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.
‣ 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 operations 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 operations 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 operations adds the given function F to the category for the basic operation KernelEmbeddingWithGivenKernelObject. F: (\alpha, K) \mapsto \iota.
‣ AddKernelLift( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation KernelLift. F: (\alpha, \tau) \mapsto u(\tau).
‣ AddKernelLiftWithGivenKernelObject( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation KernelLiftWithGivenKernelObject. F: (\alpha, \tau, K) \mapsto u.
‣ KernelObjectFunctorial( L ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \mathrm{KernelObject}( \alpha ), \mathrm{KernelObject}( \alpha' ) )
The argument is a list L = [ \alpha: A \rightarrow B, [ \mu: A \rightarrow A', \nu: B \rightarrow B' ], \alpha': A' \rightarrow B' ] of morphisms. The output is the morphism \mathrm{KernelObject}( \alpha ) \rightarrow \mathrm{KernelObject}( \alpha' ) given by the functorality of the kernel.
‣ 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 functorality 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 functorality of the kernel.
‣ AddKernelObjectFunctorialWithGivenKernelObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation 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 ).
‣ 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 operations 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 operations 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 operations adds the given function F to the category for the basic operation CokernelProjection. F: (\alpha, K) \mapsto \epsilon.
‣ AddCokernelColift( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation CokernelProjection. F: (\alpha, \tau) \mapsto u(\tau).
‣ AddCokernelColiftWithGivenCokernelObject( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation CokernelProjection. F: (\alpha, \tau, K) \mapsto u(\tau).
‣ 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 functorality 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 functorality 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 functorality of the cokernel.
‣ AddCokernelObjectFunctorialWithGivenCokernelObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations adds the given function F to the category for the basic operation InitialObjectFunctorial. F: () \rightarrow ( I \rightarrow I ).
For a given list D = (S_1, \dots, S_n) in an Ab-category, a direct sum consists of five parts:
an object S,
a list of morphisms \pi = (\pi_i: S \rightarrow S_i)_{i = 1 \dots n},
a list of morphisms \iota = (\iota_i: S_i \rightarrow S)_{i = 1 \dots n},
a dependent function u_{\mathrm{in}} mapping every list \tau = ( \tau_i: T \rightarrow S_i )_{i = 1 \dots n} to a morphism u_{\mathrm{in}}(\tau): T \rightarrow S such that \pi_i \circ u_{\mathrm{in}}(\tau) \sim_{T,S_i} \tau_i for all i = 1, \dots, n.
a dependent function u_{\mathrm{out}} mapping every list \tau = ( \tau_i: S_i \rightarrow T )_{i = 1 \dots n} to a morphism u_{\mathrm{out}}(\tau): S \rightarrow T such that u_{\mathrm{out}}(\tau) \circ \iota_i \sim_{S_i, T} \tau_i for all i = 1, \dots, n,
such that
\sum_{i=1}^{n} \iota_i \circ \pi_i = \mathrm{id}_S,
\pi_j \circ \iota_i = \delta_{i,j},
where \delta_{i,j} \in \mathrm{Hom}( S_i, S_j ) is the identity if i=j, and 0 otherwise. The 5-tuple (S, \pi, \iota, u_{\mathrm{in}}, u_{\mathrm{out}}) is called a direct sum of D. We denote the object S of such a 5-tuple by \bigoplus_{i=1}^n S_i. We say that the morphisms u_{\mathrm{in}}(\tau), u_{\mathrm{out}}(\tau) are induced by the universal property of the direct sum. \\ \mathrm{DirectSum} is a functorial operation. This means: For (\mu_i: S_i \rightarrow S'_i)_{i=1\dots n}, we obtain a morphism \bigoplus_{i=1}^n S_i \rightarrow \bigoplus_{i=1}^n S_i'.
‣ 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.
‣ 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.
‣ AddProjectionInFactorOfDirectSum( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation ProjectionInFactorOfDirectSum. F: (D,k) \mapsto \pi_{k}.
‣ AddProjectionInFactorOfDirectSumWithGivenDirectSum( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation ProjectionInFactorOfDirectSumWithGivenDirectSum. F: (D,k,S) \mapsto \pi_{k}.
‣ AddInjectionOfCofactorOfDirectSum( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation InjectionOfCofactorOfDirectSum. F: (D,k) \mapsto \iota_{k}.
‣ AddInjectionOfCofactorOfDirectSumWithGivenDirectSum( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation InjectionOfCofactorOfDirectSumWithGivenDirectSum. F: (D,k,S) \mapsto \iota_{k}.
‣ AddUniversalMorphismIntoDirectSum( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoDirectSum. F: (D,\tau) \mapsto u_{\mathrm{in}}(\tau).
‣ AddUniversalMorphismIntoDirectSumWithGivenDirectSum( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoDirectSumWithGivenDirectSum. F: (D,\tau,S) \mapsto u_{\mathrm{in}}(\tau).
‣ AddUniversalMorphismFromDirectSum( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromDirectSum. F: (D,\tau) \mapsto u_{\mathrm{out}}(\tau).
‣ AddUniversalMorphismFromDirectSumWithGivenDirectSum( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromDirectSumWithGivenDirectSum. F: (D,\tau,S) \mapsto u_{\mathrm{out}}(\tau).
‣ AddIsomorphismFromDirectSumToDirectProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromDirectSumToDirectProduct. F: D \mapsto (\bigoplus_{i=1}^n S_i \rightarrow \prod_{i=1}^{n}S_i).
‣ AddIsomorphismFromDirectProductToDirectSum( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromDirectProductToDirectSum. F: D \mapsto ( \prod_{i=1}^{n}S_i \rightarrow \bigoplus_{i=1}^n S_i ).
‣ AddIsomorphismFromDirectSumToCoproduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromDirectSumToCoproduct. F: D \mapsto ( \bigoplus_{i=1}^n S_i \rightarrow \bigsqcup_{i=1}^{n}S_i ).
‣ AddIsomorphismFromCoproductToDirectSum( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation IsomorphismFromCoproductToDirectSum. F: D \mapsto ( \bigsqcup_{i=1}^{n}S_i \rightarrow \bigoplus_{i=1}^n S_i ).
‣ AddDirectSum( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation DirectSum. F: D \mapsto \bigoplus_{i=1}^n S_i.
‣ DirectSumFunctorial( L ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \bigoplus_{i=1}^n S_i, \bigoplus_{i=1}^n S_i' )
The argument is a list of morphisms L = ( \mu_1: S_1 \rightarrow S_1', \dots, \mu_n: S_n \rightarrow S_n' ). The output is a morphism \bigoplus_{i=1}^n S_i \rightarrow \bigoplus_{i=1}^n S_i' given by the functorality of the direct sum.
‣ 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 functorality of the direct sum.
‣ AddDirectSumFunctorialWithGivenDirectSums( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation 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 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 operations 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 operations 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 operations adds the given function F to the category for the basic operation InjectionOfCofactorOfCoproductWithGivenCoproduct. F: ( (I_1, \dots, I_n), i, I ) \mapsto \iota_i.
‣ AddUniversalMorphismFromCoproduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromCoproduct. F: ( (I_1, \dots, I_n), \tau ) \mapsto u( \tau ).
‣ AddUniversalMorphismFromCoproductWithGivenCoproduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromCoproductWithGivenCoproduct. F: ( (I_1, \dots, I_n), \tau, I ) \mapsto u( \tau ).
‣ CoproductFunctorial( L ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(\bigsqcup_{i=1}^n I_i, \bigsqcup_{i=1}^n I_i')
The argument is a list L = ( \mu_1: I_1 \rightarrow I_1', \dots, \mu_n: I_n \rightarrow I_n' ). The output is a morphism \bigsqcup_{i=1}^n I_i \rightarrow \bigsqcup_{i=1}^n I_i' given by the functorality of the coproduct.
‣ 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 functorality of the coproduct.
‣ AddCoproductFunctorialWithGivenCoproducts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation 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 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 operations 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 operations 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 operations adds the given function F to the category for the basic operation ProjectionInFactorOfDirectProductWithGivenDirectProduct. F: ( (P_1, \dots, P_n),k,P ) \mapsto \pi_k
‣ AddUniversalMorphismIntoDirectProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoDirectProduct. F: ( (P_1, \dots, P_n), \tau ) \mapsto u( \tau )
‣ AddUniversalMorphismIntoDirectProductWithGivenDirectProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoDirectProductWithGivenDirectProduct. F: ( (P_1, \dots, P_n), \tau, P ) \mapsto u( \tau )
‣ DirectProductFunctorial( L ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}( \prod_{i=1}^n P_i, \prod_{i=1}^n P_i' )
The argument is a list of morphisms L = (\mu_i: P_i \rightarrow P'_i)_{i=1\dots n}. The output is a morphism \prod_{i=1}^n P_i \rightarrow \prod_{i=1}^n P_i' given by the functorality of the direct product.
‣ 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 functorality of the direct product.
‣ AddDirectProductFunctorialWithGivenDirectProducts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation 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 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 operations adds the given function F to the category for the basic operation IsomorphismFromFiberProductToKernelOfDiagonalDifference. F: ( ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} ) \mapsto \mathrm{FiberProduct}(D) \rightarrow \Delta
‣ 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 operations adds the given function F to the category for the basic operation IsomorphismFromKernelOfDiagonalDifferenceToFiberProduct. F: ( ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} ) \mapsto \Delta \rightarrow \mathrm{FiberProduct}(D)
‣ 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 operations 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 operations adds the given function F to the category for the basic operation FiberProductEmbeddingInDirectSum. F: ( ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} ) \mapsto \mathrm{FiberProduct}(D) \rightarrow \bigoplus_{i=1}^n P_i
‣ 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.
‣ UniversalMorphismIntoFiberProduct( arg ) | ( function ) |
This is a convenience method. There are three different ways to use this method:
The arguments are a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} and a list of morphisms \tau = ( \tau_i: T \rightarrow P_i ) such that \beta_i \circ \tau_i \sim_{T, B} \beta_j \circ \tau_j for all pairs i,j. The output is the morphism u( \tau ): T \rightarrow \mathrm{FiberProduct}(D) given by the universal property of the fiber product.
The arguments are a list of morphisms D = ( \beta_i: P_i \rightarrow B )_{i = 1 \dots n} and morphisms \tau_1: T \rightarrow P_1, \dots, \tau_n: T \rightarrow P_n such that \beta_i \circ \tau_i \sim_{T, B} \beta_j \circ \tau_j for all pairs i,j. The output is the morphism u( \tau ): T \rightarrow \mathrm{FiberProduct}(D) given by the universal property of the fiber product.
‣ 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 operations adds the given function F to the category for the basic operation FiberProduct. F: ( (\beta_i: P_i \rightarrow B)_{i = 1 \dots n} ) \mapsto P
‣ AddProjectionInFactorOfFiberProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation ProjectionInFactorOfFiberProduct. F: ( (\beta_i: P_i \rightarrow B)_{i = 1 \dots n}, k ) \mapsto \pi_k
‣ AddProjectionInFactorOfFiberProductWithGivenFiberProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation ProjectionInFactorOfFiberProductWithGivenFiberProduct. F: ( (\beta_i: P_i \rightarrow B)_{i = 1 \dots n}, k,P ) \mapsto \pi_k
‣ AddUniversalMorphismIntoFiberProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoFiberProduct. F: ( (\beta_i: P_i \rightarrow B)_{i = 1 \dots n}, \tau ) \mapsto u(\tau)
‣ AddUniversalMorphismIntoFiberProductWithGivenFiberProduct( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismIntoFiberProductWithGivenFiberProduct. F: ( (\beta_i: P_i \rightarrow B)_{i = 1 \dots n}, \tau, P ) \mapsto u(\tau)
‣ FiberProductFunctorial( L ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(\mathrm{FiberProduct}( ( \beta_i )_{i=1 \dots n} ), \mathrm{FiberProduct}( ( \beta_i' )_{i=1 \dots n} ))
The argument is a list of triples of morphisms L = ( (\beta_i: P_i \rightarrow B, \mu_i: P_i \rightarrow P_i', \beta_i': P_i' \rightarrow B')_{i = 1 \dots n} ) such that there exists a morphism \beta: B \rightarrow B' such that \beta_i' \circ \mu_i \sim_{P_i,B'} \beta \circ \beta_i for i = 1, \dots, n. The output is the morphism \mathrm{FiberProduct}( ( \beta_i )_{i=1 \dots n} ) \rightarrow \mathrm{FiberProduct}( ( \beta_i' )_{i=1 \dots n} ) given by the functorality of the fiber product.
‣ FiberProductFunctorialWithGivenFiberProducts( s, L, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(s, r)
The arguments are an object s = \mathrm{FiberProduct}( ( \beta_i )_{i=1 \dots n} ), a list of triples of morphisms L = ( (\beta_i: P_i \rightarrow B, \mu_i: P_i \rightarrow P_i', \beta_i': P_i' \rightarrow B')_{i = 1 \dots n} ) such that there exists a morphism \beta: B \rightarrow B' such that \beta_i' \circ \mu_i \sim_{P_i,B'} \beta \circ \beta_i for i = 1, \dots, n, and an object r = \mathrm{FiberProduct}( ( \beta_i' )_{i=1 \dots n} ). The output is the morphism s \rightarrow r given by the functorality of the fiber product.
‣ AddFiberProductFunctorialWithGivenFiberProducts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation FiberProductFunctorialWithGivenFiberProducts. F: ( \mathrm{FiberProduct}( ( \beta_i )_{i=1 \dots n} ), (\beta_i: P_i \rightarrow B, \mu_i: P_i \rightarrow P_i', \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 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 operations adds the given function F to the category for the basic operation IsomorphismFromPushoutToCokernelOfDiagonalDifference. F: ( ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} ) \mapsto (\mathrm{Pushout}(D) \rightarrow \Delta)
‣ 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 operations adds the given function F to the category for the basic operation IsomorphismFromCokernelOfDiagonalDifferenceToPushout. F: ( ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} ) \mapsto (\Delta \rightarrow \mathrm{Pushout}(D))
‣ 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 operations 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 operations adds the given function F to the category for the basic operation DirectSumProjectionInPushout. F: ( ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} ) \mapsto (\bigoplus_{i=1}^n I_i \rightarrow \mathrm{Pushout}(D))
‣ 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, \mathrm{Pushout}( D ) ).
The arguments are a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n}, an integer k, and an object I = \mathrm{Pushout}(D). The output is the k-th injection \iota_k: I_k \rightarrow \mathrm{Pushout}( D ).
‣ UniversalMorphismFromPushout( arg ) | ( function ) |
This is a convenience method. There are three different ways to use this method:
The arguments are a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} and a list of morphisms \tau = ( \tau_i: I_i \rightarrow T )_{i = 1 \dots n} such that \tau_i \circ \beta_i \sim_{B,T} \tau_j \circ \beta_j. The output is the morphism u( \tau ): \mathrm{Pushout}(D) \rightarrow T given by the universal property of the pushout.
The arguments are a list of morphisms D = ( \beta_i: B \rightarrow I_i )_{i = 1 \dots n} and morphisms \tau_1: I_1 \rightarrow T, \dots, \tau_n: I_n \rightarrow T such that \tau_i \circ \beta_i \sim_{B,T} \tau_j \circ \beta_j. The output is the morphism u( \tau ): \mathrm{Pushout}(D) \rightarrow T given by the universal property of the pushout.
‣ 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 operations adds the given function F to the category for the basic operation Pushout. F: ( (\beta_i: B \rightarrow I_i)_{i = 1 \dots n} ) \mapsto I
‣ AddInjectionOfCofactorOfPushout( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation InjectionOfCofactorOfPushout. F: ( (\beta_i: B \rightarrow I_i)_{i = 1 \dots n}, k ) \mapsto \iota_k
‣ AddInjectionOfCofactorOfPushoutWithGivenPushout( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation InjectionOfCofactorOfPushoutWithGivenPushout. F: ( (\beta_i: B \rightarrow I_i)_{i = 1 \dots n}, k, I ) \mapsto \iota_k
‣ AddUniversalMorphismFromPushout( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromPushout. F: ( (\beta_i: B \rightarrow I_i)_{i = 1 \dots n}, \tau ) \mapsto u(\tau)
‣ AddUniversalMorphismFromPushoutWithGivenPushout( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation UniversalMorphismFromPushout. F: ( (\beta_i: B \rightarrow I_i)_{i = 1 \dots n}, \tau, I ) \mapsto u(\tau)
‣ PushoutFunctorial( L ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(\mathrm{Pushout}( ( \beta_i )_{i=1}^n ), \mathrm{Pushout}( ( \beta_i' )_{i=1}^n ))
The argument is a list L = ( ( \beta_i: B \rightarrow I_i, \mu_i: I_i \rightarrow I_i', \beta_i': B' \rightarrow I_i' )_{i = 1 \dots n} ) such that there exists a morphism \beta: B \rightarrow B' such that \beta_i' \circ \beta \sim_{B, I_i'} \mu_i \circ \beta_i for i = 1, \dots n. The output is the morphism \mathrm{Pushout}( ( \beta_i )_{i=1}^n ) \rightarrow \mathrm{Pushout}( ( \beta_i' )_{i=1}^n ) given by the functorality of the pushout.
‣ PushoutFunctorialWithGivenPushouts( s, L, r ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(s, r)
The arguments are an object s = \mathrm{Pushout}( ( \beta_i )_{i=1}^n ), a list L = ( ( \beta_i: B \rightarrow I_i, \mu_i: I_i \rightarrow I_i', \beta_i': B' \rightarrow I_i' )_{i = 1 \dots n} ) such that there exists a morphism \beta: B \rightarrow B' such that \beta_i' \circ \beta \sim_{B, I_i'} \mu_i \circ \beta_i for i = 1, \dots n, and an object r = \mathrm{Pushout}( ( \beta_i' )_{i=1}^n ). The output is the morphism s \rightarrow r given by the functorality of the pushout.
‣ AddPushoutFunctorialWithGivenPushouts( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation PushoutFunctorial. F: ( \mathrm{Pushout}( ( \beta_i )_{i=1}^n ), ( \beta_i: B \rightarrow I_i, \mu_i: I_i \rightarrow I_i', \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 operations adds the given function F to the category for the basic operation IsomorphismFromImageObjectToKernelOfCokernel. F: \alpha \mapsto ( \mathrm{im}(\alpha) \rightarrow \mathrm{KernelObject}( \mathrm{CokernelProjection}( \alpha ) ) )
‣ 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 operations adds the given function F to the category for the basic operation IsomorphismFromKernelOfCokernelToImageObject. F: \alpha \mapsto ( \mathrm{KernelObject}( \mathrm{CokernelProjection}( \alpha ) ) \rightarrow \mathrm{im}(\alpha) )
‣ 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations 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( alpha ) | ( operation ) |
Returns: a morphism in \mathrm{Hom}(I,C)
The argument is an object C = \mathrm{coim}(\alpha), a morphism \alpha: A \rightarrow B, and an object I = \mathrm{im}(\alpha). The output is the inverse of the canonical morphism (in an abelian category) I \rightarrow C.
‣ AddInverseMorphismFromCoimageToImageWithGivenObjects( C, F ) | ( operation ) |
Returns: nothing
The arguments are a category C and a function F. This operations adds the given function F to the category for the basic operation MorphismFromCoimageToImageWithGivenObjects. F: (C, \alpha, I) \mapsto ( I \rightarrow C ).
‣ 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 operations adds the given function F to the category for the basic operation IsomorphismFromCoimageToCokernelOfKernel. F: \alpha \mapsto ( \mathrm{coim}( \alpha ) \rightarrow \mathrm{CokernelObject}( \mathrm{KernelEmbedding}( \alpha ) ) ).
‣ 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 operations adds the given function F to the category for the basic operation IsomorphismFromCokernelOfKernelToCoimage. F: \alpha \mapsto ( \mathrm{CokernelObject}( \mathrm{KernelEmbedding}( \alpha ) ) \rightarrow \mathrm{coim}( \alpha ) ).
‣ Coimage( alpha ) | ( attribute ) |
Returns: an object
The argument is a morphism \alpha. The output is the coimage \mathrm{coim}( \alpha ).
‣ CoimageProjection( C ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(A, C)
This is a convenience method. The argument is an object C which was created as a coimage of a morphism \alpha: A \rightarrow B. The output is the coimage projection \pi: A \twoheadrightarrow C.
‣ 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( C ) | ( attribute ) |
Returns: a morphism in \mathrm{Hom}(C,B)
This is a convenience method. The argument is an object C which was created as a coimage of a morphism \alpha: A \rightarrow B. The output is the astriction to coimage a: C \rightarrow B.
‣ 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations 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 operations adds the given function F to the category for the basic operation UniversalMorphismIntoCoimageWithGivenCoimage. F: (\alpha, \tau,C) \mapsto u(\tau)
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