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6 Universal Objects
 6.1 Kernel
 6.2 Cokernel
 6.3 Zero Object
 6.4 Terminal Object
 6.5 Initial Object
 6.6 Direct Sum

  6.6-1 DirectSumOp

  6.6-2 ProjectionInFactorOfDirectSum

  6.6-3 ProjectionInFactorOfDirectSumOp

  6.6-4 ProjectionInFactorOfDirectSumWithGivenDirectSum

  6.6-5 InjectionOfCofactorOfDirectSum

  6.6-6 InjectionOfCofactorOfDirectSumOp

  6.6-7 InjectionOfCofactorOfDirectSumWithGivenDirectSum

  6.6-8 UniversalMorphismIntoDirectSum

  6.6-9 UniversalMorphismIntoDirectSumOp

  6.6-10 UniversalMorphismIntoDirectSumWithGivenDirectSum

  6.6-11 UniversalMorphismFromDirectSum

  6.6-12 UniversalMorphismFromDirectSumOp

  6.6-13 UniversalMorphismFromDirectSumWithGivenDirectSum

  6.6-14 IsomorphismFromDirectSumToDirectProduct

  6.6-15 IsomorphismFromDirectSumToDirectProductOp

  6.6-16 IsomorphismFromDirectProductToDirectSum

  6.6-17 IsomorphismFromDirectProductToDirectSumOp

  6.6-18 IsomorphismFromDirectSumToCoproduct

  6.6-19 IsomorphismFromDirectSumToCoproductOp

  6.6-20 IsomorphismFromCoproductToDirectSum

  6.6-21 IsomorphismFromCoproductToDirectSumOp

  6.6-22 MorphismBetweenDirectSums

  6.6-23 MorphismBetweenDirectSumsOp

  6.6-24 AddProjectionInFactorOfDirectSum

  6.6-25 AddProjectionInFactorOfDirectSumWithGivenDirectSum

  6.6-26 AddInjectionOfCofactorOfDirectSum

  6.6-27 AddInjectionOfCofactorOfDirectSumWithGivenDirectSum

  6.6-28 AddUniversalMorphismIntoDirectSum

  6.6-29 AddUniversalMorphismIntoDirectSumWithGivenDirectSum

  6.6-30 AddUniversalMorphismFromDirectSum

  6.6-31 AddUniversalMorphismFromDirectSumWithGivenDirectSum

  6.6-32 AddIsomorphismFromDirectSumToDirectProduct

  6.6-33 AddIsomorphismFromDirectProductToDirectSum

  6.6-34 AddIsomorphismFromDirectSumToCoproduct

  6.6-35 AddIsomorphismFromCoproductToDirectSum

  6.6-36 AddDirectSum

  6.6-37 DirectSumFunctorial

  6.6-38 DirectSumFunctorialWithGivenDirectSums

  6.6-39 AddDirectSumFunctorialWithGivenDirectSums
 6.7 Coproduct
 6.8 Direct Product
 6.9 Fiber Product
 6.10 Pushout
 6.11 Image
 6.12 Coimage

6 Universal Objects

6.1 Kernel

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

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.

6.1-6 AddKernelObject
‣ 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).

6.1-7 AddKernelEmbedding
‣ 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.

6.1-8 AddKernelEmbeddingWithGivenKernelObject
‣ 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.

6.1-9 AddKernelLift
‣ 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).

6.1-10 AddKernelLiftWithGivenKernelObject
‣ 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.

6.1-14 AddKernelObjectFunctorialWithGivenKernelObjects
‣ 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:

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.

6.2-6 AddCokernelObject
‣ 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.

6.2-7 AddCokernelProjection
‣ 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.

6.2-8 AddCokernelProjectionWithGivenCokernelObject
‣ 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.

6.2-9 AddCokernelColift
‣ 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).

6.2-10 AddCokernelColiftWithGivenCokernelObject
‣ 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.

6.2-14 AddCokernelObjectFunctorialWithGivenCokernelObjects
‣ 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:

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}.

6.3-13 AddZeroObject
‣ 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}.

6.3-14 AddUniversalMorphismIntoZeroObject
‣ 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).

6.3-15 AddUniversalMorphismIntoZeroObjectWithGivenZeroObject
‣ 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).

6.3-16 AddUniversalMorphismFromZeroObject
‣ 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).

6.3-17 AddUniversalMorphismFromZeroObjectWithGivenZeroObject
‣ 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).

6.3-18 AddIsomorphismFromZeroObjectToInitialObject
‣ 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}).

6.3-19 AddIsomorphismFromInitialObjectToZeroObject
‣ 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}).

6.3-20 AddIsomorphismFromZeroObjectToTerminalObject
‣ 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}).

6.3-21 AddIsomorphismFromTerminalObjectToZeroObject
‣ 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}.

6.3-23 AddZeroObjectFunctorial
‣ 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:

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.

6.4-5 AddTerminalObject
‣ 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.

6.4-6 AddUniversalMorphismIntoTerminalObject
‣ 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).

6.4-7 AddUniversalMorphismIntoTerminalObjectWithGivenTerminalObject
‣ 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}.

6.4-9 AddTerminalObjectFunctorial
‣ 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:

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.

6.5-5 AddInitialObject
‣ 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.

6.5-6 AddUniversalMorphismFromInitialObject
‣ 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).

6.5-7 AddUniversalMorphismFromInitialObjectWithGivenInitialObject
‣ 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}.

6.5-9 AddInitialObjectFunctorial
‣ 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:

such that

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 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 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.

6.6-24 AddProjectionInFactorOfDirectSum
‣ 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}.

6.6-25 AddProjectionInFactorOfDirectSumWithGivenDirectSum
‣ 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}.

6.6-26 AddInjectionOfCofactorOfDirectSum
‣ 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}.

6.6-27 AddInjectionOfCofactorOfDirectSumWithGivenDirectSum
‣ 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}.

6.6-28 AddUniversalMorphismIntoDirectSum
‣ 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).

6.6-29 AddUniversalMorphismIntoDirectSumWithGivenDirectSum
‣ 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).

6.6-30 AddUniversalMorphismFromDirectSum
‣ 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).

6.6-31 AddUniversalMorphismFromDirectSumWithGivenDirectSum
‣ 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).

6.6-32 AddIsomorphismFromDirectSumToDirectProduct
‣ 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).

6.6-33 AddIsomorphismFromDirectProductToDirectSum
‣ 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 ).

6.6-34 AddIsomorphismFromDirectSumToCoproduct
‣ 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 ).

6.6-35 AddIsomorphismFromCoproductToDirectSum
‣ 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 ).

6.6-36 AddDirectSum
‣ 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.

6.6-39 AddDirectSumFunctorialWithGivenDirectSums
‣ 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:

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 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.

6.7-11 AddCoproduct
‣ 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.

6.7-12 AddInjectionOfCofactorOfCoproduct
‣ 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.

6.7-13 AddInjectionOfCofactorOfCoproductWithGivenCoproduct
‣ 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.

6.7-14 AddUniversalMorphismFromCoproduct
‣ 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 ).

6.7-15 AddUniversalMorphismFromCoproductWithGivenCoproduct
‣ 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.

6.7-18 AddCoproductFunctorialWithGivenCoproducts
‣ 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:

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 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.

6.8-8 AddDirectProduct
‣ 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

6.8-9 AddProjectionInFactorOfDirectProduct
‣ 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

6.8-10 AddProjectionInFactorOfDirectProductWithGivenDirectProduct
‣ 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

6.8-11 AddUniversalMorphismIntoDirectProduct
‣ 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 )

6.8-12 AddUniversalMorphismIntoDirectProductWithGivenDirectProduct
‣ 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.

6.8-15 AddDirectProductFunctorialWithGivenDirectProducts
‣ 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:

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.

6.9-3 AddIsomorphismFromFiberProductToKernelOfDiagonalDifference
‣ 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.

6.9-6 AddIsomorphismFromKernelOfDiagonalDifferenceToFiberProduct
‣ 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.

6.9-9 AddDirectSumDiagonalDifference
‣ 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.

6.9-12 AddFiberProductEmbeddingInDirectSum
‣ 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 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:

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.

6.9-21 AddFiberProduct
‣ 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

6.9-22 AddProjectionInFactorOfFiberProduct
‣ 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

6.9-23 AddProjectionInFactorOfFiberProductWithGivenFiberProduct
‣ 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

6.9-24 AddUniversalMorphismIntoFiberProduct
‣ 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)

6.9-25 AddUniversalMorphismIntoFiberProductWithGivenFiberProduct
‣ 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.

6.9-28 AddFiberProductFunctorialWithGivenFiberProducts
‣ 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:

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.

6.10-3 AddIsomorphismFromPushoutToCokernelOfDiagonalDifference
‣ 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.

6.10-6 AddIsomorphismFromCokernelOfDiagonalDifferenceToPushout
‣ 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.

6.10-9 AddDirectSumCodiagonalDifference
‣ 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).

6.10-12 AddDirectSumProjectionInPushout
‣ 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:

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.

6.10-22 AddPushout
‣ 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

6.10-23 AddInjectionOfCofactorOfPushout
‣ 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

6.10-24 AddInjectionOfCofactorOfPushoutWithGivenPushout
‣ 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

6.10-25 AddUniversalMorphismFromPushout
‣ 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)

6.10-26 AddUniversalMorphismFromPushoutWithGivenPushout
‣ 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.

6.10-29 AddPushoutFunctorialWithGivenPushouts
‣ 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:

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 ) ).

6.11-2 AddIsomorphismFromImageObjectToKernelOfCokernel
‣ 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).

6.11-4 AddIsomorphismFromKernelOfCokernelToImageObject
‣ 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.

6.11-12 AddImageObject
‣ 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.

6.11-13 AddImageEmbedding
‣ 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.

6.11-14 AddImageEmbeddingWithGivenImageObject
‣ 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.

6.11-15 AddCoastrictionToImage
‣ 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.

6.11-16 AddCoastrictionToImageWithGivenImageObject
‣ 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.

6.11-17 AddUniversalMorphismFromImage
‣ 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).

6.11-18 AddUniversalMorphismFromImageWithGivenImageObject
‣ 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:

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.

6.12-3 AddMorphismFromCoimageToImageWithGivenObjects
‣ 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.

6.12-6 AddInverseMorphismFromCoimageToImageWithGivenObjects
‣ 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 ) ).

6.12-8 AddIsomorphismFromCoimageToCokernelOfKernel
‣ 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 ).

6.12-10 AddIsomorphismFromCokernelOfKernelToCoimage
‣ 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.

6.12-20 AddCoimage
‣ 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

6.12-21 AddCoimageProjection
‣ 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

6.12-22 AddCoimageProjectionWithGivenCoimage
‣ 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

6.12-23 AddAstrictionToCoimage
‣ 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

6.12-24 AddAstrictionToCoimageWithGivenCoimage
‣ 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

6.12-25 AddUniversalMorphismIntoCoimage
‣ 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)

6.12-26 AddUniversalMorphismIntoCoimageWithGivenCoimage
‣ AddUniversalMorphismIntoCoimageWithGivenCoimage( 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 UniversalMorphismIntoCoimageWithGivenCoimage. F: (\alpha, \tau,C) \mapsto u(\tau)

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