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 \(\mathrm{KernelObject}( \alpha ) \rightarrow \mathrm{KernelObject}( \alpha' )\).
‣ KernelObject ( alpha ) | ( attribute ) |
Returns: an object
The argument is a morphism \(\alpha\). The output is the kernel \(K\) of \(\alpha\).
‣ 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 \(\iota: \mathrm{KernelObject}(\alpha) \rightarrow A\).
‣ 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 \(\iota: K \rightarrow A\).
‣ MorphismFromKernelObjectToSink ( alpha ) | ( operation ) |
Returns: the zero morphism in \(\mathrm{Hom}( \mathrm{KernelObject}(\alpha), B )\)
The argument is a morphism \(\alpha: A \rightarrow B\). The output is the zero morphism \(0: \mathrm{KernelObject}(\alpha) \rightarrow B\).
‣ MorphismFromKernelObjectToSinkWithGivenKernelObject ( alpha, K ) | ( operation ) |
Returns: the zero morphism in \(\mathrm{Hom}( K, B )\)
The arguments are a morphism \(\alpha: A \rightarrow B\) and an object \(K = \mathrm{KernelObject}(\alpha)\). The output is the zero morphism \(0: K \rightarrow B\).
‣ 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.
‣ 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.
‣ AddKernelObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation KernelObject
. \(F: \alpha \mapsto \mathrm{KernelObject}(\alpha)\).
‣ AddKernelEmbedding ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation KernelEmbedding
. \(F: \alpha \mapsto \iota\).
‣ AddKernelEmbeddingWithGivenKernelObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation KernelEmbeddingWithGivenKernelObject
. \(F: (\alpha, K) \mapsto \iota\).
‣ AddMorphismFromKernelObjectToSink ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromKernelObjectToSink
. \(F: \alpha \mapsto \mu\)
‣ AddMorphismFromKernelObjectToSinkWithGivenKernelObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromKernelObjectToSinkWithGivenKernelObject
. \(F: ( \alpha, K ) \mapsto \mu\)
‣ AddKernelLift ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation KernelLift
. \(F: (\alpha, \tau) \mapsto u(\tau)\).
‣ AddKernelLiftWithGivenKernelObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation KernelLiftWithGivenKernelObject
. \(F: (\alpha, \tau, K) \mapsto u\).
‣ 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 functoriality of the kernel.
‣ KernelObjectFunctorial ( alpha, mu, alpha_prime ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{KernelObject}( \alpha ), \mathrm{KernelObject}( \alpha' ) )\)
The arguments are three morphisms \(\alpha: A \rightarrow B\), \(\mu: A \rightarrow A'\), \(\alpha': A' \rightarrow B'\). The output is the morphism \(\mathrm{KernelObject}( \alpha ) \rightarrow \mathrm{KernelObject}( \alpha' )\) given by the functoriality of the kernel.
‣ KernelObjectFunctorialWithGivenKernelObjects ( s, alpha, mu, alpha_prime, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\)
The arguments are an object \(s = \mathrm{KernelObject}( \alpha )\), three morphisms \(\alpha: A \rightarrow B\), \(\mu: A \rightarrow A'\), \(\alpha': A' \rightarrow B'\), and an object \(r = \mathrm{KernelObject}( \alpha' )\). The output is the morphism \(\mathrm{KernelObject}( \alpha ) \rightarrow \mathrm{KernelObject}( \alpha' )\) given by the functoriality of the kernel.
‣ KernelObjectFunctorialWithGivenKernelObjects ( s, alpha, mu, nu, alpha_prime, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\)
The arguments are an object \(s = \mathrm{KernelObject}( \alpha )\), four morphisms \(\alpha: A \rightarrow B\), \(\mu: A \rightarrow A'\), \(\nu: B \rightarrow B'\), \(\alpha': A' \rightarrow B'\), and an object \(r = \mathrm{KernelObject}( \alpha' )\). The output is the morphism \(\mathrm{KernelObject}( \alpha ) \rightarrow \mathrm{KernelObject}( \alpha' )\) given by the functoriality of the kernel.
‣ AddKernelObjectFunctorialWithGivenKernelObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation KernelObjectFunctorialWithGivenKernelObjects
. \(F: (\mathrm{KernelObject}( \alpha ), \alpha, \mu, \alpha', \mathrm{KernelObject}( \alpha' )) \mapsto (\mathrm{KernelObject}( \alpha ) \rightarrow \mathrm{KernelObject}( \alpha' ))\).
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 \(\mathrm{CokernelObject}( \alpha ) \rightarrow \mathrm{CokernelObject}( \alpha' )\).
‣ CokernelObject ( alpha ) | ( attribute ) |
Returns: an object
The argument is a morphism \(\alpha: A \rightarrow B\). The output is the cokernel \(K\) of \(\alpha\).
‣ 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 \(\epsilon: B \rightarrow \mathrm{CokernelObject}( \alpha )\).
‣ 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 \(\epsilon: B \rightarrow \mathrm{CokernelObject}( \alpha )\).
‣ MorphismFromSourceToCokernelObject ( alpha ) | ( operation ) |
Returns: the zero morphism in \(\mathrm{Hom}( A, \mathrm{CokernelObject}( \alpha ) )\).
The argument is a morphism \(\alpha: A \rightarrow B\). The output is the zero morphism \(0: A \rightarrow \mathrm{CokernelObject}(\alpha)\).
‣ MorphismFromSourceToCokernelObjectWithGivenCokernelObject ( alpha, K ) | ( operation ) |
Returns: the zero morphism in \(\mathrm{Hom}( A, K )\).
The argument is a morphism \(\alpha: A \rightarrow B\) and an object \(K = \mathrm{CokernelObject}(\alpha)\). The output is the zero morphism \(0: A \rightarrow K\).
‣ 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.
‣ 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.
‣ AddCokernelObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CokernelObject
. \(F: \alpha \mapsto K\).
‣ AddCokernelProjection ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CokernelProjection
. \(F: \alpha \mapsto \epsilon\).
‣ AddCokernelProjectionWithGivenCokernelObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CokernelProjection
. \(F: (\alpha, K) \mapsto \epsilon\).
‣ AddMorphismFromSourceToCokernelObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromSourceToCokernelObject
. \(F: \alpha \mapsto \mu\)
‣ AddMorphismFromSourceToCokernelObjectWithGivenCokernelObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromSourceToCokernelObjectWithGivenCokernelObject
. \(F: ( \alpha, K ) \mapsto \mu\)
‣ AddCokernelColift ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CokernelProjection
. \(F: (\alpha, \tau) \mapsto u(\tau)\).
‣ AddCokernelColiftWithGivenCokernelObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CokernelProjection
. \(F: (\alpha, \tau, K) \mapsto u(\tau)\).
‣ CokernelObjectFunctorial ( L ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(\mathrm{CokernelObject}( \alpha ), \mathrm{CokernelObject}( \alpha' ))\)
The argument is a list \(L = [ \alpha: A \rightarrow B, [ \mu:A \rightarrow A', \nu: B \rightarrow B' ], \alpha': A' \rightarrow B' ]\). The output is the morphism \(\mathrm{CokernelObject}( \alpha ) \rightarrow \mathrm{CokernelObject}( \alpha' )\) given by the functoriality of the cokernel.
‣ CokernelObjectFunctorial ( alpha, nu, alpha_prime ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(\mathrm{CokernelObject}( \alpha ), \mathrm{CokernelObject}( \alpha' ))\)
The arguments are three morphisms \(\alpha: A \rightarrow B, \nu: B \rightarrow B', \alpha': A' \rightarrow B'\). The output is the morphism \(\mathrm{CokernelObject}( \alpha ) \rightarrow \mathrm{CokernelObject}( \alpha' )\) given by the functoriality of the cokernel.
‣ CokernelObjectFunctorialWithGivenCokernelObjects ( s, alpha, nu, alpha_prime, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(s, r)\)
The arguments are an object \(s = \mathrm{CokernelObject}( \alpha )\), three morphisms \(\alpha: A \rightarrow B, \nu: B \rightarrow B', \alpha': A' \rightarrow B'\), and an object \(r = \mathrm{CokernelObject}( \alpha' )\). The output is the morphism \(\mathrm{CokernelObject}( \alpha ) \rightarrow \mathrm{CokernelObject}( \alpha' )\) given by the functoriality of the cokernel.
‣ CokernelObjectFunctorialWithGivenCokernelObjects ( s, alpha, mu, nu, alpha_prime, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(s, r)\)
The arguments are an object \(s = \mathrm{CokernelObject}( \alpha )\), four morphisms \(\alpha: A \rightarrow B, \mu: A \rightarrow A', \nu: B \rightarrow B', \alpha': A' \rightarrow B'\), and an object \(r = \mathrm{CokernelObject}( \alpha' )\). The output is the morphism \(\mathrm{CokernelObject}( \alpha ) \rightarrow \mathrm{CokernelObject}( \alpha' )\) given by the functoriality of the cokernel.
‣ AddCokernelObjectFunctorialWithGivenCokernelObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CokernelObjectFunctorialWithGivenCokernelObjects
. \(F: (\mathrm{CokernelObject}( \alpha ), \alpha, \nu, \alpha', \mathrm{CokernelObject}( \alpha' )) \mapsto (\mathrm{CokernelObject}( \alpha ) \rightarrow \mathrm{CokernelObject}( \alpha' ))\).
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.
‣ ZeroObject ( C ) | ( attribute ) |
Returns: an object
The argument is a category \(C\). The output is a zero object \(Z\) of \(C\).
‣ 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 \(c \in C\).
‣ 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 \(A\).
‣ 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 \(A\).
‣ UniversalMorphismFromZeroObject ( A ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(\mathrm{ZeroObject}, A)\)
The argument is an object \(A\). The output is the universal morphism \(u_{\mathrm{out}}: \mathrm{ZeroObject} \rightarrow A\).
‣ 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 \(u_{\mathrm{out}}: Z \rightarrow A\).
‣ UniversalMorphismIntoZeroObject ( A ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(A, \mathrm{ZeroObject})\)
The argument is an object \(A\). The output is the universal morphism \(u_{\mathrm{in}}: A \rightarrow \mathrm{ZeroObject}\).
‣ 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 \(u_{\mathrm{in}}: A \rightarrow Z\).
‣ 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 \(\mathrm{ZeroObject} \rightarrow \mathrm{InitialObject}\).
‣ 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 \(\mathrm{InitialObject} \rightarrow \mathrm{ZeroObject}\).
‣ 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 \(\mathrm{ZeroObject} \rightarrow \mathrm{TerminalObject}\).
‣ 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 \(\mathrm{TerminalObject} \rightarrow \mathrm{ZeroObject}\).
‣ AddZeroObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ZeroObject
. \(F: () \mapsto \mathrm{ZeroObject}\).
‣ AddUniversalMorphismIntoZeroObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoZeroObject
. \(F: A \mapsto u_{\mathrm{in}}(A)\).
‣ AddUniversalMorphismIntoZeroObjectWithGivenZeroObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoZeroObjectWithGivenZeroObject
. \(F: (A, Z) \mapsto u_{\mathrm{in}}(A)\).
‣ AddUniversalMorphismFromZeroObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismFromZeroObject
. \(F: A \mapsto u_{\mathrm{out}}(A)\).
‣ AddUniversalMorphismFromZeroObjectWithGivenZeroObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismFromZeroObjectWithGivenZeroObject
. \(F: (A,Z) \mapsto u_{\mathrm{out}}(A)\).
‣ AddIsomorphismFromZeroObjectToInitialObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromZeroObjectToInitialObject
. \(F: () \mapsto (\mathrm{ZeroObject} \rightarrow \mathrm{InitialObject})\).
‣ AddIsomorphismFromInitialObjectToZeroObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromInitialObjectToZeroObject
. \(F: () \mapsto ( \mathrm{InitialObject} \rightarrow \mathrm{ZeroObject})\).
‣ AddIsomorphismFromZeroObjectToTerminalObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromZeroObjectToTerminalObject
. \(F: () \mapsto (\mathrm{ZeroObject} \rightarrow \mathrm{TerminalObject})\).
‣ AddIsomorphismFromTerminalObjectToZeroObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromTerminalObjectToZeroObject
. \(F: () \mapsto ( \mathrm{TerminalObject} \rightarrow \mathrm{ZeroObject})\).
‣ ZeroObjectFunctorial ( C ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(\mathrm{ZeroObject}, \mathrm{ZeroObject} )\)
The argument is a category \(C\). The output is the unique morphism \(\mathrm{ZeroObject} \rightarrow \mathrm{ZeroObject}\).
‣ AddZeroObjectFunctorial ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ZeroObjectFunctorial
. \(F: () \mapsto (T \rightarrow T)\).
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 \(T \rightarrow T\).
‣ TerminalObject ( C ) | ( attribute ) |
Returns: an object
The argument is a category \(C\). The output is a terminal object \(T\) of \(C\).
‣ 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 \(c \in C\).
‣ UniversalMorphismIntoTerminalObject ( A ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}( A, \mathrm{TerminalObject} )\)
The argument is an object \(A\). The output is the universal morphism \(u(A): A \rightarrow \mathrm{TerminalObject}\).
‣ 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 \(u(A): A \rightarrow T\).
‣ AddTerminalObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation TerminalObject
. \(F: () \mapsto T\).
‣ AddUniversalMorphismIntoTerminalObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoTerminalObject
. \(F: A \mapsto u(A)\).
‣ AddUniversalMorphismIntoTerminalObjectWithGivenTerminalObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoTerminalObjectWithGivenTerminalObject
. \(F: (A,T) \mapsto u(A)\).
‣ 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 \(\mathrm{TerminalObject} \rightarrow \mathrm{TerminalObject}\).
‣ AddTerminalObjectFunctorial ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation TerminalObjectFunctorial
. \(F: () \mapsto (T \rightarrow T)\).
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 \(I \rightarrow I\).
‣ InitialObject ( C ) | ( attribute ) |
Returns: an object
The argument is a category \(C\). The output is an initial object \(I\) of \(C\).
‣ 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 \(c \in C\).
‣ 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 \(u(A): \mathrm{InitialObject} \rightarrow A\).
‣ 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 \(u(A): \mathrm{InitialObject} \rightarrow A\).
‣ AddInitialObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InitialObject
. \(F: () \mapsto I\).
‣ AddUniversalMorphismFromInitialObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismFromInitialObject
. \(F: A \mapsto u(A)\).
‣ AddUniversalMorphismFromInitialObjectWithGivenInitialObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismFromInitialObjectWithGivenInitialObject
. \(F: (A,I) \mapsto u(A)\).
‣ 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 \(\mathrm{InitialObject} \rightarrow \mathrm{InitialObject}\).
‣ AddInitialObjectFunctorial ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InitialObjectFunctorial
. \(F: () \rightarrow ( I \rightarrow I )\).
For an integer \(n \geq 1\) and 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
\(\sum_{i=1}^{n} \iota_i \circ \pi_i \sim_{S,S} \mathrm{id}_S\),
\(\pi_j \circ \iota_i \sim_{S_i, S_j} \delta_{i,j}\),
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 \(\bigoplus_{i=1}^n S_i \rightarrow \bigoplus_{i=1}^n S_i'\).
‣ 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 \(\bigoplus_{i=1}^n S_i\).
‣ 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 \(\pi_k: \bigoplus_{i=1}^n S_i \rightarrow S_k\).
‣ 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 \(\pi_k: \bigoplus_{i=1}^n S_i \rightarrow S_k\).
‣ 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 \(\pi_k: S \rightarrow S_k\).
‣ 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 \(\iota_k: S_k \rightarrow \bigoplus_{i=1}^n S_i\).
‣ 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 \(\iota_k: S_k \rightarrow \bigoplus_{i=1}^n S_i\).
‣ 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 \(\iota_k: S_k \rightarrow S\).
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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.
‣ 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 \(\bigoplus_{i=1}^n S_i \rightarrow \prod_{i=1}^{n}S_i\).
‣ 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 \(\bigoplus_{i=1}^n S_i \rightarrow \prod_{i=1}^{n}S_i\).
‣ 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 \(\prod_{i=1}^{n}S_i \rightarrow \bigoplus_{i=1}^n S_i\).
‣ 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 \(\prod_{i=1}^{n}S_i \rightarrow \bigoplus_{i=1}^n S_i\).
‣ 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 \(\bigoplus_{i=1}^n S_i \rightarrow \bigsqcup_{i=1}^{n}S_i\).
‣ 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 \(\bigoplus_{i=1}^n S_i \rightarrow \bigsqcup_{i=1}^{n}S_i\).
‣ 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 \(\bigsqcup_{i=1}^{n}S_i \rightarrow \bigoplus_{i=1}^n S_i\).
‣ 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 \(\bigsqcup_{i=1}^{n}S_i \rightarrow \bigoplus_{i=1}^n S_i\).
‣ MorphismBetweenDirectSums ( M ) | ( operation ) |
‣ MorphismBetweenDirectSums ( S, M, T ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(\bigoplus_{i=1}^{m}A_i, \bigoplus_{j=1}^n B_j)\)
The argument \(M = ( ( \phi_{i,j}: A_i \rightarrow B_j )_{j = 1 \dots n} )_{i = 1 \dots m}\) is a list of lists of morphisms. The output is the morphism \(\bigoplus_{i=1}^{m}A_i \rightarrow \bigoplus_{j=1}^n B_j\) defined by the matrix \(M\). The extra arguments \(S = \bigoplus_{i=1}^{m}A_i\) and \(T = \bigoplus_{j=1}^n B_j\) are source and target of the output, respectively. They must be provided in case \(M\) is an empty list or a list of empty lists.
‣ AddMorphismBetweenDirectSums ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismBetweenDirectSums
. \(F: (\bigoplus_{i=1}^{m}A_i, M, \bigoplus_{j=1}^n B_j) \mapsto (\bigoplus_{i=1}^{m}A_i \rightarrow \bigoplus_{j=1}^n B_j)\).
‣ MorphismBetweenDirectSumsOp ( M, m, n, method_selection_morphism ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(\bigoplus_{i=1}^{m}A_i, \bigoplus_{j=1}^n B_j)\)
The arguments are a list \(M = ( \phi_{1,1}, \phi_{1,2}, \dots, \phi_{1,n}, \phi_{2,1}, \dots, \phi_{m,n} )\) of morphisms \(\phi_{i,j}: A_i \rightarrow B_j\), an integer \(m\), an integer \(n\), and a method selection morphism. The output is the morphism \(\bigoplus_{i=1}^{m}A_i \rightarrow \bigoplus_{j=1}^n B_j\) defined by the list \(M\) regarded as a matrix of dimension \(m \times n\).
‣ ComponentOfMorphismIntoDirectSum ( alpha, D, k ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(A, S_k)\)
The arguments are a morphism \(\alpha: A \rightarrow S\), a list \(D = (S_1, \dots, S_n)\) of objects with \(S = \bigoplus_{j=1}^n S_j\), and an integer \(k\). The output is the component morphism \(A \rightarrow S_k\).
‣ ComponentOfMorphismFromDirectSum ( alpha, D, k ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(S_k, A)\)
The arguments are a morphism \(\alpha: S \rightarrow A\), a list \(D = (S_1, \dots, S_n)\) of objects with \(S = \bigoplus_{j=1}^n S_j\), and an integer \(k\). The output is the component morphism \(S_k \rightarrow A\).
‣ AddComponentOfMorphismIntoDirectSum ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ComponentOfMorphismIntoDirectSum
. \(F: (\alpha: A \rightarrow S,D,k) \mapsto (A \rightarrow S_k)\).
‣ AddComponentOfMorphismFromDirectSum ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ComponentOfMorphismFromDirectSum
. \(F: (\alpha: S \rightarrow A,D,k) \mapsto (S_k \rightarrow A)\).
‣ AddProjectionInFactorOfDirectSum ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ProjectionInFactorOfDirectSum
. \(F: (D,k) \mapsto \pi_{k}\).
‣ AddProjectionInFactorOfDirectSumWithGivenDirectSum ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ProjectionInFactorOfDirectSumWithGivenDirectSum
. \(F: (D,k,S) \mapsto \pi_{k}\).
‣ AddInjectionOfCofactorOfDirectSum ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InjectionOfCofactorOfDirectSum
. \(F: (D,k) \mapsto \iota_{k}\).
‣ AddInjectionOfCofactorOfDirectSumWithGivenDirectSum ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InjectionOfCofactorOfDirectSumWithGivenDirectSum
. \(F: (D,k,S) \mapsto \iota_{k}\).
‣ AddUniversalMorphismIntoDirectSum ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoDirectSum
. \(F: (D,\tau) \mapsto u_{\mathrm{in}}(\tau)\).
‣ AddUniversalMorphismIntoDirectSumWithGivenDirectSum ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoDirectSumWithGivenDirectSum
. \(F: (D,\tau,S) \mapsto u_{\mathrm{in}}(\tau)\).
‣ AddUniversalMorphismFromDirectSum ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismFromDirectSum
. \(F: (D,\tau) \mapsto u_{\mathrm{out}}(\tau)\).
‣ AddUniversalMorphismFromDirectSumWithGivenDirectSum ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismFromDirectSumWithGivenDirectSum
. \(F: (D,\tau,S) \mapsto u_{\mathrm{out}}(\tau)\).
‣ AddIsomorphismFromDirectSumToDirectProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromDirectSumToDirectProduct
. \(F: D \mapsto (\bigoplus_{i=1}^n S_i \rightarrow \prod_{i=1}^{n}S_i)\).
‣ AddIsomorphismFromDirectProductToDirectSum ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromDirectProductToDirectSum
. \(F: D \mapsto ( \prod_{i=1}^{n}S_i \rightarrow \bigoplus_{i=1}^n S_i )\).
‣ AddIsomorphismFromDirectSumToCoproduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromDirectSumToCoproduct
. \(F: D \mapsto ( \bigoplus_{i=1}^n S_i \rightarrow \bigsqcup_{i=1}^{n}S_i )\).
‣ AddIsomorphismFromCoproductToDirectSum ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromCoproductToDirectSum
. \(F: D \mapsto ( \bigsqcup_{i=1}^{n}S_i \rightarrow \bigoplus_{i=1}^n S_i )\).
‣ AddDirectSum ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DirectSum
. \(F: D \mapsto \bigoplus_{i=1}^n S_i\).
‣ 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 functoriality of the direct sum.
‣ DirectSumFunctorialWithGivenDirectSums ( d_1, L, d_2 ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( d_1, d_2 )\)
The arguments are an object \(d_1 = \bigoplus_{i=1}^n S_i\), a list of morphisms \(L = ( \mu_1: S_1 \rightarrow S_1', \dots, \mu_n: S_n \rightarrow S_n' )\), and an object \(d_2 = \bigoplus_{i=1}^n S_i'\). The output is a morphism \(d_1 \rightarrow d_2\) given by the functoriality of the direct sum.
‣ AddDirectSumFunctorialWithGivenDirectSums ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DirectSumFunctorialWithGivenDirectSums
. \(F: (\bigoplus_{i=1}^n S_i, ( \mu_1, \dots, \mu_n ), \bigoplus_{i=1}^n S_i') \mapsto (\bigoplus_{i=1}^n S_i \rightarrow \bigoplus_{i=1}^n S_i')\).
For an integer \(n \geq 1\) and 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 \(\bigsqcup_{i=1}^n I_i \rightarrow \bigsqcup_{i=1}^n I_i'\).
‣ Coproduct ( D ) | ( attribute ) |
Returns: an object
The argument is a list of objects \(D = ( I_1, \dots, I_n )\). The output is the coproduct \(\bigsqcup_{i=1}^n I_i\).
‣ 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 \(I_1 \bigsqcup I_2\).
‣ 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 \(I_1 \bigsqcup I_2 \bigsqcup I_3\).
‣ 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 \(\bigsqcup_{i=1}^n I_i\).
‣ 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 \(\iota_k: I_k \rightarrow \bigsqcup_{i=1}^n I_i\).
‣ 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 \(\iota_k: I_k \rightarrow \bigsqcup_{i=1}^n I_i\).
‣ 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 \(\iota_k: I_k \rightarrow I\).
‣ 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.
‣ 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.
‣ 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.
‣ AddCoproduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation Coproduct
. \(F: ( (I_1, \dots, I_n) ) \mapsto I\).
‣ AddInjectionOfCofactorOfCoproduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InjectionOfCofactorOfCoproduct
. \(F: ( (I_1, \dots, I_n), i ) \mapsto \iota_i\).
‣ AddInjectionOfCofactorOfCoproductWithGivenCoproduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InjectionOfCofactorOfCoproductWithGivenCoproduct
. \(F: ( (I_1, \dots, I_n), i, I ) \mapsto \iota_i\).
‣ AddUniversalMorphismFromCoproduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismFromCoproduct
. \(F: ( (I_1, \dots, I_n), \tau ) \mapsto u( \tau )\).
‣ AddUniversalMorphismFromCoproductWithGivenCoproduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismFromCoproductWithGivenCoproduct
. \(F: ( (I_1, \dots, I_n), \tau, I ) \mapsto u( \tau )\).
‣ 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 functoriality of the coproduct.
‣ CoproductFunctorialWithGivenCoproducts ( s, L, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(s, r)\)
The arguments are an object \(s = \bigsqcup_{i=1}^n I_i\), a list \(L = ( \mu_1: I_1 \rightarrow I_1', \dots, \mu_n: I_n \rightarrow I_n' )\), and an object \(r = \bigsqcup_{i=1}^n I_i'\). The output is a morphism \(\bigsqcup_{i=1}^n I_i \rightarrow \bigsqcup_{i=1}^n I_i'\) given by the functoriality of the coproduct.
‣ AddCoproductFunctorialWithGivenCoproducts ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoproductFunctorialWithGivenCoproducts
. \(F: (\bigsqcup_{i=1}^n I_i, (\mu_1, \dots, \mu_n), \bigsqcup_{i=1}^n I_i') \rightarrow (\bigsqcup_{i=1}^n I_i \rightarrow \bigsqcup_{i=1}^n I_i')\).
For an integer \(n \geq 1\) and 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 \(\prod_{i=1}^n P_i \rightarrow \prod_{i=1}^n P_i'\).
‣ 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 \(\prod_{i=1}^n P_i\).
‣ 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 \(\pi_k: \prod_{i=1}^n P_i \rightarrow P_k\).
‣ 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 \(\pi_k: \prod_{i=1}^n P_i \rightarrow P_k\).
‣ 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 \(\pi_k: P \rightarrow P_k\).
‣ 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.
‣ 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.
‣ 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.
‣ AddDirectProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DirectProduct
. \(F: ( (P_1, \dots, P_n) ) \mapsto P\)
‣ AddProjectionInFactorOfDirectProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ProjectionInFactorOfDirectProduct
. \(F: ( (P_1, \dots, P_n),k ) \mapsto \pi_k\)
‣ AddProjectionInFactorOfDirectProductWithGivenDirectProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ProjectionInFactorOfDirectProductWithGivenDirectProduct
. \(F: ( (P_1, \dots, P_n),k,P ) \mapsto \pi_k\)
‣ AddUniversalMorphismIntoDirectProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoDirectProduct
. \(F: ( (P_1, \dots, P_n), \tau ) \mapsto u( \tau )\)
‣ AddUniversalMorphismIntoDirectProductWithGivenDirectProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoDirectProductWithGivenDirectProduct
. \(F: ( (P_1, \dots, P_n), \tau, P ) \mapsto u( \tau )\)
‣ 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 functoriality of the direct product.
‣ DirectProductFunctorialWithGivenDirectProducts ( s, L, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( s, r )\)
The arguments are an object \(s = \prod_{i=1}^n P_i\), a list of morphisms \(L = (\mu_i: P_i \rightarrow P'_i)_{i=1\dots n}\), and an object \(r = \prod_{i=1}^n P_i'\). The output is a morphism \(\prod_{i=1}^n P_i \rightarrow \prod_{i=1}^n P_i'\) given by the functoriality of the direct product.
‣ AddDirectProductFunctorialWithGivenDirectProducts ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DirectProductFunctorialWithGivenDirectProducts
. \(F: ( \prod_{i=1}^n P_i, (\mu_i: P_i \rightarrow P'_i)_{i=1\dots n}, \prod_{i=1}^n P_i' ) \mapsto (\prod_{i=1}^n P_i \rightarrow \prod_{i=1}^n P_i')\)
For an integer \(n \geq 1\) and a given list of morphisms \(D = ( \beta_i: A \rightarrow B )_{i = 1 \dots n}\), an equalizer of \(D\) consists of three parts:
an object \(E\),
a morphism \(\iota: E \rightarrow A \) such that \(\beta_i \circ \iota \sim_{E, B} \beta_j \circ \iota\) for all pairs \(i,j\).
a dependent function \(u\) mapping each morphism \(\tau = ( \tau: T \rightarrow A )\) such that \(\beta_i \circ \tau \sim_{T, B} \beta_j \circ \tau\) for all pairs \(i,j\) to a morphism \(u( \tau ): T \rightarrow E\) such that \(\iota \circ u( \tau ) \sim_{T, A} \tau\).
The triple \(( E, \iota, u )\) is called an equalizer of \(D\) if the morphisms \(u( \tau )\) are uniquely determined up to congruence of morphisms. We denote the object \(E\) of such a triple by \(\mathrm{Equalizer}(D)\). We say that the morphism \(u( \tau )\) is induced by the universal property of the equalizer. \(\\ \) \(\mathrm{Equalizer}\) is a functorial operation. This means: For a second diagram \(D' = (\beta_i': A' \rightarrow B')_{i = 1 \dots n}\) and a natural morphism between equalizer diagrams (i.e., a collection of morphisms \(\mu: A \rightarrow A'\) and \(\beta: B \rightarrow B'\) such that \(\beta_i' \circ \mu \sim_{A,B'} \beta \circ \beta_i\) for \(i = 1, \dots, n\)) we obtain a morphism \(\mathrm{Equalizer}( D ) \rightarrow \mathrm{Equalizer}( D' )\).
‣ Equalizer ( 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 \(D = ( \beta_i: A \rightarrow B )_{i = 1 \dots n}\).
The arguments are morphisms \(\beta_1: A \rightarrow B, \dots, \beta_n: A \rightarrow B\).
The output is the equalizer \(\mathrm{Equalizer}(D)\).
‣ EqualizerOp ( D, method_selection_morphism ) | ( operation ) |
Returns: an object
The arguments are a list of morphisms \(D = ( \beta_i: A \rightarrow B )_{i = 1 \dots n}\) and a morphism for method selection. The output is the equalizer \(\mathrm{Equalizer}(D)\).
‣ EmbeddingOfEqualizer ( D ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{Equalizer}(D), A )\)
The arguments are a list of morphisms \(D = ( \beta_i: A \rightarrow B )_{i = 1 \dots n}\). The output is the equalizer embedding \(\iota: \mathrm{Equalizer}(D) \rightarrow A\).
‣ EmbeddingOfEqualizerOp ( D, method_selection_morphism ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{Equalizer}(D), A )\)
The arguments are a list of morphisms \(D = ( \beta_i: A \rightarrow B )_{i = 1 \dots n}\). and a morphism for method selection. The output is the equalizer embedding \(\iota: \mathrm{Equalizer}(D) \rightarrow A\).
‣ EmbeddingOfEqualizerWithGivenEqualizer ( D, E ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( E, A )\)
The arguments are a list of morphisms \(D = ( \beta_i: A \rightarrow B )_{i = 1 \dots n}\), and an object \(E = \mathrm{Equalizer}(D)\). The output is the equalizer embedding \(\iota: E \rightarrow A\).
‣ MorphismFromEqualizerToSink ( D ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{Equalizer}(D), B )\)
The arguments are a list of morphisms \(D = ( \beta_i: A \rightarrow B )_{i = 1 \dots n}\). The output is the composition \(\mu: \mathrm{Equalizer}(D) \rightarrow B\) of the embedding \(\iota: \mathrm{Equalizer}(D) \rightarrow A\) and \(\beta_1\).
‣ MorphismFromEqualizerToSinkOp ( D, method_selection_morphism ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{Equalizer}(D), B )\)
The arguments are a list of morphisms \(D = ( \beta_i: A \rightarrow B )_{i = 1 \dots n}\) and a morphism for method selection. The output is the composition \(\mu: \mathrm{Equalizer}(D) \rightarrow B\) of the embedding \(\iota: \mathrm{Equalizer}(D) \rightarrow A\) and \(\beta_1\).
‣ MorphismFromEqualizerToSinkWithGivenEqualizer ( D, E ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( E, B )\)
The arguments are a list of morphisms \(D = ( \beta_i: A \rightarrow B )_{i = 1 \dots n}\) and an object \(E = \mathrm{Equalizer}(D)\). The output is the composition \(\mu: E \rightarrow B\) of the embedding \(\iota: E \rightarrow A\) and \(\beta_1\).
‣ UniversalMorphismIntoEqualizer ( D, tau ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( T, \mathrm{Equalizer}(D) )\)
The arguments are a list of morphisms \(D = ( \beta_i: A \rightarrow B )_{i = 1 \dots n}\) and a morphism \( \tau: T \rightarrow A \) such that \(\beta_i \circ \tau \sim_{T, B} \beta_j \circ \tau\) for all pairs \(i,j\). The output is the morphism \(u( \tau ): T \rightarrow \mathrm{Equalizer}(D)\) given by the universal property of the equalizer.
‣ UniversalMorphismIntoEqualizerWithGivenEqualizer ( D, tau, E ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( T, E )\)
The arguments are a list of morphisms \(D = ( \beta_i: A \rightarrow B )_{i = 1 \dots n}\), a morphism \(\tau: T \rightarrow A )\) such that \(\beta_i \circ \tau \sim_{T, B} \beta_j \circ \tau\) for all pairs \(i,j\), and an object \(E = \mathrm{Equalizer}(D)\). The output is the morphism \(u( \tau ): T \rightarrow E\) given by the universal property of the equalizer.
‣ AddEqualizer ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation Equalizer
. \(F: ( (\beta_i: A \rightarrow B)_{i = 1 \dots n} ) \mapsto E\)
‣ AddEmbeddingOfEqualizer ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation EmbeddingOfEqualizer
. \(F: ( (\beta_i: A \rightarrow B)_{i = 1 \dots n}, k ) \mapsto \iota\)
‣ AddEmbeddingOfEqualizerWithGivenEqualizer ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation EmbeddingOfEqualizerWithGivenEqualizer
. \(F: ( (\beta_i: A \rightarrow B)_{i = 1 \dots n},E ) \mapsto \iota\)
‣ AddMorphismFromEqualizerToSink ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromEqualizerToSink
. \(F: ( (\beta_i: P \rightarrow B)_{i = 1 \dots n} ) \mapsto \mu\)
‣ AddMorphismFromEqualizerToSinkWithGivenEqualizer ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromEqualizerToSinkWithGivenEqualizer
. \(F: ( (\beta_i: P \rightarrow B)_{i = 1 \dots n}, E ) \mapsto \mu\)
‣ AddUniversalMorphismIntoEqualizer ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoEqualizer
. \(F: ( (\beta_i: A \rightarrow B)_{i = 1 \dots n}, \tau ) \mapsto u(\tau)\)
‣ AddUniversalMorphismIntoEqualizerWithGivenEqualizer ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoEqualizerWithGivenEqualizer
. \(F: ( (\beta_i: A \rightarrow B)_{i = 1 \dots n}, \tau, E ) \mapsto u(\tau)\)
‣ EqualizerFunctorial ( Ls, mu, Lr ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(\mathrm{Equalizer}( ( \beta_i )_{i=1 \dots n} ), \mathrm{Equalizer}( ( \beta_i' )_{i=1 \dots n} ))\)
The arguments are a list of morphisms \(L_s = (\beta_i: A \rightarrow B)_{i = 1 \dots n}\), a morphism \(\mu: A \rightarrow A'\), and a list of morphisms \(L_r = (\beta_i': A' \rightarrow B')_{i = 1 \dots n}\) such that there exists a morphism \(\beta: B \rightarrow B'\) such that \(\beta_i' \circ \mu \sim_{A,B'} \beta \circ \beta_i\) for \(i = 1, \dots, n\). The output is the morphism \(\mathrm{Equalizer}( ( \beta_i )_{i=1 \dots n} ) \rightarrow \mathrm{Equalizer}( ( \beta_i' )_{i=1 \dots n} )\) given by the functorality of the equalizer.
‣ EqualizerFunctorialWithGivenEqualizers ( s, Ls, mu, Lr, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(s, r)\)
The arguments are an object \(s = \mathrm{Equalizer}( ( \beta_i )_{i=1 \dots n} )\), a list of morphisms \(L_s = (\beta_i: A \rightarrow B)_{i = 1 \dots n}\), a morphism \(\mu: A \rightarrow A'\), and a list of morphisms \(L_r = (\beta_i': A' \rightarrow B')_{i = 1 \dots n}\) such that there exists a morphism \(\beta: B \rightarrow B'\) such that \(\beta_i' \circ \mu \sim_{A,B'} \beta \circ \beta_i\) for \(i = 1, \dots, n\), and an object \(r = \mathrm{Equalizer}( ( \beta_i' )_{i=1 \dots n} )\). The output is the morphism \(s \rightarrow r\) given by the functorality of the equalizer.
‣ AddEqualizerFunctorialWithGivenEqualizers ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation EqualizerFunctorialWithGivenEqualizers
. \(F: ( \mathrm{Equalizer}( ( \beta_i )_{i=1 \dots n} ), ( \beta_i: A \rightarrow B )_{i = 1 \dots n}, \mu: A \rightarrow A', ( \beta_i': A' \rightarrow B' )_{i = 1 \dots n}, \mathrm{Equalizer}( ( \beta_i' )_{i=1 \dots n} ) ) \mapsto (\mathrm{Equalizer}( ( \beta_i )_{i=1 \dots n} ) \rightarrow \mathrm {Equalizer}( ( \beta_i' )_{i=1 \dots n} ) )\)
For an integer \(n \geq 1\) and a given list of morphisms \(D = ( \beta_i: B \rightarrow A )_{i = 1 \dots n}\), a coequalizer of \(D\) consists of three parts:
an object \(C\),
a morphism \(\pi: A \rightarrow C \) such that \(\pi \circ \beta_i \sim_{B,C} \pi \circ \beta_j\) for all pairs \(i,j\),
a dependent function \(u\) mapping the morphism \(\tau: A \rightarrow T \) such that \(\tau \circ \beta_i \sim_{B,T} \tau \circ \beta_j\) to a morphism \(u( \tau ): C \rightarrow T\) such that \(u( \tau ) \circ \pi \sim_{A, T} \tau\).
The triple \(( C, \pi, u )\) is called a coequalizer of \(D\) if the morphisms \(u( \tau )\) are uniquely determined up to congruence of morphisms. We denote the object \(C\) of such a triple by \(\mathrm{Coequalizer}(D)\). We say that the morphism \(u( \tau )\) is induced by the universal property of the coequalizer. \(\\ \) \(\mathrm{Coequalizer}\) is a functorial operation. This means: For a second diagram \(D' = (\beta_i': B' \rightarrow A')_{i = 1 \dots n}\) and a natural morphism between coequalizer diagrams (i.e., a collection of morphisms \(\mu: A \rightarrow A'\) and \(\beta: B \rightarrow B'\) such that \(\beta_i' \circ \beta \sim_{B, A'} \mu \circ \beta_i\) for \(i = 1, \dots n\)) we obtain a morphism \(\mathrm{Coequalizer}( D ) \rightarrow \mathrm{Coequalizer}( D' )\).
‣ Coequalizer ( 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 \(D = ( \beta_i: B \rightarrow A )_{i = 1 \dots n}\).
The arguments are morphisms \(\beta_1: B \rightarrow A, \dots, \beta_n: B \rightarrow A\).
The output is the coequalizer \(\mathrm{Coequalizer}(D)\).
‣ CoequalizerOp ( D, method_selection_morphism ) | ( operation ) |
Returns: an object
The arguments are a list of morphisms \(D = ( \beta_i: B \rightarrow A )_{i = 1 \dots n}\) and a morphism for method selection. The output is the coequalizer \(\mathrm{Coequalizer}(D)\).
‣ ProjectionOntoCoequalizer ( D ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( A, \mathrm{Coequalizer}( D ) )\).
The arguments are a list of morphisms \(D = ( \beta_i: B \rightarrow A )_{i = 1 \dots n}\). The output is the projection \(\pi: A \rightarrow \mathrm{Coequalizer}( D )\).
‣ ProjectionOntoCoequalizerOp ( D, method_selection_morphism ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( A, \mathrm{Coequalizer}( D ) )\).
The arguments are a list of morphisms \(D = ( \beta_i: B \rightarrow A )_{i = 1 \dots n}\), and a morphism for method selection. The output is the projection \(\pi: A \rightarrow \mathrm{Coequalizer}( D )\).
‣ ProjectionOntoCoequalizerWithGivenCoequalizer ( D, C ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( A, C )\).
The arguments are a list of morphisms \(D = ( \beta_i: B \rightarrow A )_{i = 1 \dots n}\), and an object \(C = \mathrm{Coequalizer}(D)\). The output is the projection \(\pi: A \rightarrow C\).
‣ MorphismFromSourceToCoequalizer ( D ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( B, \mathrm{Coequalizer}( D ) )\).
The arguments are a list of morphisms \(D = ( \beta_i: B \rightarrow A )_{i = 1 \dots n}\). The output is the composition \(\mu: B \rightarrow \mathrm{Coequalizer}(D)\) of \(\beta_1\) and the projection \(\pi: A \rightarrow \mathrm{Coequalizer}( D )\).
‣ MorphismFromSourceToCoequalizerOp ( D, method_selection_morphism ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( B, \mathrm{Coequalizer}( D ) )\).
The arguments are a list of morphisms \(D = ( \beta_i: B \rightarrow A )_{i = 1 \dots n}\) and a morphism for method selection. The output is the composition \(\mu: B \rightarrow \mathrm{Coequalizer}(D)\) of \(\beta_1\) and the projection \(\pi: A \rightarrow \mathrm{Coequalizer}( D )\).
‣ MorphismFromSourceToCoequalizerWithGivenCoequalizer ( D, C ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( B, C )\).
The arguments are a list of morphisms \(D = ( \beta_i: B \rightarrow A )_{i = 1 \dots n}\) and an object \(C = \mathrm{Coequalizer}(D)\). The output is the composition \(\mu: B \rightarrow C\) of \(\beta_1\) and the projection \(\pi: A \rightarrow C\).
‣ UniversalMorphismFromCoequalizer ( D, tau ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{Coequalizer}(D), T )\)
The arguments are a list of morphisms \(D = ( \beta_i: B \rightarrow A )_{i = 1 \dots n}\) and a morphism \(\tau: A \rightarrow T \) such that \(\tau \circ \beta_i \sim_{B,T} \tau \circ \beta_j\) for all pairs \(i,j\). The output is the morphism \(u( \tau ): \mathrm{Coequalizer}(D) \rightarrow T\) given by the universal property of the coequalizer.
‣ UniversalMorphismFromCoequalizerWithGivenCoequalizer ( D, tau, C ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( C, T )\)
The arguments are a list of morphisms \(D = ( \beta_i: B \rightarrow A )_{i = 1 \dots n}\), a morphism \(\tau: A \rightarrow T \) such that \(\tau \circ \beta_i \sim_{B,T} \tau \circ \beta_j\), and an object \(C = \mathrm{Coequalizer}(D)\). The output is the morphism \(u( \tau ): C \rightarrow T\) given by the universal property of the coequalizer.
‣ AddCoequalizer ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation Coequalizer
. \(F: ( (\beta_i: B \rightarrow A)_{i = 1 \dots n} ) \mapsto C\)
‣ AddProjectionOntoCoequalizer ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ProjectionOntoCoequalizer
. \(F: ( (\beta_i: B \rightarrow A)_{i = 1 \dots n}, k ) \mapsto \pi\)
‣ AddProjectionOntoCoequalizerWithGivenCoequalizer ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ProjectionOntoCoequalizerWithGivenCoequalizer
. \(F: ( (\beta_i: B \rightarrow A)_{i = 1 \dots n}, C) \mapsto \pi\)
‣ AddMorphismFromSourceToCoequalizer ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromSourceToCoequalizer
. \(F: ( (\beta_i: B \rightarrow A)_{i = 1 \dots n} ) \mapsto \mu\)
‣ AddMorphismFromSourceToCoequalizerWithGivenCoequalizer ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromSourceToCoequalizerWithGivenCoequalizer
. \(F: ( (\beta_i: B \rightarrow A)_{i = 1 \dots n}, C ) \mapsto \mu\)
‣ AddUniversalMorphismFromCoequalizer ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismFromCoequalizer
. \(F: ( (\beta_i: B \rightarrow A)_{i = 1 \dots n}, \tau ) \mapsto u(\tau)\)
‣ AddUniversalMorphismFromCoequalizerWithGivenCoequalizer ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismFromCoequalizerWithGivenCoequalizer
. \(F: ( (\beta_i: B \rightarrow A)_{i = 1 \dots n}, \tau, C ) \mapsto u(\tau)\)
‣ CoequalizerFunctorial ( Ls, mu, Lr ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(\mathrm{Coequalizer}( ( \beta_i )_{i=1 \dots n} ), \mathrm{Coequalizer}( ( \beta_i' )_{i=1 \dots n} ))\)
The arguments are a list of morphisms \(L_s = ( \beta_i: B \rightarrow A )_{i = 1 \dots n}\), a morphism \(\mu: A \rightarrow A'\), and a list of morphisms \(L_r = ( \beta_i': B' \rightarrow A' )_{i = 1 \dots n}\) such that there exists a morphism \(\beta: B \rightarrow B'\) such that \(\beta_i' \circ \beta \sim_{B, A'} \mu \circ \beta_i\) for \(i = 1, \dots n\). The output is the morphism \(\mathrm{Coequalizer}( ( \beta_i )_{i=1}^n ) \rightarrow \mathrm{Coequalizer}( ( \beta_i' )_{i=1}^n )\) given by the functorality of the coequalizer.
‣ CoequalizerFunctorialWithGivenCoequalizers ( s, Ls, mu, Lr, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(s, r)\)
The arguments are an object \(s = \mathrm{Coequalizer}( ( \beta_i )_{i=1}^n )\), a list of morphisms \(L_s = ( \beta_i: B \rightarrow A )_{i = 1 \dots n}\), a morphism \(\mu: A \rightarrow A'\), and a list of morphisms \(L_r = ( \beta_i': B' \rightarrow A' )_{i = 1 \dots n}\) such that there exists a morphism \(\beta: B \rightarrow B'\) such that \(\beta_i' \circ \beta \sim_{B, A'} \mu \circ \beta_i\) for \(i = 1, \dots n\), and an object \(r = \mathrm{Coequalizer}( ( \beta_i' )_{i=1}^n )\). The output is the morphism \(s \rightarrow r\) given by the functorality of the coequalizer.
‣ AddCoequalizerFunctorialWithGivenCoequalizers ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoequalizerFunctorialWithGivenCoequalizers
. \(F: ( \mathrm{Coequalizer}( ( \beta_i )_{i=1}^n ), ( \beta_i: B \rightarrow A )_{i = 1 \dots n}, \mu: A \rightarrow A', ( \beta_i': B' \rightarrow A' )_{i = 1 \dots n}, \mathrm{Coequalizer}( ( \beta_i' )_{i=1}^n ) ) \mapsto (\mathrm{Coequalizer}( ( \beta_i )_{i=1}^n ) \rightarrow \mathrm{Coequalizer}( ( \beta_i' )_{i=1}^n ) )\)
For an integer \(n \geq 1\) and 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 \(\mathrm{FiberProduct}( D ) \rightarrow \mathrm{FiberProduct}( D' )\).
‣ 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 \(\beta_i\).
‣ 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 \(\beta_i\).
‣ AddIsomorphismFromFiberProductToKernelOfDiagonalDifference ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromFiberProductToKernelOfDiagonalDifference
. \(F: ( ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} ) \mapsto \mathrm{FiberProduct}(D) \rightarrow \Delta\)
‣ 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 \(\beta_i\).
‣ 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 \(\beta_i\).
‣ AddIsomorphismFromKernelOfDiagonalDifferenceToFiberProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromKernelOfDiagonalDifferenceToFiberProduct
. \(F: ( ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} ) \mapsto \Delta \rightarrow \mathrm{FiberProduct}(D)\)
‣ IsomorphismFromFiberProductToEqualizerOfDirectProductDiagram ( 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 equalizer of the product diagram of the morphisms \(\beta_i\).
‣ IsomorphismFromFiberProductToEqualizerOfDirectProductDiagramOp ( 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 equalizer of the product diagram of the morphisms \(\beta_i\).
‣ AddIsomorphismFromFiberProductToEqualizerOfDirectProductDiagram ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromFiberProductToEqualizerOfDirectProductDiagram
. \(F: ( ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} ) \mapsto \mathrm{FiberProduct}(D) \rightarrow \Delta\)
‣ IsomorphismFromEqualizerOfDirectProductDiagramToFiberProduct ( 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 equalizer of the product diagram of the morphisms \(\beta_i\).
‣ IsomorphismFromEqualizerOfDirectProductDiagramToFiberProductOp ( 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 equalizer of the product diagram of the morphisms \(\beta_i\).
‣ AddIsomorphismFromEqualizerOfDirectProductDiagramToFiberProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromEqualizerOfDirectProductDiagramToFiberProduct
. \(F: ( ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} ) \mapsto \Delta \rightarrow \mathrm{FiberProduct}(D)\)
‣ 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 \(\beta_i\).
‣ 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 \(\beta_i\).
‣ AddDirectSumDiagonalDifference ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DirectSumDiagonalDifference
. \(F: ( D ) \mapsto \mathrm{DirectSumDiagonalDifference}(D)\)
‣ 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 \(\mathrm{FiberProduct}(D) \rightarrow \bigoplus_{i=1}^n P_i\).
‣ 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 \(\mathrm{FiberProduct}(D) \rightarrow \bigoplus_{i=1}^n P_i\).
‣ AddFiberProductEmbeddingInDirectSum ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation FiberProductEmbeddingInDirectSum
. \(F: ( ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} ) \mapsto \mathrm{FiberProduct}(D) \rightarrow \bigoplus_{i=1}^n P_i\)
‣ 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 \(\mathrm{FiberProduct}(D)\).
‣ 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 \(\mathrm{FiberProduct}(D)\).
‣ 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 \(\pi_{k}: \mathrm{FiberProduct}(D) \rightarrow P_k\).
‣ 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 \(\pi_{k}: \mathrm{FiberProduct}(D) \rightarrow P_k\).
‣ 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 \(\pi_{k}: P \rightarrow P_k\).
‣ MorphismFromFiberProductToSink ( D ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{FiberProduct}(D), B )\)
The arguments are a list of morphisms \(D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n}\). The output is the composition \(\mu: \mathrm{FiberProduct}(D) \rightarrow B\) of the \(1\)-st projection \(\pi_1: \mathrm{FiberProduct}(D) \rightarrow P_1\) and \(\beta_1\).
‣ MorphismFromFiberProductToSinkOp ( D, method_selection_morphism ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{FiberProduct}(D), B )\)
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 composition \(\mu: \mathrm{FiberProduct}(D) \rightarrow B\) of the \(1\)-st projection \(\pi_1: \mathrm{FiberProduct}(D) \rightarrow P_1\) and \(\beta_1\).
‣ MorphismFromFiberProductToSinkWithGivenFiberProduct ( D, P ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( P, B )\)
The arguments are a list of morphisms \(D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n}\) and an object \(P = \mathrm{FiberProduct}(D)\). The output is the composition \(\mu: P \rightarrow B\) of the \(1\)-st projection \(\pi_1: P \rightarrow P_1\) and \(\beta_1\).
‣ UniversalMorphismIntoFiberProduct ( arg ) | ( function ) |
This is a convenience method. There are two 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.
‣ 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.
‣ 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.
‣ AddFiberProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation FiberProduct
. \(F: ( (\beta_i: P_i \rightarrow B)_{i = 1 \dots n} ) \mapsto P\)
‣ AddProjectionInFactorOfFiberProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ProjectionInFactorOfFiberProduct
. \(F: ( (\beta_i: P_i \rightarrow B)_{i = 1 \dots n}, k ) \mapsto \pi_k\)
‣ AddProjectionInFactorOfFiberProductWithGivenFiberProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ProjectionInFactorOfFiberProductWithGivenFiberProduct
. \(F: ( (\beta_i: P_i \rightarrow B)_{i = 1 \dots n}, k,P ) \mapsto \pi_k\)
‣ AddMorphismFromFiberProductToSink ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromFiberProductToSink
. \(F: ( (\beta_i: P_i \rightarrow B)_{i = 1 \dots n} ) \mapsto \mu\)
‣ AddMorphismFromFiberProductToSinkWithGivenFiberProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromFiberProductToSinkWithGivenFiberProduct
. \(F: ( (\beta_i: P_i \rightarrow B)_{i = 1 \dots n}, P ) \mapsto \mu\)
‣ AddUniversalMorphismIntoFiberProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoFiberProduct
. \(F: ( (\beta_i: P_i \rightarrow B)_{i = 1 \dots n}, \tau ) \mapsto u(\tau)\)
‣ AddUniversalMorphismIntoFiberProductWithGivenFiberProduct ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoFiberProductWithGivenFiberProduct
. \(F: ( (\beta_i: P_i \rightarrow B)_{i = 1 \dots n}, \tau, P ) \mapsto u(\tau)\)
‣ FiberProductFunctorial ( Ls, Lm, Lr ) | ( 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 three lists of morphisms \(L_s = ( \beta_i: P_i \rightarrow B)_{i = 1 \dots n}\), \(L_m = ( \mu_i: P_i \rightarrow P_i' )_{i = 1 \dots n}\), \(L_r = ( \beta_i': P_i' \rightarrow B')_{i = 1 \dots n}\) having the same length \(n\) 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 functoriality of the fiber product.
‣ FiberProductFunctorialWithGivenFiberProducts ( s, Ls, Lm, Lr, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(s, r)\)
The arguments are an object \(s = \mathrm{FiberProduct}( ( \beta_i )_{i=1 \dots n} )\), three lists of morphisms \(L_s = ( \beta_i: P_i \rightarrow B)_{i = 1 \dots n}\), \(L_m = ( \mu_i: P_i \rightarrow P_i' )_{i = 1 \dots n}\), \(L_r = ( \beta_i': P_i' \rightarrow B')_{i = 1 \dots n}\) having the same length \(n\) such that there exists a morphism \(\beta: B \rightarrow B'\) such that \(\beta_i' \circ \mu_i \sim_{P_i,B'} \beta \circ \beta_i\) for \(i = 1, \dots, n\), and an object \(r = \mathrm{FiberProduct}( ( \beta_i' )_{i=1 \dots n} )\). The output is the morphism \(s \rightarrow r\) given by the functoriality of the fiber product.
‣ AddFiberProductFunctorialWithGivenFiberProducts ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation FiberProductFunctorialWithGivenFiberProducts
. \(F: ( \mathrm{FiberProduct}( ( \beta_i )_{i=1 \dots n} ), (\beta_i: P_i \rightarrow B)_{i = 1 \dots n}, (\mu_i: P_i \rightarrow P_i')_{i = 1 \dots n}, (\beta_i': P_i' \rightarrow B')_{i = 1 \dots n}, \mathrm{FiberProduct}( ( \beta_i' )_{i=1 \dots n} ) ) ) \mapsto (\mathrm{FiberProduct}( ( \beta_i )_{i=1 \dots n} ) \rightarrow \mathrm{FiberProduct}( ( \beta_i' )_{i=1 \dots n} ) )\)
For an integer \(n \geq 1\) and 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 \(\mathrm{Pushout}( D ) \rightarrow \mathrm{Pushout}( D' )\).
‣ 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 \(\beta_i\).
‣ 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 \(\beta_i\).
‣ AddIsomorphismFromPushoutToCokernelOfDiagonalDifference ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromPushoutToCokernelOfDiagonalDifference
. \(F: ( ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} ) \mapsto (\mathrm{Pushout}(D) \rightarrow \Delta)\)
‣ 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 \(\beta_i\).
‣ 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 \(\beta_i\).
‣ AddIsomorphismFromCokernelOfDiagonalDifferenceToPushout ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromCokernelOfDiagonalDifferenceToPushout
. \(F: ( ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} ) \mapsto (\Delta \rightarrow \mathrm{Pushout}(D))\)
‣ IsomorphismFromPushoutToCoequalizerOfCoproductDiagram ( 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 coequalizer of the coproduct diagram of the morphisms \(\beta_i\).
‣ IsomorphismFromPushoutToCoequalizerOfCoproductDiagramOp ( 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 coequalizer of the coproduct diagram of the morphisms \(\beta_i\).
‣ AddIsomorphismFromPushoutToCoequalizerOfCoproductDiagram ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromPushoutToCoequalizerOfCoproductDiagram
. \(F: ( ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} ) \mapsto (\mathrm{Pushout}(D) \rightarrow \Delta)\)
‣ IsomorphismFromCoequalizerOfCoproductDiagramToPushout ( 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 coequalizer of the coproduct diagram of the morphisms \(\beta_i\).
‣ IsomorphismFromCoequalizerOfCoproductDiagramToPushoutOp ( 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 coequalizer of the coproduct diagram of the morphisms \(\beta_i\).
‣ AddIsomorphismFromCoequalizerOfCoproductDiagramToPushout ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromCoequalizerOfCoproductDiagramToPushout
. \(F: ( ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} ) \mapsto (\Delta \rightarrow \mathrm{Pushout}(D))\)
‣ 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 \(\beta_i\).
‣ 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 \(\beta_i\).
‣ AddDirectSumCodiagonalDifference ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DirectSumCodiagonalDifference
. \(F: ( D ) \mapsto \mathrm{DirectSumCodiagonalDifference}(D)\)
‣ 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 \(\bigoplus_{i=1}^n I_i \rightarrow \mathrm{Pushout}(D)\).
‣ 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 \(\bigoplus_{i=1}^n I_i \rightarrow \mathrm{Pushout}(D)\).
‣ AddDirectSumProjectionInPushout ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation DirectSumProjectionInPushout
. \(F: ( ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} ) \mapsto (\bigoplus_{i=1}^n I_i \rightarrow \mathrm{Pushout}(D))\)
‣ 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 \(\mathrm{Pushout}(D)\).
‣ 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 \(\mathrm{Pushout}(\alpha, \beta)\).
‣ 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 \(\mathrm{Pushout}(D)\).
‣ 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 \(\iota_k: I_k \rightarrow \mathrm{Pushout}( D )\).
‣ 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 \(\iota_k: I_k \rightarrow \mathrm{Pushout}( D )\).
‣ InjectionOfCofactorOfPushoutWithGivenPushout ( D, k, I ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( I_k, I )\).
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 \(\iota_k: I_k \rightarrow I\).
‣ MorphismFromSourceToPushout ( D ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( B, \mathrm{Pushout}( D ) )\).
The arguments are a list of morphisms \(D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n}\). The output is the composition \(\mu: B \rightarrow \mathrm{Pushout}(D)\) of \(\beta_1\) and the \(1\)-st injection \(\iota_1: I_1 \rightarrow \mathrm{Pushout}( D )\).
‣ MorphismFromSourceToPushoutOp ( D, method_selection_morphism ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( B, \mathrm{Pushout}( D ) )\).
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 composition \(\mu: B \rightarrow \mathrm{Pushout}(D)\) of \(\beta_1\) and the \(1\)-st injection \(\iota_1: I_1 \rightarrow \mathrm{Pushout}( D )\).
‣ MorphismFromSourceToPushoutWithGivenPushout ( D, I ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( B, I )\).
The arguments are a list of morphisms \(D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n}\) and an object \(I = \mathrm{Pushout}(D)\). The output is the composition \(\mu: B \rightarrow I\) of \(\beta_1\) and the \(1\)-st injection \(\iota_1: I_1 \rightarrow I\).
‣ UniversalMorphismFromPushout ( arg ) | ( function ) |
This is a convenience method. There are two 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.
‣ 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.
‣ 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.
‣ AddPushout ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation Pushout
. \(F: ( (\beta_i: B \rightarrow I_i)_{i = 1 \dots n} ) \mapsto I\)
‣ AddInjectionOfCofactorOfPushout ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InjectionOfCofactorOfPushout
. \(F: ( (\beta_i: B \rightarrow I_i)_{i = 1 \dots n}, k ) \mapsto \iota_k\)
‣ AddInjectionOfCofactorOfPushoutWithGivenPushout ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation InjectionOfCofactorOfPushoutWithGivenPushout
. \(F: ( (\beta_i: B \rightarrow I_i)_{i = 1 \dots n}, k, I ) \mapsto \iota_k\)
‣ AddMorphismFromSourceToPushout ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromSourceToPushout
. \(F: ( (\beta_i: B \rightarrow I_i)_{i = 1 \dots n} ) \mapsto \mu\)
‣ AddMorphismFromSourceToPushoutWithGivenPushout ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromSourceToPushoutWithGivenPushout
. \(F: ( (\beta_i: B \rightarrow I_i)_{i = 1 \dots n}, I ) \mapsto \mu\)
‣ AddUniversalMorphismFromPushout ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismFromPushout
. \(F: ( (\beta_i: B \rightarrow I_i)_{i = 1 \dots n}, \tau ) \mapsto u(\tau)\)
‣ AddUniversalMorphismFromPushoutWithGivenPushout ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismFromPushout
. \(F: ( (\beta_i: B \rightarrow I_i)_{i = 1 \dots n}, \tau, I ) \mapsto u(\tau)\)
‣ PushoutFunctorial ( Ls, Lm, Lr ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(\mathrm{Pushout}( ( \beta_i )_{i=1}^n ), \mathrm{Pushout}( ( \beta_i' )_{i=1}^n ))\)
The arguments are three lists of morphisms \(L_s = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n}\), \(L_m = ( \mu_i: I_i \rightarrow I_i' )_{i = 1 \dots n}\), \(L_r = ( \beta_i': B' \rightarrow I_i' )_{i = 1 \dots n}\) having the same length \(n\) 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 functoriality of the pushout.
‣ PushoutFunctorialWithGivenPushouts ( s, Ls, Lm, Lr, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(s, r)\)
The arguments are an object \(s = \mathrm{Pushout}( ( \beta_i )_{i=1}^n )\), three lists of morphisms \(L_s = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n}\), \(L_m = ( \mu_i: I_i \rightarrow I_i' )_{i = 1 \dots n}\), \(L_r = ( \beta_i': B' \rightarrow I_i' )_{i = 1 \dots n}\) having the same length \(n\) such that there exists a morphism \(\beta: B \rightarrow B'\) such that \(\beta_i' \circ \beta \sim_{B, I_i'} \mu_i \circ \beta_i\) for \(i = 1, \dots n\), and an object \(r = \mathrm{Pushout}( ( \beta_i' )_{i=1}^n )\). The output is the morphism \(s \rightarrow r\) given by the functoriality of the pushout.
‣ AddPushoutFunctorialWithGivenPushouts ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation PushoutFunctorial
. \(F: ( \mathrm{Pushout}( ( \beta_i )_{i=1}^n ), ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n}, ( \mu_i: I_i \rightarrow I_i' )_{i = 1 \dots n}, ( \beta_i': B' \rightarrow I_i' )_{i = 1 \dots n}, \mathrm{Pushout}( ( \beta_i' )_{i=1}^n ) ) ) \mapsto (\mathrm{Pushout}( ( \beta_i )_{i=1}^n ) \rightarrow \mathrm{Pushout}( ( \beta_i' )_{i=1}^n ) )\)
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.
‣ 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 \(\mathrm{im}(\alpha) \rightarrow \mathrm{KernelObject}( \mathrm{CokernelProjection}( \alpha ) )\).
‣ AddIsomorphismFromImageObjectToKernelOfCokernel ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromImageObjectToKernelOfCokernel
. \(F: \alpha \mapsto ( \mathrm{im}(\alpha) \rightarrow \mathrm{KernelObject}( \mathrm{CokernelProjection}( \alpha ) ) )\)
‣ 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 \(\mathrm{KernelObject}( \mathrm{CokernelProjection}( \alpha ) ) \rightarrow \mathrm{im}(\alpha)\).
‣ AddIsomorphismFromKernelOfCokernelToImageObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromKernelOfCokernelToImageObject
. \(F: \alpha \mapsto ( \mathrm{KernelObject}( \mathrm{CokernelProjection}( \alpha ) ) \rightarrow \mathrm{im}(\alpha) )\)
‣ ImageObject ( alpha ) | ( attribute ) |
Returns: an object
The argument is a morphism \(\alpha\). The output is the image \(\mathrm{im}( \alpha )\).
‣ 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 \(\iota: \mathrm{im}(\alpha) \hookrightarrow B\).
‣ 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 \(\iota: I \hookrightarrow B\).
‣ 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 \(c: A \rightarrow \mathrm{im}( \alpha )\).
‣ 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 \(c: A \rightarrow I\).
‣ 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.
‣ 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.
‣ AddImageObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ImageObject
. \(F: \alpha \mapsto I\).
‣ AddImageEmbedding ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ImageEmbedding
. \(F: \alpha \mapsto \iota\).
‣ AddImageEmbeddingWithGivenImageObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation ImageEmbeddingWithGivenImageObject
. \(F: (\alpha,I) \mapsto \iota\).
‣ AddCoastrictionToImage ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoastrictionToImage
. \(F: \alpha \mapsto c\).
‣ AddCoastrictionToImageWithGivenImageObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoastrictionToImageWithGivenImageObject
. \(F: (\alpha,I) \mapsto c\).
‣ AddUniversalMorphismFromImage ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismFromImage
. \(F: (\alpha, \tau) \mapsto u(\tau)\).
‣ AddUniversalMorphismFromImageWithGivenImageObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismFromImageWithGivenImageObject
. \(F: (\alpha, \tau, I) \mapsto u(\tau)\).
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.
‣ 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) \(\mathrm{coim}(\alpha) \rightarrow \mathrm{im}(\alpha)\).
‣ 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) \(C \rightarrow I\).
‣ AddMorphismFromCoimageToImageWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromCoimageToImageWithGivenObjects
. \(F: (C, \alpha, I) \mapsto ( C \rightarrow I )\).
‣ 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) \(\mathrm{im}(\alpha) \rightarrow \mathrm{coim}(\alpha)\).
‣ InverseMorphismFromCoimageToImageWithGivenObjects ( C, alpha, I ) | ( 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) \(I \rightarrow C\).
‣ AddInverseMorphismFromCoimageToImageWithGivenObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation MorphismFromCoimageToImageWithGivenObjects
. \(F: (C, \alpha, I) \mapsto ( I \rightarrow C )\).
‣ 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 \(\mathrm{coim}( \alpha ) \rightarrow \mathrm{CokernelObject}( \mathrm{KernelEmbedding}( \alpha ) )\).
‣ AddIsomorphismFromCoimageToCokernelOfKernel ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromCoimageToCokernelOfKernel
. \(F: \alpha \mapsto ( \mathrm{coim}( \alpha ) \rightarrow \mathrm{CokernelObject}( \mathrm{KernelEmbedding}( \alpha ) ) )\).
‣ 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 \(\mathrm{CokernelObject}( \mathrm{KernelEmbedding}( \alpha ) ) \rightarrow \mathrm{coim}( \alpha )\).
‣ AddIsomorphismFromCokernelOfKernelToCoimage ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation IsomorphismFromCokernelOfKernelToCoimage
. \(F: \alpha \mapsto ( \mathrm{CokernelObject}( \mathrm{KernelEmbedding}( \alpha ) ) \rightarrow \mathrm{coim}( \alpha ) )\).
‣ Coimage ( alpha ) | ( attribute ) |
Returns: an object
The argument is a morphism \(\alpha\). The output is the coimage \(\mathrm{coim}( \alpha )\).
‣ 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 \(\pi: A \twoheadrightarrow \mathrm{coim}( \alpha )\).
‣ 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 \(\pi: A \twoheadrightarrow C\).
‣ 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 \(a: \mathrm{coim}( \alpha ) \rightarrow B\).
‣ 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 \(a: C \rightarrow B\).
‣ 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.
‣ 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.
‣ AddCoimage ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation Coimage
. \(F: \alpha \mapsto C\)
‣ AddCoimageProjection ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoimageProjection
. \(F: \alpha \mapsto \pi\)
‣ AddCoimageProjectionWithGivenCoimage ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CoimageProjectionWithGivenCoimage
. \(F: (\alpha,C) \mapsto \pi\)
‣ AddAstrictionToCoimage ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation AstrictionToCoimage
. \(F: \alpha \mapsto a\)
‣ AddAstrictionToCoimageWithGivenCoimage ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation AstrictionToCoimageWithGivenCoimage
. \(F: (\alpha,C) \mapsto a\)
‣ AddUniversalMorphismIntoCoimage ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoCoimage
. \(F: (\alpha, \tau) \mapsto u(\tau)\)
‣ AddUniversalMorphismIntoCoimageWithGivenCoimage ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation UniversalMorphismIntoCoimageWithGivenCoimage
. \(F: (\alpha, \tau,C) \mapsto u(\tau)\)
Whenever the CoastrictionToImage
is an epi, or the AstrictionToCoimage
is a mono, there is a canonical morphism from the image to the coimage. If this canonical morphism is an isomorphism, we call it the canonical identification (between image and coimage).
‣ CanonicalIdentificationFromImageObjectToCoimage ( 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 canonical identification \(c: \mathrm{im}( \alpha ) \rightarrow \mathrm{coim}( \alpha )\).
‣ AddCanonicalIdentificationFromImageObjectToCoimage ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CanonicalIdentificationFromImageObjectToCoimage
. \(F: \alpha \mapsto c\)
‣ CanonicalIdentificationFromCoimageToImageObject ( 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 identification \(c: \mathrm{coim}( \alpha ) \rightarrow \mathrm{im}( \alpha )\).
‣ AddCanonicalIdentificationFromCoimageToImageObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation CanonicalIdentificationFromCoimageToImageObject
. \(F: \alpha \mapsto c\)
In an abelian category, we can define the operation that takes as an input a pair of morphisms \(\alpha: A \rightarrow B\), \(\beta: B \rightarrow C\) and outputs the subquotient of \(B\) given by
\(H := \mathrm{KernelObject}( \beta )/ (\mathrm{KernelObject}( \beta ) \cap \mathrm{ImageObject( \alpha )}\)).
This object is called a homology object of the pair \(\alpha, \beta\). Note that we do not need the precomposition of \(\alpha\) and \(\beta\) to be zero in order to make sense of this notion. Moreover, given a second pair \(\gamma: D \rightarrow E\), \(\delta: E \rightarrow F\) of morphisms, and a morphism \(\epsilon: B \rightarrow E\) such that there exists \(\omega_1: A \rightarrow D\), \(\omega_2: C \rightarrow F\) with \(\epsilon \circ \alpha \sim_{A,E} \gamma \circ \omega_1\) and \(\omega_2 \circ \beta \sim_{B,F} \delta \circ \epsilon\) there is a functorial way to obtain from these data a morphism between the two corresponding homology objects.
‣ HomologyObject ( alpha, beta ) | ( operation ) |
Returns: an object
The arguments are two morphisms \(\alpha: A \rightarrow B, \beta: B \rightarrow C\). The output is the homology object \(H\) of this pair.
‣ HomologyObjectFunctorial ( alpha, beta, epsilon, gamma, delta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( H_1, H_2 )\)
The argument are five morphisms \(\alpha: A \rightarrow B\), \(\beta: B \rightarrow C\), \(\epsilon: B \rightarrow E\), \(\gamma: D \rightarrow E, \delta: E \rightarrow F\) such that there exists \(\omega_1: A \rightarrow D\), \(\omega_2: C \rightarrow F\) with \(\epsilon \circ \alpha \sim_{A,E} \gamma \circ \omega_1\) and \(\omega_2 \circ \beta \sim_{B,F} \delta \circ \epsilon\). The output is the functorial morphism induced by \(\epsilon\) between the corresponding homology objects \(H_1\) and \(H_2\), where \(H_1\) denotes the homology object of the pair \(\alpha, \beta\), and \(H_2\) denotes the homology object of the pair \(\gamma, \delta\).
‣ HomologyObjectFunctorialWithGivenHomologyObjects ( H_1, L, H_2 ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( H_1, H_2 )\)
The arguments are an object \(H_1\), a list \(L\) consisting of five morphisms \(\alpha: A \rightarrow B\), \(\beta: B \rightarrow C\), \(\epsilon: B \rightarrow E\), \(\gamma: D \rightarrow E, \delta: E \rightarrow F\), and an object \(H_2\), such that \(H_1 = \mathrm{HomologyObject}( \alpha, \beta )\) and \(H_2 = \mathrm{HomologyObject}( \gamma, \delta )\), and such that there exists \(\omega_1: A \rightarrow D\), \(\omega_2: C \rightarrow F\) with \(\epsilon \circ \alpha \sim_{A,E} \gamma \circ \omega_1\) and \(\omega_2 \circ \beta \sim_{B,F} \delta \circ \epsilon\). The output is the functorial morphism induced by \(\epsilon\) between the corresponding homology objects \(H_1\) and \(H_2\), where \(H_1\) denotes the homology object of the pair \(\alpha, \beta\), and \(H_2\) denotes the homology object of the pair \(\gamma, \delta\).
‣ IsomorphismFromHomologyObjectToItsConstructionAsAnImageObject ( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( \mathrm{HomologyObject}( \alpha, \beta ), I )\)
The arguments are two morphisms \(\alpha: A \rightarrow B, \beta: B \rightarrow C\). The output is the natural isomorphism from the homology object \(H\) of \(\alpha\) and \(\beta\) to the construction of the homology object as \(\mathrm{ImageObject}( \mathrm{PreCompose}( \mathrm{KernelEmbedding}( \beta ), \mathrm{CokernelProjection}( \alpha ) ) )\), denoted by \(I\).
‣ IsomorphismFromItsConstructionAsAnImageObjectToHomologyObject ( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( I, \mathrm{HomologyObject}( \alpha, \beta ) )\)
The arguments are two morphisms \(\alpha: A \rightarrow B, \beta: B \rightarrow C\). The output is the natural isomorphism from the construction of the homology object as \(\mathrm{ImageObject}( \mathrm{PreCompose}( \mathrm{KernelEmbedding}( \beta ), \mathrm{CokernelProjection}( \alpha ) ) )\), denoted by \(I\), to the homology object \(H\) of \(\alpha\) and \(\beta\).
‣ AddHomologyObject ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation HomologyObject
. \(F: (\alpha, \beta) \mapsto \mathrm{HomologyObject}(\alpha, \beta)\).
‣ AddHomologyObjectFunctorialWithGivenHomologyObjects ( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation AddHomologyObjectFunctorialWithGivenHomologyObjects
. \(F: (H_1, \alpha, \beta, \epsilon, \gamma, \delta, H_2) \mapsto (H_1 \rightarrow H_2)\)
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