For a given morphism \(\alpha: A \rightarrow B\), a weak 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 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.
‣ WeakKernelObject( alpha ) | ( attribute ) |
Returns: an object
The argument is a morphism \(\alpha\). The output is the weak kernel \(K\) of \(\alpha\).
‣ 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 \(\iota: \mathrm{WeakKernelObject}(\alpha) \rightarrow A\).
‣ 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 \(\iota: K \rightarrow A\).
‣ 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.
‣ 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.
For a given morphism \(\alpha: A \rightarrow B\), a weak 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 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.
‣ WeakCokernelObject( alpha ) | ( attribute ) |
Returns: an object
The argument is a morphism \(\alpha: A \rightarrow B\). The output is the weak cokernel \(K\) of \(\alpha\).
‣ 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 \(\epsilon: B \rightarrow \mathrm{WeakCokernelObject}( \alpha )\).
‣ 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 \(\epsilon: B \rightarrow \mathrm{WeakCokernelObject}( \alpha )\).
‣ 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.
‣ 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.
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:
an object \(P\),
morphisms \(\pi_1: P \rightarrow A\), \(\pi_2: P \rightarrow B\) such that \(\alpha \circ \pi_1 \sim_{P,B} \beta \circ \pi_2\),
a dependent function \(u\) mapping each pair \(\tau = ( \tau_1, \tau_2 )\) of morphisms \(\tau_1: T \rightarrow A\), \(\tau_2: T \rightarrow C\) with the property \(\alpha \circ \tau_1 \sim_{T,B} \beta \circ \tau_2\) to a morphism \(u(\tau):T \rightarrow P\) such that \(\pi_1 \circ u( \tau ) \sim_{A,T} \tau_1\) and \(\pi_2 \circ u( \tau ) \sim_{C,T} \tau_2\).
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.
‣ 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 \(\beta\).
‣ 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 \(\pi_1: P \rightarrow A\).
‣ 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 \(\pi_1: P \rightarrow A\).
‣ 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 \(\pi_2: P \rightarrow C\).
‣ 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 \(\pi_2: P \rightarrow C\).
‣ 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 \(\beta\).
‣ 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 \(P\).
‣ 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.
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:
an object \(P\),
a morphism \(\pi: P \rightarrow A\) such that there exists a morphism \(\delta: P \rightarrow C\) such that \(\beta \circ \delta \sim_{P,B} \alpha \circ \pi\),
a dependent function \(u\) mapping each \(\tau: T \rightarrow A\), which admits a morphism \(\mu \colon T \rightarrow C\) with \(\beta \circ \mu \sim_{T,B} \alpha \circ \tau\), to a morphism \(u(\tau):T \rightarrow P\) such that \(\pi \circ u(\tau) \sim_{T,A} \tau\).
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.
‣ 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 \(\beta\).
‣ 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 \(\pi: P \rightarrow A\).
‣ 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 \(\pi: P \rightarrow A\).
‣ 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 \(\beta\).
‣ 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 \(\beta\).
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:
an object \(P\),
morphisms \(\iota_1: B \rightarrow P\), \(\iota_2: C \rightarrow P\) such that \(\iota_1 \circ \alpha \sim_{A,P} \iota_2 \circ \beta\),
a dependent function \(u\) mapping each pair \(\tau = (\tau_1, \tau_2)\) of morphisms \(\tau_1: B \rightarrow T\), \(\tau_2: C \rightarrow T\) with the property \(\tau_1 \circ \alpha \sim_{A,T} \tau_2 \circ \beta\) to a morphism \(u(\tau): P \rightarrow T\) such that \(u( \tau ) \circ \iota_1 \sim_{B,T} \tau_1\) and \(u( \tau ) \circ \iota_2 \sim_{C,T} \tau_2\).
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.
‣ 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 \(\beta\).
‣ 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 \(\iota_1: B \rightarrow P\).
‣ 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 \(\iota_2: C \rightarrow P\).
‣ 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 \(\iota_1: B \rightarrow P\).
‣ 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 \(\iota_2: C \rightarrow P\).
‣ 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 \(\beta\).
‣ 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 \(\beta\).
‣ 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.
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:
an object \(P\),
a morphism \(\iota: B \rightarrow P\) such that there exists a morphism \(\delta: C \rightarrow P\) such that \(\delta \circ \beta \sim_{A,P} \iota \circ \alpha\),
a dependent function \(u\) mapping each \(\tau: B \rightarrow T\), which admits a morphism \(\mu \colon C \rightarrow T\) with \(\mu \circ \beta \sim_{B,T} \tau \circ \alpha\), to a morphism \(u(\tau):P \rightarrow T\) such that \(u(\tau) \circ \iota \sim_{A,T} \tau\).
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.
‣ 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 \(\beta\).
‣ 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 \(\iota: B \rightarrow P\).
‣ 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 \(\iota: B \rightarrow P\).
‣ 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 \(\beta\).
‣ 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 \(\beta\).
‣ SomeProjectiveObjectForKernelObject( alpha ) | ( attribute ) |
Returns: an object
The argument is a morphism \(\alpha\). The output is the source of EpimorphismFromSomeProjectiveObjectForKernelObject applied to \(\alpha\).
‣ 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.
‣ 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 \(\pi: P \rightarrow \mathrm{KernelObject}( \alpha )\).
‣ SomeInjectiveObjectForCokernelObject( alpha ) | ( attribute ) |
Returns: an object
The argument is a morphism \(\alpha\). The output is the range of MonomorphismToSomeInjectiveObjectForCokernelObject applied to \(\alpha\).
‣ 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.
‣ 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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