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1 Basic operations
 1.1 Weak kernel
 1.2 Weak cokernel
 1.3 Weak bi-fiber product
 1.4 Biased weak fiber product
 1.5 Weak bi-pushout
 1.6 Biased weak pushout
 1.7 Abelian constructions

1 Basic operations

1.1 Weak kernel

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

The triple \(( K, \iota, u )\) is called a weak kernel of \(\alpha\). We denote the object \(K\) of such a triple by \(\mathrm{WeakKernelObject}(\alpha)\). We say that the morphism \(u(\tau)\) is induced by the universal property of the weak kernel.

1.1-1 WeakKernelObject
‣ WeakKernelObject( alpha )( attribute )

Returns: an object

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

1.1-2 WeakKernelEmbedding
‣ WeakKernelEmbedding( alpha )( attribute )

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

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

1.1-3 WeakKernelEmbeddingWithGivenWeakKernelObject
‣ WeakKernelEmbeddingWithGivenWeakKernelObject( alpha, K )( operation )

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

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

1.1-4 WeakKernelLift
‣ WeakKernelLift( alpha, tau )( operation )

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

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 \(u(\tau): T \rightarrow \mathrm{WeakKernelObject}(\alpha)\) given by the universal property of the weak kernel.

1.1-5 WeakKernelLiftWithGivenWeakKernelObject
‣ WeakKernelLiftWithGivenWeakKernelObject( 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 \(K = \mathrm{WeakKernelObject}(\alpha)\). The output is the morphism \(u(\tau): T \rightarrow K\) given by the universal property of the weak kernel.

1.2 Weak cokernel

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

The triple \(( K, \epsilon, u )\) is called a weak cokernel of \(\alpha\). We denote the object \(K\) of such a triple by \(\mathrm{WeakCokernelObject}(\alpha)\). We say that the morphism \(u(\tau)\) is induced by the universal property of the weak cokernel.

1.2-1 WeakCokernelObject
‣ WeakCokernelObject( alpha )( attribute )

Returns: an object

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

1.2-2 WeakCokernelProjection
‣ WeakCokernelProjection( alpha )( attribute )

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

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

1.2-3 WeakCokernelProjectionWithGivenWeakCokernelObject
‣ WeakCokernelProjectionWithGivenWeakCokernelObject( alpha, K )( operation )

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

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

1.2-4 WeakCokernelColift
‣ WeakCokernelColift( alpha, tau )( operation )

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

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 \(u(\tau): \mathrm{WeakCokernelObject}(\alpha) \rightarrow T\) given by the universal property of the weak cokernel.

1.2-5 WeakCokernelColiftWithGivenWeakCokernelObject
‣ WeakCokernelColiftWithGivenWeakCokernelObject( 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 \(K = \mathrm{WeakCokernelObject}(\alpha)\). The output is the morphism \(u(\tau): K \rightarrow T\) given by the universal property of the weak cokernel.

1.3 Weak bi-fiber product

For a given pair of morphisms \((\alpha: A \rightarrow B, \beta \colon C \rightarrow B)\), a weak bi-fiber product of \((\alpha, \beta)\) consists of three parts:

The quadrupel \(( P, \pi_1, \pi_2, u )\) is called a weak bi-fiber product of \((\alpha,\beta)\). We denote the object \(P\) of such a quadrupel by \(\mathrm{WeakBiFiberProduct}(\alpha,\beta)\). We say that the morphism \(u(\tau)\) is induced by the universal property of the weak bi-fiber product.

1.3-1 WeakBiFiberProduct
‣ WeakBiFiberProduct( alpha, beta )( operation )

Returns: an object

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\). The output is the weak bi-fiber product \(P\) of \(\alpha\) and \(\beta\).

1.3-2 ProjectionInFirstFactorOfWeakBiFiberProduct
‣ ProjectionInFirstFactorOfWeakBiFiberProduct( alpha, beta )( operation )

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

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\). The output is the first weak bi-fiber product projection \(\pi_1: P \rightarrow A\).

1.3-3 ProjectionInFirstFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct
‣ ProjectionInFirstFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct( alpha, beta, P )( operation )

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

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\) and an object \(P = \mathrm{WeakBiFiberProduct}( \alpha, \beta )\). The output is the first weak bi-fiber product projection \(\pi_1: P \rightarrow A\).

1.3-4 ProjectionInSecondFactorOfWeakBiFiberProduct
‣ ProjectionInSecondFactorOfWeakBiFiberProduct( alpha, beta )( operation )

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

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\). The output is the second weak bi-fiber product projection \(\pi_2: P \rightarrow C\).

1.3-5 ProjectionInSecondFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct
‣ ProjectionInSecondFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct( alpha, beta, P )( operation )

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

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\) and an object \(P = \mathrm{WeakBiFiberProduct}( \alpha, \beta )\). The output is the second weak bi-fiber product projection \(\pi_2: P \rightarrow C\).

1.3-6 UniversalMorphismIntoWeakBiFiberProduct
‣ UniversalMorphismIntoWeakBiFiberProduct( alpha, beta, tau_1, tau_2 )( operation )

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

The arguments are four morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\), \(\tau_1: T \rightarrow A\), \(\tau_2: T \rightarrow C\). The output is the morphism \(u( \tau )\) induced by the universal property of the weak bi-fiber product \(P\) of \(\alpha\) and \(\beta\).

1.3-7 UniversalMorphismIntoWeakBiFiberProductWithGivenWeakBiFiberProduct
‣ UniversalMorphismIntoWeakBiFiberProductWithGivenWeakBiFiberProduct( alpha, beta, tau_1, tau_2, P )( operation )

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

The arguments are four morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\), \(\tau_1: T \rightarrow A\), \(\tau_2: T \rightarrow C\) and an object \(P = \mathrm{WeakBiFiberProduct}( \alpha, \beta )\). The output is the morphism \(u( \tau )\) induced by the universal property of the weak bi-fiber product \(P\).

1.3-8 WeakBiFiberProductMorphismToDirectSum
‣ WeakBiFiberProductMorphismToDirectSum( alpha, beta )( operation )

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

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\). The output is the morphism \(P \rightarrow A \oplus C\) obtained from the two weak bi-fiber product projections \(\pi_1\) and \(\pi_2\) and the universal property of the direct sum.

1.4 Biased weak fiber product

For a given pair of morphisms \((\alpha: A \rightarrow B, \beta \colon C \rightarrow B)\), a biased weak fiber product of \((\alpha, \beta)\) consists of three parts:

The triple \(( P, \pi, u )\) is called a biased weak fiber product of \((\alpha,\beta)\). We denote the object \(P\) of such a triple by \(\mathrm{BiasedWeakFiberProduct}(\alpha,\beta)\). We say that the morphism \(u(\tau)\) is induced by the universal property of the biased weak fiber product.

1.4-1 BiasedWeakFiberProduct
‣ BiasedWeakFiberProduct( alpha, beta )( operation )

Returns: an object

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\). The output is the biased weak fiber product \(P\) of \(\alpha\) and \(\beta\).

1.4-2 ProjectionOfBiasedWeakFiberProduct
‣ ProjectionOfBiasedWeakFiberProduct( alpha, beta )( operation )

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

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\). The output is the biased weak fiber product projection \(\pi: P \rightarrow A\).

1.4-3 ProjectionOfBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct
‣ ProjectionOfBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct( alpha, beta, P )( operation )

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

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\), and an object \(P = \mathrm{BiasedWeakFiberProduct}( \alpha, \beta )\). The output is the biased weak fiber product projection \(\pi: P \rightarrow A\).

1.4-4 UniversalMorphismIntoBiasedWeakFiberProduct
‣ UniversalMorphismIntoBiasedWeakFiberProduct( alpha, beta, tau )( operation )

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

The arguments are three morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\), \(\tau: T \rightarrow A\). The output is the morphism \(u( \tau )\) induced by the universal property of the biased weak fiber product \(P\) of \(\alpha\) and \(\beta\).

1.4-5 UniversalMorphismIntoBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct
‣ UniversalMorphismIntoBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct( alpha, beta, tau, P )( operation )

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

The arguments are three morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\), \(\tau: T \rightarrow A\) and an object \(P = \mathrm{BiasedWeakFiberProduct}( \alpha, \beta )\). The output is the morphism \(u( \tau )\) induced by the universal property of the biased weak fiber product \(P\) of \(\alpha\) and \(\beta\).

1.5 Weak bi-pushout

For a given pair of morphisms \((\alpha: A \rightarrow B, \beta \colon A \rightarrow C)\), a weak bi-pushout of \((\alpha, \beta)\) consists of three parts:

The quadrupel \(( P, \iota_1, \iota_2, u )\) is called a weak bi-pushout of \((\alpha,\beta)\). We denote the object \(P\) of such a quadrupel by \(\mathrm{WeakBiPushout}(\alpha,\beta)\). We say that the morphism \(u(\tau)\) is induced by the universal property of the weak bi-pushout.

1.5-1 WeakBiPushout
‣ WeakBiPushout( alpha, beta )( operation )

Returns: an object

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: A \rightarrow C\). The output is the weak bi-pushout \(P\) of \(\alpha\) and \(\beta\).

1.5-2 InjectionOfFirstCofactorOfWeakBiPushout
‣ InjectionOfFirstCofactorOfWeakBiPushout( alpha, beta )( operation )

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

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: A \rightarrow C\). The output is the first weak bi-pushout injection \(\iota_1: B \rightarrow P\).

1.5-3 InjectionOfSecondCofactorOfWeakBiPushout
‣ InjectionOfSecondCofactorOfWeakBiPushout( alpha, beta )( operation )

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

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: A \rightarrow C\). The output is the second weak bi-pushout injection \(\iota_2: C \rightarrow P\).

1.5-4 InjectionOfFirstCofactorOfWeakBiPushoutWithGivenWeakBiPushout
‣ InjectionOfFirstCofactorOfWeakBiPushoutWithGivenWeakBiPushout( alpha, beta, P )( operation )

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

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: A \rightarrow C\) and an object \(P = \mathrm{WeakBiPushout}( \alpha, \beta )\). The output is the first weak bi-pushout injection \(\iota_1: B \rightarrow P\).

1.5-5 InjectionOfSecondCofactorOfWeakBiPushoutWithGivenWeakBiPushout
‣ InjectionOfSecondCofactorOfWeakBiPushoutWithGivenWeakBiPushout( alpha, beta, P )( operation )

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

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: A \rightarrow C\) and an object \(P = \mathrm{WeakBiPushout}( \alpha, \beta )\). The output is the second weak bi-pushout injection \(\iota_2: C \rightarrow P\).

1.5-6 UniversalMorphismFromWeakBiPushout
‣ UniversalMorphismFromWeakBiPushout( alpha, beta, tau_1, tau_2 )( operation )

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

The arguments are four morphisms \(\alpha: A \rightarrow B\), \(\beta: A \rightarrow C\), \(\tau_1: B \rightarrow T\), \(\tau_2: C \rightarrow T\). The output is the morphism \(u( \tau )\) induced by the universal property of the weak bi-pushout \(P\) of \(\alpha\) and \(\beta\).

1.5-7 UniversalMorphismFromWeakBiPushoutWithGivenWeakBiPushout
‣ UniversalMorphismFromWeakBiPushoutWithGivenWeakBiPushout( alpha, beta, tau_1, tau_2, P )( operation )

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

The arguments are four morphisms \(\alpha: A \rightarrow B\), \(\beta: A \rightarrow C\), \(\tau_1: B \rightarrow T\), \(\tau_2: C \rightarrow T\), and an object \(P = \mathrm{WeakBiPushout}( \alpha, \beta )\). The output is the morphism \(u( \tau )\) induced by the universal property of the weak bi-pushout \(P\) of \(\alpha\) and \(\beta\).

1.5-8 DirectSumMorphismToWeakBiPushout
‣ DirectSumMorphismToWeakBiPushout( alpha, beta )( operation )

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

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: C \rightarrow B\). The output is the morphism \(B \oplus C \rightarrow P\) obtained from the two weak bi-fiber product injections \(\iota_1\) and \(\iota_2\) and the universal property of the direct sum.

1.6 Biased weak pushout

For a given pair of morphisms \((\alpha: A \rightarrow B, \beta: A \rightarrow C)\), a biased weak pushout of \((\alpha, \beta)\) consists of three parts:

The triple \(( P, \iota, u )\) is called a biased weak pushout of \((\alpha,\beta)\). We denote the object \(P\) of such a triple by \(\mathrm{BiasedWeakPushout}(\alpha,\beta)\). We say that the morphism \(u(\tau)\) is induced by the universal property of the biased weak pushout.

1.6-1 BiasedWeakPushout
‣ BiasedWeakPushout( alpha, beta )( operation )

Returns: an object

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: A \rightarrow C\). The output is the biased weak pushout \(P\) of \(\alpha\) and \(\beta\).

1.6-2 InjectionOfBiasedWeakPushout
‣ InjectionOfBiasedWeakPushout( alpha, beta )( operation )

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

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: A \rightarrow C\). The output is the biased weak pushout injection \(\iota: B \rightarrow P\).

1.6-3 InjectionOfBiasedWeakPushoutWithGivenBiasedWeakPushout
‣ InjectionOfBiasedWeakPushoutWithGivenBiasedWeakPushout( alpha, beta, P )( operation )

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

The arguments are two morphisms \(\alpha: A \rightarrow B\), \(\beta: A \rightarrow C\) and an object \(P = \mathrm{BiasedWeakPushout}( \alpha, \beta )\). The output is the biased weak pushout injection \(\iota: B \rightarrow P\).

1.6-4 UniversalMorphismFromBiasedWeakPushout
‣ UniversalMorphismFromBiasedWeakPushout( alpha, beta, tau )( operation )

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

The arguments are three morphisms \(\alpha: A \rightarrow B\), \(\beta: A \rightarrow C\), \(\tau: B \rightarrow T\). The output is the morphism \(u( \tau )\) induced by the universal property of the biased weak pushout \(P\) of \(\alpha\) and \(\beta\).

1.6-5 UniversalMorphismFromBiasedWeakPushoutWithGivenBiasedWeakPushout
‣ UniversalMorphismFromBiasedWeakPushoutWithGivenBiasedWeakPushout( alpha, beta, tau, P )( operation )

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

The arguments are three morphisms \(\alpha: A \rightarrow B\), \(\beta: A \rightarrow C\), \(\tau: B \rightarrow T\) and an object \(P = \mathrm{BiasedWeakPushout}( \alpha, \beta )\). The output is the morphism \(u( \tau )\) induced by the universal property of the biased weak pushout \(P\) of \(\alpha\) and \(\beta\).

1.7 Abelian constructions

1.7-1 SomeProjectiveObjectForKernelObject
‣ SomeProjectiveObjectForKernelObject( alpha )( attribute )

Returns: an object

The argument is a morphism \(\alpha\). The output is the source of EpimorphismFromSomeProjectiveObjectForKernelObject applied to \(\alpha\).

1.7-2 EpimorphismFromSomeProjectiveObjectForKernelObject
‣ EpimorphismFromSomeProjectiveObjectForKernelObject( alpha )( attribute )

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

The argument is a morphism \(\alpha\). The output is an epimorphism \(\pi: P \rightarrow \mathrm{KernelObject}( \alpha )\) with \(P\) a projective object.

1.7-3 EpimorphismFromSomeProjectiveObjectForKernelObjectWithGivenSomeProjectiveObjectForKernelObject
‣ EpimorphismFromSomeProjectiveObjectForKernelObjectWithGivenSomeProjectiveObjectForKernelObject( alpha )( operation )

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

The arguments are a morphism \(\alpha\) and an object \(P = \mathrm{SomeProjectiveObjectForKernelObject}( \alpha )\). The output is an epimorphism \(\pi: P \rightarrow \mathrm{KernelObject}( \alpha )\).

1.7-4 SomeInjectiveObjectForCokernelObject
‣ SomeInjectiveObjectForCokernelObject( alpha )( attribute )

Returns: an object

The argument is a morphism \(\alpha\). The output is the range of MonomorphismToSomeInjectiveObjectForCokernelObject applied to \(\alpha\).

1.7-5 MonomorphismToSomeInjectiveObjectForCokernelObject
‣ MonomorphismToSomeInjectiveObjectForCokernelObject( alpha )( attribute )

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

The argument is a morphism \(\alpha\). The output is a monomorphism \(\iota: \mathrm{CokernelObject}( \alpha ) \rightarrow I\) with \(I\) an injective object.

1.7-6 MonomorphismToSomeInjectiveObjectForCokernelObjectWithGivenSomeInjectiveObjectForCokernelObject
‣ MonomorphismToSomeInjectiveObjectForCokernelObjectWithGivenSomeInjectiveObjectForCokernelObject( alpha )( operation )

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

The arguments are a morphism \(\alpha\) and an object \(I = \mathrm{SomeInjectiveObjectForCokernelObject}( \alpha )\). The output is a monomorphism \(\iota: \mathrm{CokernelObject}( \alpha ) \rightarrow I\).

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