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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