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1 Basic operations

1.1 Weak kernel

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

• an object K,

• a morphism \iota: K \rightarrow A such that \alpha \circ \iota \sim_{K,B} 0,

• a dependent function u mapping each morphism \tau: T \rightarrow A satisfying \alpha \circ \tau \sim_{T,B} 0 to a morphism u(\tau): T \rightarrow K such that \iota \circ u( \tau ) \sim_{T,A} \tau.

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

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

Returns: an object

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

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

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

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

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

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

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

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

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

The arguments are a morphism \alpha: A \rightarrow B and a test morphism \tau: T \rightarrow A satisfying \alpha \circ \tau \sim_{T,B} 0. The output is the morphism u(\tau): T \rightarrow \mathrm{WeakKernelObject}(\alpha) given by the universal property of the weak kernel.

1.1-5 WeakKernelLiftWithGivenWeakKernelObject
 ‣ WeakKernelLiftWithGivenWeakKernelObject( alpha, tau, K ) ( operation )

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

The arguments are a morphism \alpha: A \rightarrow B, a test morphism \tau: T \rightarrow A satisfying \alpha \circ \tau \sim_{T,B} 0, and an object K = \mathrm{WeakKernelObject}(\alpha). The output is the morphism u(\tau): T \rightarrow K given by the universal property of the weak kernel.

 ‣ AddWeakKernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation WeakKernelObject. F: \alpha \mapsto \mathrm{WeakKernelObject}(\alpha).

 ‣ AddWeakKernelEmbedding( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation WeakKernelEmbedding. F: \alpha \mapsto \iota.

 ‣ AddWeakKernelEmbeddingWithGivenWeakKernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation WeakKernelEmbeddingWithGivenWeakKernelObject. F: (\alpha, K) \mapsto \iota.

 ‣ AddWeakKernelLift( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation WeakKernelLift. F: (\alpha, \tau) \mapsto u(\tau).

 ‣ AddWeakKernelLiftWithGivenWeakKernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation WeakKernelLiftWithGivenWeakKernelObject. F: (\alpha, \tau, K) \mapsto u.

1.2 Weak cokernel

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

• an object K,

• a morphism \epsilon: B \rightarrow K such that \epsilon \circ \alpha \sim_{A,K} 0,

• a dependent function u mapping each \tau: B \rightarrow T satisfying \tau \circ \alpha \sim_{A, T} 0 to a morphism u(\tau):K \rightarrow T such that u(\tau) \circ \epsilon \sim_{B,T} \tau.

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

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

Returns: an object

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

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

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

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

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

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

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

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

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

The arguments are a morphism \alpha: A \rightarrow B and a test morphism \tau: B \rightarrow T satisfying \tau \circ \alpha \sim_{A, T} 0. The output is the morphism u(\tau): \mathrm{WeakCokernelObject}(\alpha) \rightarrow T given by the universal property of the weak cokernel.

1.2-5 WeakCokernelColiftWithGivenWeakCokernelObject
 ‣ WeakCokernelColiftWithGivenWeakCokernelObject( alpha, tau, K ) ( operation )

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

The arguments are a morphism \alpha: A \rightarrow B, a test morphism \tau: B \rightarrow T satisfying \tau \circ \alpha \sim_{A, T} 0, and an object K = \mathrm{WeakCokernelObject}(\alpha). The output is the morphism u(\tau): K \rightarrow T given by the universal property of the weak cokernel.

 ‣ AddWeakCokernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation WeakCokernelObject. F: \alpha \mapsto K.

 ‣ AddWeakCokernelProjection( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation WeakCokernelProjection. F: \alpha \mapsto \epsilon.

 ‣ AddWeakCokernelProjectionWithGivenWeakCokernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation WeakCokernelProjectionWithGivenWeakCokernelObject. F: (\alpha, K) \mapsto \epsilon.

 ‣ AddWeakCokernelColift( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation WeakCokernelColift. F: (\alpha, \tau) \mapsto u(\tau).

 ‣ AddWeakCokernelColiftWithGivenWeakCokernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation WeakCokernelColiftWithGivenWeakCokernelObject. F: (\alpha, \tau, K) \mapsto u(\tau).

1.3 Weak bi-fiber product

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

• an object P,

• morphisms \pi_1: P \rightarrow A, \pi_2: P \rightarrow B such that \alpha \circ \pi_1 \sim_{P,B} \beta \circ \pi_2,

• a dependent function u mapping each pair \tau = ( \tau_1, \tau_2 ) of morphisms \tau_1: T \rightarrow A, \tau_2: T \rightarrow C with the property \alpha \circ \tau_1 \sim_{T,B} \beta \circ \tau_2 to a morphism u(\tau):T \rightarrow P such that \pi_1 \circ u( \tau ) \sim_{A,T} \tau_1 and \pi_2 \circ u( \tau ) \sim_{C,T} \tau_2.

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

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

Returns: an object

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 ‣ AddWeakBiFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation WeakBiFiberProduct. F: ( \alpha, \beta ) \mapsto P

 ‣ AddProjectionInFirstFactorOfWeakBiFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation ProjectionInFirstFactorOfWeakBiFiberProduct. F: ( \alpha, \beta ) \mapsto \pi_1

 ‣ AddProjectionInSecondFactorOfWeakBiFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation ProjectionInSecondFactorOfWeakBiFiberProduct. F: ( \alpha, \beta ) \mapsto \pi_2

 ‣ AddProjectionInFirstFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation ProjectionInFirstFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct. F: ( \alpha, \beta, P ) \mapsto \pi_1

 ‣ AddProjectionInSecondFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation ProjectionInSecondFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct. F: ( \alpha, \beta, P ) \mapsto \pi_2

 ‣ AddUniversalMorphismIntoWeakBiFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation UniversalMorphismIntoWeakBiFiberProduct. F: ( \alpha, \beta, \tau_1, \tau_2 ) \mapsto u(\tau)

 ‣ AddUniversalMorphismIntoWeakBiFiberProductWithGivenWeakBiFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation UniversalMorphismIntoWeakBiFiberProductWithGivenWeakBiFiberProduct. F: ( \alpha, \beta, \tau_1, \tau_2, P ) \mapsto u(\tau)

 ‣ AddWeakBiFiberProductMorphismToDirectSum( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation WeakBiFiberProductMorphismToDirectSum. F: ( \alpha, \beta ) \mapsto \mathrm{WeakBiFiberProductMorphismToDirectSum}( \alpha, \beta )

1.4 Biased weak fiber product

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

• an object P,

• a morphism \pi: P \rightarrow A such that there exists a morphism \delta: P \rightarrow C such that \beta \circ \delta \sim_{P,B} \alpha \circ \pi,

• a dependent function u mapping each \tau: T \rightarrow A, which admits a morphism \mu \colon T \rightarrow C with \beta \circ \mu \sim_{T,B} \alpha \circ \tau, to a morphism u(\tau):T \rightarrow P such that \pi \circ u(\tau) \sim_{T,A} \tau.

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

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

Returns: an object

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

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

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

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

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

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

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

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

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

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

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

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

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

 ‣ AddBiasedWeakFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation BiasedWeakFiberProduct. F: ( \alpha, \beta ) \mapsto P

 ‣ AddProjectionOfBiasedWeakFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation ProjectionOfBiasedWeakFiberProduct. F: ( \alpha, \beta ) \mapsto \pi

 ‣ AddProjectionOfBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation ProjectionOfBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct. F: ( \alpha, \beta, P ) \mapsto \pi

 ‣ AddUniversalMorphismIntoBiasedWeakFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation UniversalMorphismIntoBiasedWeakFiberProduct. F: ( \alpha, \beta, \tau ) \mapsto u(\tau)

 ‣ AddUniversalMorphismIntoBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation UniversalMorphismIntoBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct. F: ( \alpha, \beta, \tau, P ) \mapsto u(\tau)

1.5 Weak bi-pushout

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

• an object P,

• morphisms \iota_1: B \rightarrow P, \iota_2: C \rightarrow P such that \iota_1 \circ \alpha \sim_{A,P} \iota_2 \circ \beta,

• a dependent function u mapping each pair \tau = (\tau_1, \tau_2) of morphisms \tau_1: B \rightarrow T, \tau_2: C \rightarrow T with the property \tau_1 \circ \alpha \sim_{A,T} \tau_2 \circ \beta to a morphism u(\tau): P \rightarrow T such that u( \tau ) \circ \iota_1 \sim_{B,T} \tau_1 and u( \tau ) \circ \iota_2 \sim_{C,T} \tau_2.

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

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

Returns: an object

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 ‣ AddWeakBiPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation WeakBiPushout. F: ( \alpha, \beta ) \mapsto P

 ‣ AddInjectionOfFirstCofactorOfWeakBiPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InjectionOfFirstCofactorOfWeakBiPushout. F: ( \alpha, \beta ) \mapsto \iota_1

 ‣ AddInjectionOfSecondCofactorOfWeakBiPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InjectionOfSecondCofactorOfWeakBiPushout. F: ( \alpha, \beta ) \mapsto \iota_2

 ‣ AddInjectionOfFirstCofactorOfWeakBiPushoutWithGivenWeakBiPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InjectionOfFirstCofactorOfWeakBiPushoutWithGivenWeakBiPushout. F: ( \alpha, \beta, P ) \mapsto \iota_1

 ‣ AddInjectionOfSecondCofactorOfWeakBiPushoutWithGivenWeakBiPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InjectionOfSecondCofactorOfWeakBiPushoutWithGivenWeakBiPushout. F: ( \alpha, \beta, P ) \mapsto \iota_2

 ‣ AddUniversalMorphismFromWeakBiPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation UniversalMorphismFromWeakBiPushout. F: ( \alpha, \beta, \tau_1, \tau_2 ) \mapsto u(\tau)

 ‣ AddUniversalMorphismFromWeakBiPushoutWithGivenWeakBiPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation UniversalMorphismFromWeakBiPushoutWithGivenWeakBiPushout. F: ( \alpha, \beta, \tau_1, \tau_2, P ) \mapsto u(\tau)

 ‣ AddDirectSumMorphismToWeakBiPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation DirectSumMorphismToWeakBiPushout. F: ( \alpha, \beta ) \mapsto \mathrm{DirectSumMorphismToWeakBiPushout}( \alpha, \beta )

1.6 Biased weak pushout

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

• an object P,

• a morphism \iota: B \rightarrow P such that there exists a morphism \delta: C \rightarrow P such that \delta \circ \beta \sim_{A,P} \iota \circ \alpha,

• a dependent function u mapping each \tau: B \rightarrow T, which admits a morphism \mu \colon C \rightarrow T with \mu \circ \beta \sim_{B,T} \tau \circ \alpha, to a morphism u(\tau):P \rightarrow T such that u(\tau) \circ \iota \sim_{A,T} \tau.

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

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

Returns: an object

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

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

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

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

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

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

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

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

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

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

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

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

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

 ‣ AddBiasedWeakPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation BiasedWeakPushout. F: ( \alpha, \beta ) \mapsto P

 ‣ AddInjectionOfBiasedWeakPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InjectionOfBiasedWeakPushout. F: ( \alpha, \beta ) \mapsto \iota

 ‣ AddInjectionOfBiasedWeakPushoutWithGivenBiasedWeakPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation InjectionOfBiasedWeakPushoutWithGivenBiasedWeakPushout. F: ( \alpha, \beta, P ) \mapsto \iota

 ‣ AddUniversalMorphismFromBiasedWeakPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation UniversalMorphismFromBiasedWeakPushout. F: ( \alpha, \beta, \tau ) \mapsto u(\tau)

 ‣ AddUniversalMorphismFromBiasedWeakPushoutWithGivenBiasedWeakPushout( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation UniversalMorphismFromBiasedWeakPushoutWithGivenBiasedWeakPushout. F: ( \alpha, \beta, \tau, P ) \mapsto u(\tau)

1.7 Abelian constructions

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

Returns: an object

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

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

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

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

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

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

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

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

Returns: an object

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

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

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

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

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

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

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

 ‣ AddSomeProjectiveObjectForKernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation SomeProjectiveObjectForKernelObject. F: \alpha \mapsto P.

 ‣ AddSomeInjectiveObjectForCokernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation SomeInjectiveObjectForCokernelObject. F: \alpha \mapsto I.

 ‣ AddEpimorphismFromSomeProjectiveObjectForKernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation EpimorphismFromSomeProjectiveObjectForKernelObject. F: \alpha \mapsto \pi.

 ‣ AddMonomorphismToSomeInjectiveObjectForCokernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MonomorphismToSomeInjectiveObjectForCokernelObject. F: \alpha \mapsto \iota.

 ‣ AddEpimorphismFromSomeProjectiveObjectForKernelObjectWithGivenSomeProjectiveObjectForKernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation EpimorphismFromSomeProjectiveObjectForKernelObjectWithGivenSomeProjectiveObjectForKernelObject. F: (\alpha, P) \mapsto \pi.

 ‣ AddMonomorphismToSomeInjectiveObjectForCokernelObjectWithGivenSomeInjectiveObjectForCokernelObject( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation MonomorphismToSomeInjectiveObjectForCokernelObjectWithGivenSomeInjectiveObjectForCokernelObject. F: (\alpha, I) \mapsto \iota.

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