirectSum. F: ( ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} ) \mapsto \mathrm{FiberProduct}(D) \rightarrow \bigoplus_{i=1}^n P_i

##### 6.9-13 FiberProduct
 ‣ FiberProduct( arg ) ( function )

Returns: an object

This is a convenience method. There are two different ways to use this method:

• The argument is a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n}.

• The arguments are morphisms \beta_1: P_1 \rightarrow B, \dots, \beta_n: P_n \rightarrow B.

The output is the fiber product \mathrm{FiberProduct}(D).

##### 6.9-14 FiberProductOp
 ‣ FiberProductOp( D, method_selection_morphism ) ( operation )

Returns: an object

The arguments are a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} and a morphism for method selection. The output is the fiber product \mathrm{FiberProduct}(D).

##### 6.9-15 ProjectionInFactorOfFiberProduct
 ‣ ProjectionInFactorOfFiberProduct( D, k ) ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{FiberProduct}(D), P_k )

The arguments are a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} and an integer k. The output is the k-th projection \pi_{k}: \mathrm{FiberProduct}(D) \rightarrow P_k.

##### 6.9-16 ProjectionInFactorOfFiberProductOp
 ‣ ProjectionInFactorOfFiberProductOp( D, k, method_selection_morphism ) ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{FiberProduct}(D), P_k )

The arguments are a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n}, an integer k, and a morphism for method selection. The output is the k-th projection \pi_{k}: \mathrm{FiberProduct}(D) \rightarrow P_k.

##### 6.9-17 ProjectionInFactorOfFiberProductWithGivenFiberProduct
 ‣ ProjectionInFactorOfFiberProductWithGivenFiberProduct( D, k, P ) ( operation )

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

The arguments are a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n}, an integer k, and an object P = \mathrm{FiberProduct}(D). The output is the k-th projection \pi_{k}: P \rightarrow P_k.

##### 6.9-18 UniversalMorphismIntoFiberProduct
 ‣ UniversalMorphismIntoFiberProduct( arg ) ( function )

This is a convenience method. There are three different ways to use this method:

• The arguments are a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} and a list of morphisms \tau = ( \tau_i: T \rightarrow P_i ) such that \beta_i \circ \tau_i \sim_{T, B} \beta_j \circ \tau_j for all pairs i,j. The output is the morphism u( \tau ): T \rightarrow \mathrm{FiberProduct}(D) given by the universal property of the fiber product.

• The arguments are a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} and morphisms \tau_1: T \rightarrow P_1, \dots, \tau_n: T \rightarrow P_n such that \beta_i \circ \tau_i \sim_{T, B} \beta_j \circ \tau_j for all pairs i,j. The output is the morphism u( \tau ): T \rightarrow \mathrm{FiberProduct}(D) given by the universal property of the fiber product.

##### 6.9-19 UniversalMorphismIntoFiberProductOp
 ‣ UniversalMorphismIntoFiberProductOp( D, tau, method_selection_morphism ) ( operation )

Returns: a morphism in \mathrm{Hom}( T, \mathrm{FiberProduct}(D) )

The arguments are a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n}, a list of morphisms \tau = ( \tau_i: T \rightarrow P_i ) such that \beta_i \circ \tau_i \sim_{T, B} \beta_j \circ \tau_j for all pairs i,j, and a morphism for method selection. The output is the morphism u( \tau ): T \rightarrow \mathrm{FiberProduct}(D) given by the universal property of the fiber product.

##### 6.9-20 UniversalMorphismIntoFiberProductWithGivenFiberProduct
 ‣ UniversalMorphismIntoFiberProductWithGivenFiberProduct( D, tau, P ) ( operation )

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

The arguments are a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n}, a list of morphisms \tau = ( \tau_i: T \rightarrow P_i ) such that \beta_i \circ \tau_i \sim_{T, B} \beta_j \circ \tau_j for all pairs i,j, and an object P = \mathrm{FiberProduct}(D). The output is the morphism u( \tau ): T \rightarrow P given by the universal property of the fiber product.

 ‣ AddFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation FiberProduct. F: ( (\beta_i: P_i \rightarrow B)_{i = 1 \dots n} ) \mapsto P

 ‣ AddProjectionInFactorOfFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation ProjectionInFactorOfFiberProduct. F: ( (\beta_i: P_i \rightarrow B)_{i = 1 \dots n}, k ) \mapsto \pi_k

 ‣ AddProjectionInFactorOfFiberProductWithGivenFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation ProjectionInFactorOfFiberProductWithGivenFiberProduct. F: ( (\beta_i: P_i \rightarrow B)_{i = 1 \dots n}, k,P ) \mapsto \pi_k

 ‣ AddUniversalMorphismIntoFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoFiberProduct. F: ( (\beta_i: P_i \rightarrow B)_{i = 1 \dots n}, \tau ) \mapsto u(\tau)

 ‣ AddUniversalMorphismIntoFiberProductWithGivenFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoFiberProductWithGivenFiberProduct. F: ( (\beta_i: P_i \rightarrow B)_{i = 1 \dots n}, \tau, P ) \mapsto u(\tau)

##### 6.9-26 FiberProductFunctorial
 ‣ FiberProductFunctorial( L ) ( operation )

Returns: a morphism in \mathrm{Hom}(\mathrm{FiberProduct}( ( \beta_i )_{i=1 \dots n} ), \mathrm{FiberProduct}( ( \beta_i' )_{i=1 \dots n} ))

The argument is a list of triples of morphisms L = ( (\beta_i: P_i \rightarrow B, \mu_i: P_i \rightarrow P_i', \beta_i': P_i' \rightarrow B')_{i = 1 \dots n} ) such that there exists a morphism \beta: B \rightarrow B' such that \beta_i' \circ \mu_i \sim_{P_i,B'} \beta \circ \beta_i for i = 1, \dots, n. The output is the morphism \mathrm{FiberProduct}( ( \beta_i )_{i=1 \dots n} ) \rightarrow \mathrm{FiberProduct}( ( \beta_i' )_{i=1 \dots n} ) given by the functorality of the fiber product.

##### 6.9-27 FiberProductFunctorialWithGivenFiberProducts
 ‣ FiberProductFunctorialWithGivenFiberProducts( s, L, r ) ( operation )

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

The arguments are an object s = \mathrm{FiberProduct}( ( \beta_i )_{i=1 \dots n} ), a list of triples of morphisms L = ( (\beta_i: P_i \rightarrow B, \mu_i: P_i \rightarrow P_i', \beta_i': P_i' \rightarrow B')_{i = 1 \dots n} ) such that there exists a morphism \beta: B \rightarrow B' such that \beta_i' \circ \mu_i \sim_{P_i,B'} \beta \circ \beta_i for i = 1, \dots, n, and an object r = \mathrm{FiberProduct}( ( \beta_i' )_{i=1 \dots n} ). The output is the morphism s \rightarrow r given by the functorality of the fiber product.

 ‣ AddFiberProductFunctorialWithGivenFiberProducts( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation FiberProductFunctorialWithGivenFiberProducts. F: ( \mathrm{FiberProduct}( ( \beta_i )_{i=1 \dots n} ), (\beta_i: P_i \rightarrow B, \mu_i: P_i \rightarrow P_i', \beta_i': P_i' \rightarrow B')_{i = 1 \dots n}, \mathrm{FiberProduct}( ( \beta_i' )_{i=1 \dots n} ) ) \mapsto (\mathrm{FiberProduct}( ( \beta_i )_{i=1 \dots n} ) \rightarrow \mathrm{FiberProduct}( ( \beta_i' )_{i=1 \dots n} ) )

#### 6.10 Pushout

For a given list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n}, a pushout of D consists of three parts:

• an object I,

• a list of morphisms \iota = ( \iota_i: I_i \rightarrow I )_{i = 1 \dots n} such that \iota_i \circ \beta_i \sim_{B,I} \iota_j \circ \beta_j for all pairs i,j,

• a dependent function u mapping each list of morphisms \tau = ( \tau_i: I_i \rightarrow T )_{i = 1 \dots n} such that \tau_i \circ \beta_i \sim_{B,T} \tau_j \circ \beta_j to a morphism u( \tau ): I \rightarrow T such that u( \tau ) \circ \iota_i \sim_{I_i, T} \tau_i for all i = 1, \dots, n.

The triple ( I, \iota, u ) is called a pushout of D if the morphisms u( \tau ) are uniquely determined up to congruence of morphisms. We denote the object I of such a triple by \mathrm{Pushout}(D). We say that the morphism u( \tau ) is induced by the universal property of the pushout. \\ \mathrm{Pushout} is a functorial operation. This means: For a second diagram D' = (\beta_i': B' \rightarrow I_i')_{i = 1 \dots n} and a natural morphism between pushout diagrams (i.e., a collection of morphisms (\mu_i: I_i \rightarrow I'_i)_{i=1\dots n} and \beta: B \rightarrow B' such that \beta_i' \circ \beta \sim_{B, I_i'} \mu_i \circ \beta_i for i = 1, \dots n) we obtain a morphism \mathrm{Pushout}( D ) \rightarrow \mathrm{Pushout}( D' ).

##### 6.10-1 IsomorphismFromPushoutToCokernelOfDiagonalDifference
 ‣ IsomorphismFromPushoutToCokernelOfDiagonalDifference( D ) ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{Pushout}(D), \Delta)

The argument is a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n}. The output is a morphism \mathrm{Pushout}(D) \rightarrow \Delta, where \Delta denotes the cokernel object coequalizing the morphisms \beta_i.

##### 6.10-2 IsomorphismFromPushoutToCokernelOfDiagonalDifferenceOp
 ‣ IsomorphismFromPushoutToCokernelOfDiagonalDifferenceOp( D, method_selection_morphism ) ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{Pushout}(D), \Delta)

The argument is a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} and a morphism for method selection. The output is a morphism \mathrm{Pushout}(D) \rightarrow \Delta, where \Delta denotes the cokernel object coequalizing the morphisms \beta_i.

 ‣ AddIsomorphismFromPushoutToCokernelOfDiagonalDifference( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromPushoutToCokernelOfDiagonalDifference. F: ( ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} ) \mapsto (\mathrm{Pushout}(D) \rightarrow \Delta)

##### 6.10-4 IsomorphismFromCokernelOfDiagonalDifferenceToPushout
 ‣ IsomorphismFromCokernelOfDiagonalDifferenceToPushout( D ) ( operation )

Returns: a morphism in \mathrm{Hom}( \Delta, \mathrm{Pushout}(D))

The argument is a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n}. The output is a morphism \Delta \rightarrow \mathrm{Pushout}(D), where \Delta denotes the cokernel object coequalizing the morphisms \beta_i.

##### 6.10-5 IsomorphismFromCokernelOfDiagonalDifferenceToPushoutOp
 ‣ IsomorphismFromCokernelOfDiagonalDifferenceToPushoutOp( D, method_selection_morphism ) ( operation )

Returns: a morphism in \mathrm{Hom}( \Delta, \mathrm{Pushout}(D))

The argument is a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} and a morphism for method selection. The output is a morphism \Delta \rightarrow \mathrm{Pushout}(D), where \Delta denotes the cokernel object coequalizing the morphisms \beta_i.

 ‣ AddIsomorphismFromCokernelOfDiagonalDifferenceToPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromCokernelOfDiagonalDifferenceToPushout. F: ( ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} ) \mapsto (\Delta \rightarrow \mathrm{Pushout}(D))

##### 6.10-7 DirectSumCodiagonalDifference
 ‣ DirectSumCodiagonalDifference( D ) ( operation )

Returns: a morphism in \mathrm{Hom}(B, \bigoplus_{i=1}^n I_i)

The argument is a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n}. The output is a morphism B \rightarrow \bigoplus_{i=1}^n I_i such that its cokernel coequalizes the \beta_i.

##### 6.10-8 DirectSumCodiagonalDifferenceOp
 ‣ DirectSumCodiagonalDifferenceOp( D, method_selection_morphism ) ( operation )

Returns: a morphism in \mathrm{Hom}(B, \bigoplus_{i=1}^n I_i)

The argument is a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} and a morphism for method selection. The output is a morphism B \rightarrow \bigoplus_{i=1}^n I_i such that its cokernel coequalizes the \beta_i.

 ‣ AddDirectSumCodiagonalDifference( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation DirectSumCodiagonalDifference. F: ( D ) \mapsto \mathrm{DirectSumCodiagonalDifference}(D)

##### 6.10-10 DirectSumProjectionInPushout
 ‣ DirectSumProjectionInPushout( D ) ( operation )

Returns: a morphism in \mathrm{Hom}( \bigoplus_{i=1}^n I_i, \mathrm{Pushout}(D) )

The argument is a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n}. The output is the natural projection \bigoplus_{i=1}^n I_i \rightarrow \mathrm{Pushout}(D).

##### 6.10-11 DirectSumProjectionInPushoutOp
 ‣ DirectSumProjectionInPushoutOp( D, method_selection_morphism ) ( operation )

Returns: a morphism in \mathrm{Hom}( \bigoplus_{i=1}^n I_i, \mathrm{Pushout}(D) )

The argument is a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} and a morphism for method selection. The output is the natural projection \bigoplus_{i=1}^n I_i \rightarrow \mathrm{Pushout}(D).

 ‣ AddDirectSumProjectionInPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation DirectSumProjectionInPushout. F: ( ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} ) \mapsto (\bigoplus_{i=1}^n I_i \rightarrow \mathrm{Pushout}(D))

##### 6.10-13 Pushout
 ‣ Pushout( D ) ( operation )

Returns: an object

The argument is a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} The output is the pushout \mathrm{Pushout}(D).

##### 6.10-14 Pushout
 ‣ Pushout( D ) ( operation )

Returns: an object

This is a convenience method. The arguments are a morphism \alpha and a morphism \beta. The output is the pushout \mathrm{Pushout}(\alpha, \beta).

##### 6.10-15 PushoutOp
 ‣ PushoutOp( D ) ( operation )

Returns: an object

The arguments are a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} and a morphism for method selection. The output is the pushout \mathrm{Pushout}(D).

##### 6.10-16 InjectionOfCofactorOfPushout
 ‣ InjectionOfCofactorOfPushout( D, k ) ( operation )

Returns: a morphism in \mathrm{Hom}( I_k, \mathrm{Pushout}( D ) ).

The arguments are a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} and an integer k. The output is the k-th injection \iota_k: I_k \rightarrow \mathrm{Pushout}( D ).

##### 6.10-17 InjectionOfCofactorOfPushoutOp
 ‣ InjectionOfCofactorOfPushoutOp( D, k, method_selection_morphism ) ( operation )

Returns: a morphism in \mathrm{Hom}( I_k, \mathrm{Pushout}( D ) ).

The arguments are a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n}, an integer k, and a morphism for method selection. The output is the k-th injection \iota_k: I_k \rightarrow \mathrm{Pushout}( D ).

##### 6.10-18 InjectionOfCofactorOfPushoutWithGivenPushout
 ‣ InjectionOfCofactorOfPushoutWithGivenPushout( D, k, I ) ( operation )

Returns: a morphism in \mathrm{Hom}( I_k, \mathrm{Pushout}( D ) ).

The arguments are a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n}, an integer k, and an object I = \mathrm{Pushout}(D). The output is the k-th injection \iota_k: I_k \rightarrow \mathrm{Pushout}( D ).

##### 6.10-19 UniversalMorphismFromPushout
 ‣ UniversalMorphismFromPushout( arg ) ( function )

This is a convenience method. There are three different ways to use this method:

• The arguments are a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} and a list of morphisms \tau = ( \tau_i: I_i \rightarrow T )_{i = 1 \dots n} such that \tau_i \circ \beta_i \sim_{B,T} \tau_j \circ \beta_j. The output is the morphism u( \tau ): \mathrm{Pushout}(D) \rightarrow T given by the universal property of the pushout.

• The arguments are a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} and morphisms \tau_1: I_1 \rightarrow T, \dots, \tau_n: I_n \rightarrow T such that \tau_i \circ \beta_i \sim_{B,T} \tau_j \circ \beta_j. The output is the morphism u( \tau ): \mathrm{Pushout}(D) \rightarrow T given by the universal property of the pushout.

##### 6.10-20 UniversalMorphismFromPushoutOp
 ‣ UniversalMorphismFromPushoutOp( D, tau, method_selection_morphism ) ( operation )

Returns: a morphism in \mathrm{Hom}( \mathrm{Pushout}(D), T )

The arguments are a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n}, a list of morphisms \tau = ( \tau_i: I_i \rightarrow T )_{i = 1 \dots n} such that \tau_i \circ \beta_i \sim_{B,T} \tau_j \circ \beta_j, and a morphism for method selection. The output is the morphism u( \tau ): \mathrm{Pushout}(D) \rightarrow T given by the universal property of the pushout.

##### 6.10-21 UniversalMorphismFromPushoutWithGivenPushout
 ‣ UniversalMorphismFromPushoutWithGivenPushout( D, tau, I ) ( operation )

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

The arguments are a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n}, a list of morphisms \tau = ( \tau_i: I_i \rightarrow T )_{i = 1 \dots n} such that \tau_i \circ \beta_i \sim_{B,T} \tau_j \circ \beta_j, and an object I = \mathrm{Pushout}(D). The output is the morphism u( \tau ): I \rightarrow T given by the universal property of the pushout.

 ‣ AddPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation Pushout. F: ( (\beta_i: B \rightarrow I_i)_{i = 1 \dots n} ) \mapsto I

 ‣ AddInjectionOfCofactorOfPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation InjectionOfCofactorOfPushout. F: ( (\beta_i: B \rightarrow I_i)_{i = 1 \dots n}, k ) \mapsto \iota_k

 ‣ AddInjectionOfCofactorOfPushoutWithGivenPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation InjectionOfCofactorOfPushoutWithGivenPushout. F: ( (\beta_i: B \rightarrow I_i)_{i = 1 \dots n}, k, I ) \mapsto \iota_k

 ‣ AddUniversalMorphismFromPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromPushout. F: ( (\beta_i: B \rightarrow I_i)_{i = 1 \dots n}, \tau ) \mapsto u(\tau)

 ‣ AddUniversalMorphismFromPushoutWithGivenPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromPushout. F: ( (\beta_i: B \rightarrow I_i)_{i = 1 \dots n}, \tau, I ) \mapsto u(\tau)

##### 6.10-27 PushoutFunctorial
 ‣ PushoutFunctorial( L ) ( operation )

Returns: a morphism in \mathrm{Hom}(\mathrm{Pushout}( ( \beta_i )_{i=1}^n ), \mathrm{Pushout}( ( \beta_i' )_{i=1}^n ))

The argument is a list L = ( ( \beta_i: B \rightarrow I_i, \mu_i: I_i \rightarrow I_i', \beta_i': B' \rightarrow I_i' )_{i = 1 \dots n} ) such that there exists a morphism \beta: B \rightarrow B' such that \beta_i' \circ \beta \sim_{B, I_i'} \mu_i \circ \beta_i for i = 1, \dots n. The output is the morphism \mathrm{Pushout}( ( \beta_i )_{i=1}^n ) \rightarrow \mathrm{Pushout}( ( \beta_i' )_{i=1}^n ) given by the functorality of the pushout.

##### 6.10-28 PushoutFunctorialWithGivenPushouts
 ‣ PushoutFunctorialWithGivenPushouts( s, L, r ) ( operation )

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

The arguments are an object s = \mathrm{Pushout}( ( \beta_i )_{i=1}^n ), a list L = ( ( \beta_i: B \rightarrow I_i, \mu_i: I_i \rightarrow I_i', \beta_i': B' \rightarrow I_i' )_{i = 1 \dots n} ) such that there exists a morphism \beta: B \rightarrow B' such that \beta_i' \circ \beta \sim_{B, I_i'} \mu_i \circ \beta_i for i = 1, \dots n, and an object r = \mathrm{Pushout}( ( \beta_i' )_{i=1}^n ). The output is the morphism s \rightarrow r given by the functorality of the pushout.

 ‣ AddPushoutFunctorialWithGivenPushouts( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation PushoutFunctorial. F: ( \mathrm{Pushout}( ( \beta_i )_{i=1}^n ), ( \beta_i: B \rightarrow I_i, \mu_i: I_i \rightarrow I_i', \beta_i': B' \rightarrow I_i' )_{i = 1 \dots n}, \mathrm{Pushout}( ( \beta_i' )_{i=1}^n ) ) \mapsto (\mathrm{Pushout}( ( \beta_i )_{i=1}^n ) \rightarrow \mathrm{Pushout}( ( \beta_i' )_{i=1}^n ) )

#### 6.11 Image

For a given morphism \alpha: A \rightarrow B, an image of \alpha consists of four parts:

• an object I,

• a morphism c: A \rightarrow I,

• a monomorphism \iota: I \hookrightarrow B such that \iota \circ c \sim_{A,B} \alpha,

• a dependent function u mapping each pair of morphisms \tau = ( \tau_1: A \rightarrow T, \tau_2: T \hookrightarrow B ) where \tau_2 is a monomorphism such that \tau_2 \circ \tau_1 \sim_{A,B} \alpha to a morphism u(\tau): I \rightarrow T such that \tau_2 \circ u(\tau) \sim_{I,B} \iota and u(\tau) \circ c \sim_{A,T} \tau_1.

The 4-tuple ( I, c, \iota, u ) is called an image of \alpha if the morphisms u( \tau ) are uniquely determined up to congruence of morphisms. We denote the object I of such a 4-tuple by \mathrm{im}(\alpha). We say that the morphism u( \tau ) is induced by the universal property of the image.

##### 6.11-1 IsomorphismFromImageObjectToKernelOfCokernel
 ‣ IsomorphismFromImageObjectToKernelOfCokernel( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}( \mathrm{im}(\alpha), \mathrm{KernelObject}( \mathrm{CokernelProjection}( \alpha ) ) )

The argument is a morphism \alpha. The output is the canonical morphism \mathrm{im}(\alpha) \rightarrow \mathrm{KernelObject}( \mathrm{CokernelProjection}( \alpha ) ).

 ‣ AddIsomorphismFromImageObjectToKernelOfCokernel( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromImageObjectToKernelOfCokernel. F: \alpha \mapsto ( \mathrm{im}(\alpha) \rightarrow \mathrm{KernelObject}( \mathrm{CokernelProjection}( \alpha ) ) )

##### 6.11-3 IsomorphismFromKernelOfCokernelToImageObject
 ‣ IsomorphismFromKernelOfCokernelToImageObject( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}( \mathrm{KernelObject}( \mathrm{CokernelProjection}( \alpha ) ), \mathrm{im}(\alpha) )

The argument is a morphism \alpha. The output is the canonical morphism \mathrm{KernelObject}( \mathrm{CokernelProjection}( \alpha ) ) \rightarrow \mathrm{im}(\alpha).

 ‣ AddIsomorphismFromKernelOfCokernelToImageObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromKernelOfCokernelToImageObject. F: \alpha \mapsto ( \mathrm{KernelObject}( \mathrm{CokernelProjection}( \alpha ) ) \rightarrow \mathrm{im}(\alpha) )

##### 6.11-5 ImageObject
 ‣ ImageObject( alpha ) ( attribute )

Returns: an object

The argument is a morphism \alpha. The output is the image \mathrm{im}( \alpha ).

##### 6.11-6 ImageEmbedding
 ‣ ImageEmbedding( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{im}(\alpha), B)

The argument is a morphism \alpha: A \rightarrow B. The output is the image embedding \iota: \mathrm{im}(\alpha) \hookrightarrow B.

##### 6.11-7 ImageEmbeddingWithGivenImageObject
 ‣ ImageEmbeddingWithGivenImageObject( alpha, I ) ( operation )

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

The argument is a morphism \alpha: A \rightarrow B and an object I = \mathrm{im}( \alpha ). The output is the image embedding \iota: I \hookrightarrow B.

##### 6.11-8 CoastrictionToImage
 ‣ CoastrictionToImage( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}(A, \mathrm{im}( \alpha ))

The argument is a morphism \alpha: A \rightarrow B. The output is the coastriction to image c: A \rightarrow \mathrm{im}( \alpha ).

##### 6.11-9 CoastrictionToImageWithGivenImageObject
 ‣ CoastrictionToImageWithGivenImageObject( alpha, I ) ( operation )

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

The argument is a morphism \alpha: A \rightarrow B and an object I = \mathrm{im}( \alpha ). The output is the coastriction to image c: A \rightarrow I.

##### 6.11-10 UniversalMorphismFromImage
 ‣ UniversalMorphismFromImage( alpha, tau ) ( operation )

Returns: a morphism in \mathrm{Hom}(\mathrm{im}(\alpha), T)

The arguments are a morphism \alpha: A \rightarrow B and a pair of morphisms \tau = ( \tau_1: A \rightarrow T, \tau_2: T \hookrightarrow B ) where \tau_2 is a monomorphism such that \tau_2 \circ \tau_1 \sim_{A,B} \alpha. The output is the morphism u(\tau): \mathrm{im}(\alpha) \rightarrow T given by the universal property of the image.

##### 6.11-11 UniversalMorphismFromImageWithGivenImageObject
 ‣ UniversalMorphismFromImageWithGivenImageObject( alpha, tau, I ) ( operation )

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

The arguments are a morphism \alpha: A \rightarrow B, a pair of morphisms \tau = ( \tau_1: A \rightarrow T, \tau_2: T \hookrightarrow B ) where \tau_2 is a monomorphism such that \tau_2 \circ \tau_1 \sim_{A,B} \alpha, and an object I = \mathrm{im}( \alpha ). The output is the morphism u(\tau): \mathrm{im}(\alpha) \rightarrow T given by the universal property of the image.

 ‣ AddImageObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation ImageObject. F: \alpha \mapsto I.

 ‣ AddImageEmbedding( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation ImageEmbedding. F: \alpha \mapsto \iota.

 ‣ AddImageEmbeddingWithGivenImageObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation ImageEmbeddingWithGivenImageObject. F: (\alpha,I) \mapsto \iota.

 ‣ AddCoastrictionToImage( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation CoastrictionToImage. F: \alpha \mapsto c.

 ‣ AddCoastrictionToImageWithGivenImageObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation CoastrictionToImageWithGivenImageObject. F: (\alpha,I) \mapsto c.

 ‣ AddUniversalMorphismFromImage( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromImage. F: (\alpha, \tau) \mapsto u(\tau).

 ‣ AddUniversalMorphismFromImageWithGivenImageObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromImageWithGivenImageObject. F: (\alpha, \tau, I) \mapsto u(\tau).

#### 6.12 Coimage

For a given morphism \alpha: A \rightarrow B, a coimage of \alpha consists of four parts:

• an object C,

• an epimorphism \pi: A \twoheadrightarrow C,

• a morphism a: C \rightarrow B such that a \circ \pi \sim_{A,B} \alpha,

• a dependent function u mapping each pair of morphisms \tau = ( \tau_1: A \twoheadrightarrow T, \tau_2: T \rightarrow B ) where \tau_1 is an epimorphism such that \tau_2 \circ \tau_1 \sim_{A,B} \alpha to a morphism u(\tau): T \rightarrow C such that u( \tau ) \circ \tau_1 \sim_{A,C} \pi and a \circ u( \tau ) \sim_{T,B} \tau_2.

The 4-tuple ( C, \pi, a, u ) is called a coimage of \alpha if the morphisms u( \tau ) are uniquely determined up to congruence of morphisms. We denote the object C of such a 4-tuple by \mathrm{coim}(\alpha). We say that the morphism u( \tau ) is induced by the universal property of the coimage.

##### 6.12-1 MorphismFromCoimageToImage
 ‣ MorphismFromCoimageToImage( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{coim}(\alpha), \mathrm{im}(\alpha))

The argument is a morphism \alpha: A \rightarrow B. The output is the canonical morphism (in a preabelian category) \mathrm{coim}(\alpha) \rightarrow \mathrm{im}(\alpha).

##### 6.12-2 MorphismFromCoimageToImageWithGivenObjects
 ‣ MorphismFromCoimageToImageWithGivenObjects( alpha ) ( operation )

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

The argument is an object C = \mathrm{coim}(\alpha), a morphism \alpha: A \rightarrow B, and an object I = \mathrm{im}(\alpha). The output is the canonical morphism (in a preabelian category) C \rightarrow I.

 ‣ AddMorphismFromCoimageToImageWithGivenObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation MorphismFromCoimageToImageWithGivenObjects. F: (C, \alpha, I) \mapsto ( C \rightarrow I ).

##### 6.12-4 InverseMorphismFromCoimageToImage
 ‣ InverseMorphismFromCoimageToImage( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{im}(\alpha), \mathrm{coim}(\alpha))

The argument is a morphism \alpha: A \rightarrow B. The output is the inverse of the canonical morphism (in an abelian category) \mathrm{im}(\alpha) \rightarrow \mathrm{coim}(\alpha).

##### 6.12-5 InverseMorphismFromCoimageToImageWithGivenObjects
 ‣ InverseMorphismFromCoimageToImageWithGivenObjects( alpha ) ( operation )

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

The argument is an object C = \mathrm{coim}(\alpha), a morphism \alpha: A \rightarrow B, and an object I = \mathrm{im}(\alpha). The output is the inverse of the canonical morphism (in an abelian category) I \rightarrow C.

 ‣ AddInverseMorphismFromCoimageToImageWithGivenObjects( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation MorphismFromCoimageToImageWithGivenObjects. F: (C, \alpha, I) \mapsto ( I \rightarrow C ).

##### 6.12-7 IsomorphismFromCoimageToCokernelOfKernel
 ‣ IsomorphismFromCoimageToCokernelOfKernel( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}( \mathrm{coim}( \alpha ), \mathrm{CokernelObject}( \mathrm{KernelEmbedding}( \alpha ) ) ).

The argument is a morphism \alpha: A \rightarrow B. The output is the canonical morphism \mathrm{coim}( \alpha ) \rightarrow \mathrm{CokernelObject}( \mathrm{KernelEmbedding}( \alpha ) ).

 ‣ AddIsomorphismFromCoimageToCokernelOfKernel( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromCoimageToCokernelOfKernel. F: \alpha \mapsto ( \mathrm{coim}( \alpha ) \rightarrow \mathrm{CokernelObject}( \mathrm{KernelEmbedding}( \alpha ) ) ).

##### 6.12-9 IsomorphismFromCokernelOfKernelToCoimage
 ‣ IsomorphismFromCokernelOfKernelToCoimage( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}( \mathrm{CokernelObject}( \mathrm{KernelEmbedding}( \alpha ) ), \mathrm{coim}( \alpha ) ).

The argument is a morphism \alpha: A \rightarrow B. The output is the canonical morphism \mathrm{CokernelObject}( \mathrm{KernelEmbedding}( \alpha ) ) \rightarrow \mathrm{coim}( \alpha ).

 ‣ AddIsomorphismFromCokernelOfKernelToCoimage( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromCokernelOfKernelToCoimage. F: \alpha \mapsto ( \mathrm{CokernelObject}( \mathrm{KernelEmbedding}( \alpha ) ) \rightarrow \mathrm{coim}( \alpha ) ).

##### 6.12-11 Coimage
 ‣ Coimage( alpha ) ( attribute )

Returns: an object

The argument is a morphism \alpha. The output is the coimage \mathrm{coim}( \alpha ).

##### 6.12-12 CoimageProjection
 ‣ CoimageProjection( C ) ( attribute )

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

This is a convenience method. The argument is an object C which was created as a coimage of a morphism \alpha: A \rightarrow B. The output is the coimage projection \pi: A \twoheadrightarrow C.

##### 6.12-13 CoimageProjection
 ‣ CoimageProjection( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}(A, \mathrm{coim}( \alpha ))

The argument is a morphism \alpha: A \rightarrow B. The output is the coimage projection \pi: A \twoheadrightarrow \mathrm{coim}( \alpha ).

##### 6.12-14 CoimageProjectionWithGivenCoimage
 ‣ CoimageProjectionWithGivenCoimage( alpha, C ) ( operation )

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

The arguments are a morphism \alpha: A \rightarrow B and an object C = \mathrm{coim}(\alpha). The output is the coimage projection \pi: A \twoheadrightarrow C.

##### 6.12-15 AstrictionToCoimage
 ‣ AstrictionToCoimage( C ) ( attribute )

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

This is a convenience method. The argument is an object C which was created as a coimage of a morphism \alpha: A \rightarrow B. The output is the astriction to coimage a: C \rightarrow B.

##### 6.12-16 AstrictionToCoimage
 ‣ AstrictionToCoimage( alpha ) ( attribute )

Returns: a morphism in \mathrm{Hom}(\mathrm{coim}( \alpha ),B)

The argument is a morphism \alpha: A \rightarrow B. The output is the astriction to coimage a: \mathrm{coim}( \alpha ) \rightarrow B.

##### 6.12-17 AstrictionToCoimageWithGivenCoimage
 ‣ AstrictionToCoimageWithGivenCoimage( alpha, C ) ( operation )

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

The argument are a morphism \alpha: A \rightarrow B and an object C = \mathrm{coim}( \alpha ). The output is the astriction to coimage a: C \rightarrow B.

##### 6.12-18 UniversalMorphismIntoCoimage
 ‣ UniversalMorphismIntoCoimage( alpha, tau ) ( operation )

Returns: a morphism in \mathrm{Hom}(T, \mathrm{coim}( \alpha ))

The arguments are a morphism \alpha: A \rightarrow B and a pair of morphisms \tau = ( \tau_1: A \twoheadrightarrow T, \tau_2: T \rightarrow B ) where \tau_1 is an epimorphism such that \tau_2 \circ \tau_1 \sim_{A,B} \alpha. The output is the morphism u(\tau): T \rightarrow \mathrm{coim}( \alpha ) given by the universal property of the coimage.

##### 6.12-19 UniversalMorphismIntoCoimageWithGivenCoimage
 ‣ UniversalMorphismIntoCoimageWithGivenCoimage( alpha, tau, C ) ( operation )

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

The arguments are a morphism \alpha: A \rightarrow B, a pair of morphisms \tau = ( \tau_1: A \twoheadrightarrow T, \tau_2: T \rightarrow B ) where \tau_1 is an epimorphism such that \tau_2 \circ \tau_1 \sim_{A,B} \alpha, and an object C = \mathrm{coim}( \alpha ). The output is the morphism u(\tau): T \rightarrow C given by the universal property of the coimage.

 ‣ AddCoimage( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation Coimage. F: \alpha \mapsto C

 ‣ AddCoimageProjection( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation CoimageProjection. F: \alpha \mapsto \pi

 ‣ AddCoimageProjectionWithGivenCoimage( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation CoimageProjectionWithGivenCoimage. F: (\alpha,C) \mapsto \pi

 ‣ AddAstrictionToCoimage( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation AstrictionToCoimage. F: \alpha \mapsto a

 ‣ AddAstrictionToCoimageWithGivenCoimage( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation AstrictionToCoimageWithGivenCoimage. F: (\alpha,C) \mapsto a

 ‣ AddUniversalMorphismIntoCoimage( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoCoimage. F: (\alpha, \tau) \mapsto u(\tau)

 ‣ AddUniversalMorphismIntoCoimageWithGivenCoimage( C, F ) ( operation )
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoCoimageWithGivenCoimage. F: (\alpha, \tau,C) \mapsto u(\tau)