GAP (CAP) - Chapter 6: Universal Objects

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6 Universal Objects
6.1 Kernel
6.1-1 KernelObject
6.1-2 KernelEmbedding
6.1-3 KernelEmbedding
6.1-4 KernelEmbeddingWithGivenKernelObject
6.1-5 KernelLift
6.1-6 KernelLift
6.1-7 KernelLiftWithGivenKernelObject
6.1-8 AddKernelObject
6.1-9 AddKernelEmbedding
6.1-10 AddKernelEmbeddingWithGivenKernelObject
6.1-11 AddKernelLift
6.1-12 AddKernelLiftWithGivenKernelObject
6.1-13 KernelObjectFunctorial
6.1-14 KernelObjectFunctorial
6.1-15 AddKernelObjectFunctorial
6.2 Cokernel
6.2-1 CokernelObject
6.2-2 CokernelProjection
6.2-3 CokernelProjection
6.2-4 CokernelProjectionWithGivenCokernelObject
6.2-5 CokernelColift
6.2-6 CokernelColift
6.2-7 CokernelColiftWithGivenCokernelObject
6.2-8 AddCokernelObject
6.2-9 AddCokernelProjection
6.2-10 AddCokernelProjectionWithGivenCokernelObject
6.2-11 AddCokernelColift
6.2-12 AddCokernelColiftWithGivenCokernelObject
6.2-13 CokernelFunctorial
6.2-14 CokernelFunctorial
6.2-15 AddCokernelFunctorial
6.3 Zero Object
6.3-1 ZeroObject
6.3-2 ZeroObject
6.3-3 MorphismFromZeroObject
6.3-4 MorphismIntoZeroObject
6.3-5 UniversalMorphismFromZeroObject
6.3-6 UniversalMorphismFromZeroObjectWithGivenZeroObject
6.3-7 UniversalMorphismIntoZeroObject
6.3-8 UniversalMorphismIntoZeroObjectWithGivenZeroObject
6.3-9 IsomorphismFromZeroObjectToInitialObject
6.3-10 IsomorphismFromInitialObjectToZeroObject
6.3-11 IsomorphismFromZeroObjectToTerminalObject
6.3-12 IsomorphismFromTerminalObjectToZeroObject
6.3-13 AddZeroObject
6.3-14 AddUniversalMorphismIntoZeroObject
6.3-15 AddUniversalMorphismIntoZeroObjectWithGivenZeroObject
6.3-16 AddUniversalMorphismFromZeroObject
6.3-17 AddUniversalMorphismFromZeroObjectWithGivenZeroObject
6.3-18 AddIsomorphismFromZeroObjectToInitialObject
6.3-19 AddIsomorphismFromInitialObjectToZeroObject
6.3-20 AddIsomorphismFromZeroObjectToTerminalObject
6.3-21 AddIsomorphismFromTerminalObjectToZeroObject
6.4 Terminal Object
6.4-1 TerminalObject
6.4-2 TerminalObject
6.4-3 UniversalMorphismIntoTerminalObject
6.4-4 UniversalMorphismIntoTerminalObjectWithGivenTerminalObject
6.4-5 AddTerminalObject
6.4-6 AddUniversalMorphismIntoTerminalObject
6.4-7 AddUniversalMorphismIntoTerminalObjectWithGivenTerminalObject
6.4-8 TerminalObjectFunctorial
6.4-9 AddTerminalObjectFunctorial
6.5 Initial Object
6.5-1 InitialObject
6.5-2 InitialObject
6.5-3 UniversalMorphismFromInitialObject
6.5-4 UniversalMorphismFromInitialObjectWithGivenInitialObject
6.5-5 AddInitialObject
6.5-6 AddUniversalMorphismFromInitialObject
6.5-7 AddUniversalMorphismFromInitialObjectWithGivenInitialObject
6.5-8 InitialObjectFunctorial
6.5-9 AddInitialObjectFunctorial
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 AddDirectSumFunctorial
6.6-39 DirectSumFunctorialOp
6.7 Coproduct
6.7-1 Coproduct
6.7-2 Coproduct
6.7-3 Coproduct
6.7-4 CoproductOp
6.7-5 InjectionOfCofactorOfCoproduct
6.7-6 InjectionOfCofactorOfCoproductOp
6.7-7 InjectionOfCofactorOfCoproductWithGivenCoproduct
6.7-8 UniversalMorphismFromCoproduct
6.7-9 UniversalMorphismFromCoproductOp
6.7-10 UniversalMorphismFromCoproductWithGivenCoproduct
6.7-11 AddCoproduct
6.7-12 AddInjectionOfCofactorOfCoproduct
6.7-13 AddInjectionOfCofactorOfCoproductWithGivenCoproduct
6.7-14 AddUniversalMorphismFromCoproduct
6.7-15 AddUniversalMorphismFromCoproductWithGivenCoproduct
6.7-16 CoproductFunctorial
6.7-17 AddCoproductFunctorial
6.7-18 CoproductFunctorialOp
6.8 Direct Product
6.8-1 DirectProductOp
6.8-2 ProjectionInFactorOfDirectProduct
6.8-3 ProjectionInFactorOfDirectProductOp
6.8-4 ProjectionInFactorOfDirectProductWithGivenDirectProduct
6.8-5 UniversalMorphismIntoDirectProduct
6.8-6 UniversalMorphismIntoDirectProductOp
6.8-7 UniversalMorphismIntoDirectProductWithGivenDirectProduct
6.8-8 AddDirectProduct
6.8-9 AddProjectionInFactorOfDirectProduct
6.8-10 AddProjectionInFactorOfDirectProductWithGivenDirectProduct
6.8-11 AddUniversalMorphismIntoDirectProduct
6.8-12 AddUniversalMorphismIntoDirectProductWithGivenDirectProduct
6.8-13 DirectProductFunctorial
6.8-14 AddDirectProductFunctorial
6.8-15 DirectProductFunctorialOp
6.9 Fiber Product
6.9-1 IsomorphismFromFiberProductToKernelOfDiagonalDifference
6.9-2 IsomorphismFromFiberProductToKernelOfDiagonalDifferenceOp
6.9-3 AddIsomorphismFromFiberProductToKernelOfDiagonalDifference
6.9-4 IsomorphismFromKernelOfDiagonalDifferenceToFiberProduct
6.9-5 IsomorphismFromKernelOfDiagonalDifferenceToFiberProductOp
6.9-6 AddIsomorphismFromKernelOfDiagonalDifferenceToFiberProduct
6.9-7 DirectSumDiagonalDifference
6.9-8 DirectSumDiagonalDifferenceOp
6.9-9 AddDirectSumDiagonalDifference
6.9-10 FiberProductEmbeddingInDirectSum
6.9-11 FiberProductEmbeddingInDirectSumOp
6.9-12 AddFiberProductEmbeddingInDirectSum
6.9-13 FiberProduct
6.9-14 FiberProductOp
6.9-15 ProjectionInFactorOfFiberProduct
6.9-16 ProjectionInFactorOfFiberProductOp
6.9-17 ProjectionInFactorOfFiberProductWithGivenFiberProduct
6.9-18 UniversalMorphismIntoFiberProduct
6.9-19 UniversalMorphismIntoFiberProductOp
6.9-20 UniversalMorphismIntoFiberProductWithGivenFiberProduct
6.9-21 AddFiberProduct
6.9-22 AddProjectionInFactorOfFiberProduct
6.9-23 AddProjectionInFactorOfFiberProductWithGivenFiberProduct
6.9-24 AddUniversalMorphismIntoFiberProduct
6.9-25 AddUniversalMorphismIntoFiberProductWithGivenFiberProduct
6.9-26 FiberProductFunctorial
6.9-27 AddFiberProductFunctorial
6.9-28 FiberProductFunctorialOp
6.10 Pushout
6.10-1 IsomorphismFromPushoutToCokernelOfDiagonalDifference
6.10-2 IsomorphismFromPushoutToCokernelOfDiagonalDifferenceOp
6.10-3 AddIsomorphismFromPushoutToCokernelOfDiagonalDifference
6.10-4 IsomorphismFromCokernelOfDiagonalDifferenceToPushout
6.10-5 IsomorphismFromCokernelOfDiagonalDifferenceToPushoutOp
6.10-6 AddIsomorphismFromCokernelOfDiagonalDifferenceToPushout
6.10-7 DirectSumCodiagonalDifference
6.10-8 DirectSumCodiagonalDifferenceOp
6.10-9 AddDirectSumCodiagonalDifference
6.10-10 DirectSumProjectionInPushout
6.10-11 DirectSumProjectionInPushoutOp
6.10-12 AddDirectSumProjectionInPushout
6.10-13 Pushout
6.10-14 Pushout
6.10-15 PushoutOp
6.10-16 InjectionOfCofactorOfPushout
6.10-17 InjectionOfCofactorOfPushoutOp
6.10-18 InjectionOfCofactorOfPushoutWithGivenPushout
6.10-19 UniversalMorphismFromPushout
6.10-20 UniversalMorphismFromPushoutOp
6.10-21 UniversalMorphismFromPushoutWithGivenPushout
6.10-22 AddPushout
6.10-23 AddInjectionOfCofactorOfPushout
6.10-24 AddInjectionOfCofactorOfPushoutWithGivenPushout
6.10-25 AddUniversalMorphismFromPushout
6.10-26 AddUniversalMorphismFromPushoutWithGivenPushout
6.10-27 PushoutFunctorial
6.10-28 AddPushoutFunctorial
6.10-29 PushoutFunctorialOp
6.11 Image
6.11-1 IsomorphismFromImageObjectToKernelOfCokernel
6.11-2 AddIsomorphismFromImageObjectToKernelOfCokernel
6.11-3 IsomorphismFromKernelOfCokernelToImageObject
6.11-4 AddIsomorphismFromKernelOfCokernelToImageObject
6.11-5 ImageObject
6.11-6 ImageEmbedding
6.11-7 ImageEmbedding
6.11-8 ImageEmbeddingWithGivenImageObject
6.11-9 CoastrictionToImage
6.11-10 CoastrictionToImage
6.11-11 CoastrictionToImageWithGivenImageObject
6.11-12 UniversalMorphismFromImage
6.11-13 UniversalMorphismFromImageWithGivenImageObject
6.11-14 AddImageObject
6.11-15 AddImageEmbedding
6.11-16 AddImageEmbeddingWithGivenImageObject
6.11-17 AddCoastrictionToImage
6.11-18 AddCoastrictionToImageWithGivenImageObject
6.11-19 AddUniversalMorphismFromImage
6.11-20 AddUniversalMorphismFromImageWithGivenImageObject
6.12 Coimage
6.12-1 MorphismFromCoimageToImage
6.12-2 MorphismFromCoimageToImageWithGivenObjects
6.12-3 AddMorphismFromCoimageToImageWithGivenObjects
6.12-4 InverseMorphismFromCoimageToImage
6.12-5 InverseMorphismFromCoimageToImageWithGivenObjects
6.12-6 AddInverseMorphismFromCoimageToImageWithGivenObjects
6.12-7 IsomorphismFromCoimageToCokernelOfKernel
6.12-8 AddIsomorphismFromCoimageToCokernelOfKernel
6.12-9 IsomorphismFromCokernelOfKernelToCoimage
6.12-10 AddIsomorphismFromCokernelOfKernelToCoimage
6.12-11 Coimage
6.12-12 CoimageProjection
6.12-13 CoimageProjection
6.12-14 CoimageProjectionWithGivenCoimage
6.12-15 AstrictionToCoimage
6.12-16 AstrictionToCoimage
6.12-17 AstrictionToCoimageWithGivenCoimage
6.12-18 UniversalMorphismIntoCoimage
6.12-19 UniversalMorphismIntoCoimageWithGivenCoimage
6.12-20 AddCoimage
6.12-21 AddCoimageProjection
6.12-22 AddCoimageProjectionWithGivenCoimage
6.12-23 AddAstrictionToCoimage
6.12-24 AddAstrictionToCoimageWithGivenCoimage
6.12-25 AddUniversalMorphismIntoCoimage
6.12-26 AddUniversalMorphismIntoCoimageWithGivenCoimage
6.13 Convenience Methods
6.13-1 InjectionOfCofactor
6.13-2 ProjectionInFactor

### 6 Universal Objects

#### 6.1 Kernel

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

- an object K,

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

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

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

##### 6.1-1 KernelObject

`‣ KernelObject`( alpha ) | ( attribute )

Returns: an object

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

##### 6.1-2 KernelEmbedding

`‣ KernelEmbedding`( alpha ) | ( attribute )

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

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

##### 6.1-3 KernelEmbedding

`‣ KernelEmbedding`( K ) | ( attribute )

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

This is a convenience method. The argument is an object K that was created as a kernel. The output is the kernel embedding \iota: K \rightarrow A.

##### 6.1-4 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.

##### 6.1-5 KernelLift

`‣ KernelLift`( K, tau ) | ( operation )

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

This is a convenience method. The arguments are an object K which was created as a kernel, 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 K given by the universal property of the kernel.

##### 6.1-6 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.

##### 6.1-7 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.

##### 6.1-8 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).

##### 6.1-9 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.

##### 6.1-10 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.

##### 6.1-11 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).

##### 6.1-12 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.

##### 6.1-13 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.

##### 6.1-14 KernelObjectFunctorial

`‣ KernelObjectFunctorial`( alpha, mu, alpha_prime ) | ( operation )

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

The arguments are three morphism \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.

##### 6.1-15 AddKernelObjectFunctorial

`‣ AddKernelObjectFunctorial`( 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 `KernelObjectFunctorial`. F: (\alpha, \mu, \alpha') \mapsto (\mathrm{KernelObject}( \alpha ) \rightarrow \mathrm{KernelObject}( \alpha' )).

#### 6.2 Cokernel

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

- an object K,

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

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

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

##### 6.2-1 CokernelObject

`‣ CokernelObject`( alpha ) | ( attribute )

Returns: an object

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

##### 6.2-2 CokernelProjection

`‣ CokernelProjection`( alpha ) | ( attribute )

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

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

##### 6.2-3 CokernelProjection

`‣ CokernelProjection`( K ) | ( attribute )

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

This is a convenience method. The argument is an object K which was created as a cokernel. The output is the cokernel projection \epsilon: B \rightarrow K.

##### 6.2-4 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 ).

##### 6.2-5 CokernelColift

`‣ CokernelColift`( K, tau ) | ( operation )

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

This is a convenience method. The arguments are an object K which was created as a cokernel, and a test morphism \tau: B \rightarrow T satisfying \tau \circ \alpha \sim_{A, T} 0. The output is the morphism u(\tau): K \rightarrow T given by the universal property of the cokernel.

##### 6.2-6 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.

##### 6.2-7 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.

##### 6.2-8 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.

##### 6.2-9 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.

##### 6.2-10 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.

##### 6.2-11 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).

##### 6.2-12 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).

##### 6.2-13 CokernelFunctorial

`‣ CokernelFunctorial`( 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.

##### 6.2-14 CokernelFunctorial

`‣ CokernelFunctorial`( 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.

##### 6.2-15 AddCokernelFunctorial

`‣ AddCokernelFunctorial`( 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 `CokernelFunctorial`. F: (\alpha, \nu, \alpha') \mapsto (\mathrm{CokernelObject}( \alpha ) \rightarrow \mathrm{CokernelObject}( \alpha' )).

#### 6.3 Zero Object

A zero object consists of three parts:

- an object Z,

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

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

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

##### 6.3-1 ZeroObject

`‣ ZeroObject`( C ) | ( attribute )

Returns: an object

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

##### 6.3-2 ZeroObject

`‣ ZeroObject`( c ) | ( attribute )

Returns: an object

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

##### 6.3-3 MorphismFromZeroObject

`‣ MorphismFromZeroObject`( A ) | ( attribute )

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

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

##### 6.3-4 MorphismIntoZeroObject

`‣ MorphismIntoZeroObject`( A ) | ( attribute )

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

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

##### 6.3-5 UniversalMorphismFromZeroObject

`‣ UniversalMorphismFromZeroObject`( A ) | ( attribute )

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

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

##### 6.3-6 UniversalMorphismFromZeroObjectWithGivenZeroObject

`‣ UniversalMorphismFromZeroObjectWithGivenZeroObject`( A, Z ) | ( operation )

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

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

##### 6.3-7 UniversalMorphismIntoZeroObject

`‣ UniversalMorphismIntoZeroObject`( A ) | ( attribute )

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

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

##### 6.3-8 UniversalMorphismIntoZeroObjectWithGivenZeroObject

`‣ UniversalMorphismIntoZeroObjectWithGivenZeroObject`( A, Z ) | ( operation )

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

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

##### 6.3-9 IsomorphismFromZeroObjectToInitialObject

`‣ IsomorphismFromZeroObjectToInitialObject`( C ) | ( attribute )

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

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

##### 6.3-10 IsomorphismFromInitialObjectToZeroObject

`‣ IsomorphismFromInitialObjectToZeroObject`( C ) | ( attribute )

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

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

##### 6.3-11 IsomorphismFromZeroObjectToTerminalObject

`‣ IsomorphismFromZeroObjectToTerminalObject`( C ) | ( attribute )

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

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

##### 6.3-12 IsomorphismFromTerminalObjectToZeroObject

`‣ IsomorphismFromTerminalObjectToZeroObject`( C ) | ( attribute )

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

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

##### 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}).

#### 6.4 Terminal Object

A terminal object consists of two parts:

- an object T,

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

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

##### 6.4-1 TerminalObject

`‣ TerminalObject`( C ) | ( attribute )

Returns: an object

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

##### 6.4-2 TerminalObject

`‣ TerminalObject`( c ) | ( attribute )

Returns: an object

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

##### 6.4-3 UniversalMorphismIntoTerminalObject

`‣ UniversalMorphismIntoTerminalObject`( A ) | ( attribute )

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

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

##### 6.4-4 UniversalMorphismIntoTerminalObjectWithGivenTerminalObject

`‣ UniversalMorphismIntoTerminalObjectWithGivenTerminalObject`( A, T ) | ( operation )

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

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

##### 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:

- an object I,

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

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

##### 6.5-1 InitialObject

`‣ InitialObject`( C ) | ( attribute )

Returns: an object

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

##### 6.5-2 InitialObject

`‣ InitialObject`( c ) | ( attribute )

Returns: an object

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

##### 6.5-3 UniversalMorphismFromInitialObject

`‣ UniversalMorphismFromInitialObject`( A ) | ( attribute )

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

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

##### 6.5-4 UniversalMorphismFromInitialObjectWithGivenInitialObject

`‣ UniversalMorphismFromInitialObjectWithGivenInitialObject`( A, I ) | ( operation )

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

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

##### 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:

- an object S,

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

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

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

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

such that

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

- \pi_j \circ \iota_i = \delta_{i,j},

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

##### 6.6-1 DirectSumOp

`‣ DirectSumOp`( D, method_selection_object ) | ( operation )

Returns: an object

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

##### 6.6-2 ProjectionInFactorOfDirectSum

`‣ ProjectionInFactorOfDirectSum`( D, k ) | ( operation )

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

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

##### 6.6-3 ProjectionInFactorOfDirectSumOp

`‣ ProjectionInFactorOfDirectSumOp`( D, k, method_selection_object ) | ( operation )

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

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

##### 6.6-4 ProjectionInFactorOfDirectSumWithGivenDirectSum

`‣ ProjectionInFactorOfDirectSumWithGivenDirectSum`( D, k, S ) | ( operation )

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

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

##### 6.6-5 InjectionOfCofactorOfDirectSum

`‣ InjectionOfCofactorOfDirectSum`( D, k ) | ( operation )

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

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

##### 6.6-6 InjectionOfCofactorOfDirectSumOp

`‣ InjectionOfCofactorOfDirectSumOp`( D, k, method_selection_object ) | ( operation )

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

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

##### 6.6-7 InjectionOfCofactorOfDirectSumWithGivenDirectSum

`‣ InjectionOfCofactorOfDirectSumWithGivenDirectSum`( D, k, S ) | ( operation )

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

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

##### 6.6-8 UniversalMorphismIntoDirectSum

`‣ UniversalMorphismIntoDirectSum`( arg ) | ( function )

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

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

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

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

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

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

##### 6.6-9 UniversalMorphismIntoDirectSumOp

`‣ UniversalMorphismIntoDirectSumOp`( D, tau, method_selection_object ) | ( operation )

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

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

##### 6.6-10 UniversalMorphismIntoDirectSumWithGivenDirectSum

`‣ UniversalMorphismIntoDirectSumWithGivenDirectSum`( D, tau, S ) | ( operation )

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

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

##### 6.6-11 UniversalMorphismFromDirectSum

`‣ UniversalMorphismFromDirectSum`( arg ) | ( function )

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

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

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

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

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

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

##### 6.6-12 UniversalMorphismFromDirectSumOp

`‣ UniversalMorphismFromDirectSumOp`( D, tau, method_selection_object ) | ( operation )

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

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

##### 6.6-13 UniversalMorphismFromDirectSumWithGivenDirectSum

`‣ UniversalMorphismFromDirectSumWithGivenDirectSum`( D, tau, S ) | ( operation )

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

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

##### 6.6-14 IsomorphismFromDirectSumToDirectProduct

`‣ IsomorphismFromDirectSumToDirectProduct`( D ) | ( operation )

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

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

##### 6.6-15 IsomorphismFromDirectSumToDirectProductOp

`‣ IsomorphismFromDirectSumToDirectProductOp`( D, method_selection_object ) | ( operation )

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

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

##### 6.6-16 IsomorphismFromDirectProductToDirectSum

`‣ IsomorphismFromDirectProductToDirectSum`( D ) | ( operation )

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

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

##### 6.6-17 IsomorphismFromDirectProductToDirectSumOp

`‣ IsomorphismFromDirectProductToDirectSumOp`( D, method_selection_object ) | ( operation )

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

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

##### 6.6-18 IsomorphismFromDirectSumToCoproduct

`‣ IsomorphismFromDirectSumToCoproduct`( D ) | ( operation )

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

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

##### 6.6-19 IsomorphismFromDirectSumToCoproductOp

`‣ IsomorphismFromDirectSumToCoproductOp`( D, method_selection_object ) | ( operation )

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

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

##### 6.6-20 IsomorphismFromCoproductToDirectSum

`‣ IsomorphismFromCoproductToDirectSum`( D ) | ( operation )

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

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

##### 6.6-21 IsomorphismFromCoproductToDirectSumOp

`‣ IsomorphismFromCoproductToDirectSumOp`( D, method_selection_object ) | ( operation )

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

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

##### 6.6-22 MorphismBetweenDirectSums

`‣ MorphismBetweenDirectSums`( M ) | ( operation )

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

The argument is a list of lists of morphisms M = ( ( \phi_{i,j}: A_i \rightarrow A_j )_{j = 1 \dots n} )_{i = 1 \dots m}. The output is the morphism \bigoplus_{i=1}^{m}A_i \rightarrow \bigoplus_{j=1}^n A_j defined by the matrix M.

##### 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 A_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 A_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 A_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 AddDirectSumFunctorial

`‣ AddDirectSumFunctorial`( 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 `DirectSumFunctorial`. F: (( \mu_1, \dots, \mu_n )) \mapsto (\bigoplus_{i=1}^n S_i \rightarrow \bigoplus_{i=1}^n S_i').

##### 6.6-39 DirectSumFunctorialOp

`‣ DirectSumFunctorialOp`( L, method_selection_morphism ) | ( operation )

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

The arguments are a list of morphisms L = ( \mu_1: S_1 \rightarrow S_1', \dots, \mu_n: S_n \rightarrow S_n' ) and a method selection morphism. 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.

#### 6.7 Coproduct

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

- an object I,

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

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

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

##### 6.7-1 Coproduct

`‣ Coproduct`( D ) | ( attribute )

Returns: an object

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

##### 6.7-2 Coproduct

`‣ Coproduct`( I1, I2 ) | ( operation )

Returns: an object

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

##### 6.7-3 Coproduct

`‣ Coproduct`( I1, I2 ) | ( operation )

Returns: an object

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

##### 6.7-4 CoproductOp

`‣ CoproductOp`( D, method_selection_object ) | ( operation )

Returns: an object

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

##### 6.7-5 InjectionOfCofactorOfCoproduct

`‣ InjectionOfCofactorOfCoproduct`( D, k ) | ( operation )

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

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

##### 6.7-6 InjectionOfCofactorOfCoproductOp

`‣ InjectionOfCofactorOfCoproductOp`( D, k, method_selection_object ) | ( operation )

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

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

##### 6.7-7 InjectionOfCofactorOfCoproductWithGivenCoproduct

`‣ InjectionOfCofactorOfCoproductWithGivenCoproduct`( D, k, I ) | ( operation )

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

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

##### 6.7-8 UniversalMorphismFromCoproduct

`‣ UniversalMorphismFromCoproduct`( arg ) | ( function )

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

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

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

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

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

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

##### 6.7-9 UniversalMorphismFromCoproductOp

`‣ UniversalMorphismFromCoproductOp`( D, tau, method_selection_object ) | ( operation )

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

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

##### 6.7-10 UniversalMorphismFromCoproductWithGivenCoproduct

`‣ UniversalMorphismFromCoproductWithGivenCoproduct`( D, tau, I ) | ( operation )

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

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

##### 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 AddCoproductFunctorial

`‣ AddCoproductFunctorial`( 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 `CoproductFunctorial`. F: ( \mu_1, \dots, \mu_n) \rightarrow (\bigsqcup_{i=1}^n I_i \rightarrow \bigsqcup_{i=1}^n I_i').

##### 6.7-18 CoproductFunctorialOp

`‣ CoproductFunctorialOp`( L, method_selection_morphism ) | ( operation )

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

The arguments are a list L = ( \mu_1: I_1 \rightarrow I_1', \dots, \mu_n: I_n \rightarrow I_n' ) and a method selection morphism. 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.

#### 6.8 Direct Product

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

- an object P,

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

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

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

##### 6.8-1 DirectProductOp

`‣ DirectProductOp`( D ) | ( operation )

Returns: an object

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

##### 6.8-2 ProjectionInFactorOfDirectProduct

`‣ ProjectionInFactorOfDirectProduct`( D, k ) | ( operation )

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

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

##### 6.8-3 ProjectionInFactorOfDirectProductOp

`‣ ProjectionInFactorOfDirectProductOp`( D, k, method_selection_object ) | ( operation )

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

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

##### 6.8-4 ProjectionInFactorOfDirectProductWithGivenDirectProduct

`‣ ProjectionInFactorOfDirectProductWithGivenDirectProduct`( D, k, P ) | ( operation )

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

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

##### 6.8-5 UniversalMorphismIntoDirectProduct

`‣ UniversalMorphismIntoDirectProduct`( arg ) | ( function )

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

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

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

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

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

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

##### 6.8-6 UniversalMorphismIntoDirectProductOp

`‣ UniversalMorphismIntoDirectProductOp`( D, tau, method_selection_object ) | ( operation )

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

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

##### 6.8-7 UniversalMorphismIntoDirectProductWithGivenDirectProduct

`‣ UniversalMorphismIntoDirectProductWithGivenDirectProduct`( D, tau, P ) | ( operation )

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

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

##### 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 AddDirectProductFunctorial

`‣ AddDirectProductFunctorial`( 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 `DirectProductFunctorial`. F: ( (\mu_i: P_i \rightarrow P'_i)_{i=1\dots n} ) \mapsto (\prod_{i=1}^n P_i \rightarrow \prod_{i=1}^n P_i')

##### 6.8-15 DirectProductFunctorialOp

`‣ DirectProductFunctorialOp`( L, method_selection_morphism ) | ( operation )

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

The arguments are a list of morphisms L = (\mu_i: P_i \rightarrow P'_i)_{i=1\dots n}, and a morphism for method selection. 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.

#### 6.9 Fiber Product

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

- an object P,

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

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

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

##### 6.9-1 IsomorphismFromFiberProductToKernelOfDiagonalDifference

`‣ IsomorphismFromFiberProductToKernelOfDiagonalDifference`( D ) | ( operation )

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

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

##### 6.9-2 IsomorphismFromFiberProductToKernelOfDiagonalDifferenceOp

`‣ IsomorphismFromFiberProductToKernelOfDiagonalDifferenceOp`( D, method_selection_morphism ) | ( operation )

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

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

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

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

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

##### 6.9-14 FiberProductOp

`‣ FiberProductOp`( D, method_selection_morphism ) | ( operation )

Returns: an object

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

##### 6.9-15 ProjectionInFactorOfFiberProduct

`‣ ProjectionInFactorOfFiberProduct`( D, k ) | ( operation )

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

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

##### 6.9-16 ProjectionInFactorOfFiberProductOp

`‣ ProjectionInFactorOfFiberProductOp`( D, k, method_selection_morphism ) | ( operation )

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

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

##### 6.9-17 ProjectionInFactorOfFiberProductWithGivenFiberProduct

`‣ ProjectionInFactorOfFiberProductWithGivenFiberProduct`( D, k, P ) | ( operation )

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

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

##### 6.9-18 UniversalMorphismIntoFiberProduct

`‣ UniversalMorphismIntoFiberProduct`( arg ) | ( function )

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

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

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

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

##### 6.9-19 UniversalMorphismIntoFiberProductOp

`‣ UniversalMorphismIntoFiberProductOp`( D, tau, method_selection_morphism ) | ( operation )

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

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

##### 6.9-20 UniversalMorphismIntoFiberProductWithGivenFiberProduct

`‣ UniversalMorphismIntoFiberProductWithGivenFiberProduct`( D, tau, P ) | ( operation )

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

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

##### 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 AddFiberProductFunctorial

`‣ AddFiberProductFunctorial`( 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 `FiberProductFunctorial`. F: ( (\beta_i: P_i \rightarrow B, \mu_i: P_i \rightarrow P_i', \beta_i': P_i' \rightarrow B')_{i = 1 \dots n} ) \mapsto (\mathrm{FiberProduct}( ( \beta_i )_{i=1 \dots n} ) \rightarrow \mathrm{FiberProduct}( ( \beta_i' )_{i=1 \dots n} ) )

##### 6.9-28 FiberProductFunctorialOp

`‣ FiberProductFunctorialOp`( L, method_selection_morphism ) | ( operation )

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

The arguments are 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 a morphism for method selection. 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.

#### 6.10 Pushout

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

- an object I,

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

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

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

##### 6.10-1 IsomorphismFromPushoutToCokernelOfDiagonalDifference

`‣ IsomorphismFromPushoutToCokernelOfDiagonalDifference`( D ) | ( operation )

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

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

##### 6.10-2 IsomorphismFromPushoutToCokernelOfDiagonalDifferenceOp

`‣ IsomorphismFromPushoutToCokernelOfDiagonalDifferenceOp`( D, method_selection_morphism ) | ( operation )

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

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

##### 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:

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

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

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

##### 6.10-20 UniversalMorphismFromPushoutOp

`‣ UniversalMorphismFromPushoutOp`( D, tau, method_selection_morphism ) | ( operation )

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

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

##### 6.10-21 UniversalMorphismFromPushoutWithGivenPushout

`‣ UniversalMorphismFromPushoutWithGivenPushout`( D, tau, I ) | ( operation )

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

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

##### 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 AddPushoutFunctorial

`‣ AddPushoutFunctorial`( 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: ( ( \beta_i: B \rightarrow I_i, \mu_i: I_i \rightarrow I_i', \beta_i': B' \rightarrow I_i' )_{i = 1 \dots n} ) \mapsto (\mathrm{Pushout}( ( \beta_i )_{i=1}^n ) \rightarrow \mathrm{Pushout}( ( \beta_i' )_{i=1}^n ) )

##### 6.10-29 PushoutFunctorialOp

`‣ PushoutFunctorialOp`( L, method_selection_morphism ) | ( 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, and a morphism for method selection. 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.

#### 6.11 Image

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

- an object I,

- a morphism c: A \rightarrow I,

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

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

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

##### 6.11-1 IsomorphismFromImageObjectToKernelOfCokernel

`‣ IsomorphismFromImageObjectToKernelOfCokernel`( alpha ) | ( attribute )

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

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

##### 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`( I ) | ( attribute )

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

This is a convenience method. The argument is an object I which was created as an image object of a morphism \alpha: A \rightarrow B. The output is the image embedding \iota: I \hookrightarrow B.

##### 6.11-7 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.

##### 6.11-8 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.

##### 6.11-9 CoastrictionToImage

`‣ CoastrictionToImage`( I ) | ( attribute )

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

This is a convenience method. The argument is an object I which was created as an image object of a morphism \alpha: A \rightarrow B. The output is the coastriction to image c: A \rightarrow I.

##### 6.11-10 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 ).

##### 6.11-11 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.

##### 6.11-12 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.

##### 6.11-13 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.

##### 6.11-14 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.

##### 6.11-15 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.

##### 6.11-16 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.

##### 6.11-17 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.

##### 6.11-18 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.

##### 6.11-19 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).

##### 6.11-20 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:

- an object C,

- an epimorphism \pi: A \twoheadrightarrow C,

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

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

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

##### 6.12-1 MorphismFromCoimageToImage

`‣ MorphismFromCoimageToImage`( alpha ) | ( attribute )

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

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

##### 6.12-2 MorphismFromCoimageToImageWithGivenObjects

`‣ MorphismFromCoimageToImageWithGivenObjects`( alpha ) | ( operation )

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

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

##### 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)

#### 6.13 Convenience Methods

##### 6.13-1 InjectionOfCofactor

`‣ InjectionOfCofactor`( arg ) | ( function )

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

- The arguments are an object S which was created as a direct sum and an integer k. The output is the k-th injection \iota_{k}: S_k \rightarrow S.

- The arguments are an object I which was created as a coproduct and an integer k. The output is the k-th injection \iota_{k}: I_k \rightarrow I.

- The arguments are an object I which was created as a pushout and an integer k. The output is the k-th injection \iota_{k}: I_k \rightarrow 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.

- 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).

##### 6.13-2 ProjectionInFactor

`‣ ProjectionInFactor`( arg ) | ( function )

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

- The arguments are an object S which was created as a direct sum and an integer k. The output is the k-th projection \pi_{k}: S \rightarrow S_k.

- The arguments are an object P which was created as a direct product and an integer k. The output is the k-th projection \pi_{k}: P \rightarrow P_k.

- The arguments are an object P which was created as a fiber product and an integer k. The output is the k-th projection \pi_{k}: P \rightarrow 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.

- 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.

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