3.7-10 \*
Any GAP object satisfying IsCapCategoryMorphism
can be added to a category and then becomes a morphism in this category. Any morphism can belong to one or no category. After a GAP object is added to the category, it knows which things can be computed in its category and to which category it belongs. It knows categorical properties and attributes, and the functions for existential quantifiers can be applied to the morphism.
‣ CapCategory ( alpha ) | ( attribute ) |
Returns: a category
The argument is a morphism \(\alpha\). The output is the category \(\mathbf{C}\) to which \(\alpha\) was added.
‣ Source ( alpha ) | ( attribute ) |
Returns: an object
The argument is a morphism \(\alpha: a \rightarrow b\). The output is its source \(a\).
‣ Range ( alpha ) | ( attribute ) |
Returns: an object
The argument is a morphism \(\alpha: a \rightarrow b\). The output is its range \(b\).
‣ IsMonomorphism ( alpha ) | ( property ) |
Returns: a boolean
The argument is a morphism \(\alpha\). The output is true
if \(\alpha\) is a monomorphism, otherwise the output is false
.
‣ AddIsMonomorphism ( 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 IsMonomorphism
. \(F: \alpha \mapsto \mathtt{IsMonomorphism}(\alpha)\).
‣ IsEpimorphism ( alpha ) | ( property ) |
Returns: a boolean
The argument is a morphism \(\alpha\). The output is true
if \(\alpha\) is an epimorphism, otherwise the output is false
.
‣ AddIsEpimorphism ( 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 IsEpimorphism
. \(F: \alpha \mapsto \mathtt{IsEpimorphism}(\alpha)\).
‣ IsIsomorphism ( alpha ) | ( property ) |
Returns: a boolean
The argument is a morphism \(\alpha\). The output is true
if \(\alpha\) is an isomorphism, otherwise the output is false
.
‣ AddIsIsomorphism ( 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 IsIsomorphism
. \(F: \alpha \mapsto \mathtt{IsIsomorphism}(\alpha)\).
‣ IsSplitMonomorphism ( alpha ) | ( property ) |
Returns: a boolean
The argument is a morphism \(\alpha\). The output is true
if \(\alpha\) is a split monomorphism, otherwise the output is false
.
‣ AddIsSplitMonomorphism ( 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 IsSplitMonomorphism
. \(F: \alpha \mapsto \mathtt{IsSplitMonomorphism}(\alpha)\).
‣ IsSplitEpimorphism ( alpha ) | ( property ) |
Returns: a boolean
The argument is a morphism \(\alpha\). The output is true
if \(\alpha\) is a split epimorphism, otherwise the output is false
.
‣ AddIsSplitEpimorphism ( 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 IsSplitEpimorphism
. \(F: \alpha \mapsto \mathtt{IsSplitEpimorphism}(\alpha)\).
‣ IsOne ( alpha ) | ( property ) |
Returns: a boolean
The argument is a morphism \(\alpha: a \rightarrow a\). The output is true
if \(\alpha\) is congruent to the identity of \(a\), otherwise the output is false
.
‣ AddIsOne ( 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 IsOne
. \(F: \alpha \mapsto \mathtt{IsOne}(\alpha)\).
‣ IsIdempotent ( alpha ) | ( property ) |
Returns: a boolean
The argument is a morphism \(\alpha: a \rightarrow a\). The output is true
if \(\alpha^2 \sim_{a,a} \alpha\), otherwise the output is false
.
‣ AddIsIdempotent ( 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 IsIdempotent
. \(F: \alpha \mapsto \mathtt{IsIdempotent}(\alpha)\).
CAP provides two principal methods to generate random morphisms with or without fixed source and range:
By integers: The integer is simply a parameter that can be used to create a random morphism.
By lists: The list is used when creating a random morphism would need more than one parameter. Lists offer more flexibility at the expense of the genericity of the methods. This happens because lists that are valid as input in some category may be not valid for other categories. Hence, these operations are not thought to be used in generic categorical algorithms.
‣ RandomMorphismWithFixedSourceByInteger ( a, n ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\) or fail
The arguments are an object \(a\) in a category \(C\) and an integer \(n\). The output is a random morphism \(\alpha: a \rightarrow b\) for some object \(b\) in \(C\) or fail
.
‣ AddRandomMorphismWithFixedSourceByInteger ( 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 RandomMorphismWithFixedSourceByInteger
. The function \(F\) maps \((a,n)\) to fail
or to a random morphism in \(C\) whose source is \(a\).
‣ RandomMorphismWithFixedSourceByList ( a, L ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\) or fail
The arguments are an object \(a\) in a category \(C\) and a list \(L\). The output is a random morphism \(\alpha: a \rightarrow b\) for some object \(b\) in \(C\) or fail
.
‣ AddRandomMorphismWithFixedSourceByList ( 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 RandomMorphismWithFixedSourceByList
. The function \(F\) maps \((a,L)\) to fail
or to a random morphism in \(C\) whose source is \(a\).
‣ RandomMorphismWithFixedRangeByInteger ( b, n ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\) or fail
The arguments are an object \(b\) in a category \(C\) and an integer \(n\). The output is a random morphism \(\alpha: a \rightarrow b\) for some object \(a\) in \(C\) or fail
.
‣ AddRandomMorphismWithFixedRangeByInteger ( 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 RandomMorphismWithFixedRangeByInteger
. The function \(F\) maps \((b,n)\) to fail
or to a random morphism in \(C\) whose range is \(b\).
‣ RandomMorphismWithFixedRangeByList ( b, L ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\) or fail
The arguments are an object \(b\) in a category \(C\) and a list \(L\). The output is a random morphism \(\alpha: a \rightarrow b\) for some object \(a\) in \(C\) or fail
.
‣ AddRandomMorphismWithFixedRangeByList ( 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 RandomMorphismWithFixedRangeByList
. The function \(F\) maps \((b,L)\) to fail
or to a random morphism in \(C\) whose range is \(b\).
‣ RandomMorphismWithFixedSourceAndRangeByInteger ( a, b, n ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\) or fail
The arguments are two objects \(a\) and \(b\) in a category \(C\) and an integer \(n\). The output is a random morphism \(\alpha: a \rightarrow b\) in \(C\) or fail
.
‣ AddRandomMorphismWithFixedSourceAndRangeByInteger ( 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 RandomMorphismWithFixedSourceAndRangeByInteger
. The function \(F\) maps \((a,b,n)\) to fail
or to a random morphism in \(C\) from \(a\) to \(b\).
‣ RandomMorphismWithFixedSourceAndRangeByList ( a, b, L ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\) or fail
This operation is not a CAP basic operation The arguments are two objects \(a\) and \(b\) in a category \(C\) and a list \(L\). The output is a random morphism \(\alpha: a \rightarrow b\) in \(C\) or fail
.
‣ AddRandomMorphismWithFixedSourceAndRangeByList ( 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 RandomMorphismWithFixedSourceAndRangeByList
. The function \(F\) maps \((a,b,L)\) to fail
or to a random morphism in \(C\) from \(a\) to \(b\).
‣ RandomMorphismByInteger ( C, n ) | ( operation ) |
Returns: a morphism or fail
The arguments are a category \(C\) and an integer \(n\). The output is a random morphism in \(C\) or fail
. If the methods RandomObjectByInteger
and RandomMorphismWithFixedSourceByInteger
(RandomMorphismWithFixedRangeByInteger
) are added to the category \(C\), then RandomMorphismByInteger
can be derived to return a random morphism of complexity \(n\) with random source(range) of complexity \(n\).
‣ AddRandomMorphismByInteger ( 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 RandomMorphismByInteger
. The function \(F\) maps \((C,n)\) to fail
or to a random morphism in \(C\).
‣ RandomMorphismByList ( C, L ) | ( operation ) |
Returns: a morphism or fail
The arguments are a category \(C\) and a list \(L\). The output is a random morphism in \(C\) or fail
.
‣ AddRandomMorphismByList ( 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 RandomMorphismByList
. The function \(F\) maps \((C,L)\) to fail
or to a random morphism in \(C\).
‣ RandomMorphismWithFixedSource ( a, n ) | ( operation ) |
‣ RandomMorphismWithFixedSource ( a, L ) | ( operation ) |
‣ RandomMorphismWithFixedRange ( b, n ) | ( operation ) |
‣ RandomMorphismWithFixedRange ( b, L ) | ( operation ) |
‣ RandomMorphismWithFixedSourceAndRange ( a, b, n ) | ( operation ) |
‣ RandomMorphismWithFixedSourceAndRange ( a, b, L ) | ( operation ) |
‣ RandomMorphism ( C, n ) | ( operation ) |
‣ RandomMorphism ( C, L ) | ( operation ) |
These are convenient methods and they, depending on the input, delegate to one of the above methods.
Non-categorical properties are not stable under equivalences of categories.
‣ IsIdenticalToIdentityMorphism ( alpha ) | ( property ) |
Returns: a boolean
The argument is a morphism \(\alpha: a \rightarrow b\). The output is true
if \(\alpha = \mathrm{id}_a\), otherwise the output is false
.
‣ AddIsIdenticalToIdentityMorphism ( 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 IsIdenticalToIdentityMorphism
. \(F: \alpha \mapsto \mathtt{IsIdenticalToIdentityMorphism}(\alpha)\).
‣ IsIdenticalToZeroMorphism ( alpha ) | ( property ) |
Returns: a boolean
The argument is a morphism \(\alpha: a \rightarrow b\). The output is true
if \(\alpha = 0\), otherwise the output is false
.
‣ AddIsIdenticalToZeroMorphism ( 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 IsIdenticalToZeroMorphism
. \(F: \alpha \mapsto \mathtt{IsIdenticalToZeroMorphism }(\alpha)\).
‣ IsEndomorphism ( alpha ) | ( property ) |
Returns: a boolean
The argument is a morphism \(\alpha\). The output is true
if \(\alpha\) is an endomorphism, otherwise the output is false
.
‣ AddIsEndomorphism ( 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 IsEndomorphism
. \(F: \alpha \mapsto \mathtt{IsEndomorphism}(\alpha)\).
‣ IsAutomorphism ( alpha ) | ( property ) |
Returns: a boolean
The argument is a morphism \(\alpha\). The output is true
if \(\alpha\) is an automorphism, otherwise the output is false
.
‣ AddIsAutomorphism ( 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 IsAutomorphism
. \(F: \alpha \mapsto \mathtt{IsAutomorphism}(\alpha)\).
‣ Add ( category, morphism ) | ( operation ) |
Adds morphism as a morphism to category.
‣ AddMorphism ( category, morphism ) | ( operation ) |
Adds morphism as a morphism to category. If morphism already lies in the filter IsCapCategoryMorphism
, the operation Add
(3.5-1) can be used instead.
‣ AddMorphismRepresentation ( category, filter ) | ( operation ) |
The argument filter is used to create a morphism type for the category category, which is then used in ObjectifyMorphismForCAPWithAttributes
to objectify morphisms for this category.
‣ ObjectifyMorphismWithSourceAndRangeForCAPWithAttributes ( morphism, category, source, range[, attr1, val1, attr2, val2, ...] ) | ( function ) |
Returns: a morphism
Objectifies the morphism morphism with the type created for morphisms in the category category. The type is created by passing a representation to AddMorphismRepresentation
. Morphisms which are objectified using this method do not have to be passed to the AddMorphism
function. The arguments source
and range
are assumed to be objectified. The optional arguments behave like the corresponding arguments in ObjectifyWithAttributes
. Also returns the objectified morphism.
‣ IsCongruentForMorphisms ( alpha, beta ) | ( operation ) |
Returns: a boolean
The arguments are two morphisms \(\alpha, \beta: a \rightarrow b\). The output is true
if \(\alpha \sim_{a,b} \beta\), otherwise the output is false
.
‣ AddIsCongruentForMorphisms ( 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 IsCongruentForMorphisms
. \(F: (\alpha, \beta) \mapsto \mathtt{IsCongruentForMorphisms}(\alpha, \beta)\).
‣ IsEqualForMorphisms ( alpha, beta ) | ( operation ) |
Returns: a boolean
The arguments are two morphisms \(\alpha, \beta: a \rightarrow b\). The output is true
if \(\alpha = \beta\), otherwise the output is false
.
‣ AddIsEqualForMorphisms ( 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 IsEqualForMorphisms
. \(F: (\alpha, \beta) \mapsto \mathtt{IsEqualForMorphisms}(\alpha, \beta)\).
‣ IsEqualForMorphismsOnMor ( alpha, beta ) | ( operation ) |
Returns: a boolean
The arguments are two morphisms \(\alpha: a \rightarrow b, \beta: c \rightarrow d\). The output is true
if \(\alpha = \beta\), otherwise the output is false
.
‣ AddIsEqualForMorphismsOnMor ( 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 IsEqualForMorphismsOnMor
. \(F: (\alpha, \beta) \mapsto \mathtt{IsEqualForMorphismsOnMor}(\alpha, \beta)\).
‣ IsZeroForMorphisms ( alpha ) | ( property ) |
Returns: a boolean
The argument is a morphism \(\alpha: a \rightarrow b\). The output is true
if \(\alpha \sim_{a,b} 0\), otherwise the output is false
.
‣ AddIsZeroForMorphisms ( 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 IsZeroForMorphisms
. \(F: \alpha \mapsto \mathtt{IsZeroForMorphisms}(\alpha)\).
‣ AdditionForMorphisms ( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\)
The arguments are two morphisms \(\alpha, \beta: a \rightarrow b\). The output is the addition \(\alpha + \beta\).
‣ AddAdditionForMorphisms ( 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 AdditionForMorphisms
. \(F: (\alpha, \beta) \mapsto \alpha + \beta\).
‣ SubtractionForMorphisms ( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\)
The arguments are two morphisms \(\alpha, \beta: a \rightarrow b\). The output is the addition \(\alpha - \beta\).
‣ AddSubtractionForMorphisms ( 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 SubtractionForMorphisms
. \(F: (\alpha, \beta) \mapsto \alpha - \beta\).
‣ AdditiveInverseForMorphisms ( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\)
The argument is a morphism \(\alpha: a \rightarrow b\). The output is its additive inverse \(-\alpha\).
‣ AddAdditiveInverseForMorphisms ( 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 AdditiveInverseForMorphisms
. \(F: \alpha \mapsto -\alpha\).
‣ MultiplyWithElementOfCommutativeRingForMorphisms ( r, alpha ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\)
The arguments are an element \(r\) of a commutative ring and a morphism \(\alpha: a \rightarrow b\). The output is the multiplication with the ring element \(r \cdot \alpha\).
3.7-10 \*
‣ \* ( r, alpha ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\)
This is a convenience method. It has two arguments. The first argument is either a rational number \(q\) or an element \(r\) of a commutative ring \(R\). The second argument is a morphism \(\alpha: a \rightarrow b\) in a linear category over the commutative ring \(R\). In the case where the first element is a rational number, this method tries to interpret \(q\) as an element \(r\) of \(R\) via R!.interpret_rationals_func
. If no such interpretation exists, this method throws an error. The output is the multiplication with the ring element \(r \cdot \alpha\).
‣ AddMultiplyWithElementOfCommutativeRingForMorphisms ( 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 MultiplyWithElementOfCommutativeRingForMorphisms
. \(F: (r, \alpha) \mapsto r \cdot \alpha\).
‣ ZeroMorphism ( a, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b)\)
The arguments are two objects \(a\) and \(b\). The output is the zero morphism \(0: a \rightarrow b\).
‣ AddZeroMorphism ( 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 ZeroMorphism
. \(F: (a,b) \mapsto (0: a \rightarrow b)\).
Subobjects of an object \(c\) are monomorphisms with range \(c\) and a special function for comparision. Similarly, factorobjects of an object \(c\) are epimorphisms with source \(c\) and a special function for comparision.
‣ IsEqualAsSubobjects ( alpha, beta ) | ( operation ) |
Returns: a boolean
The arguments are two subobjects \(\alpha: a \rightarrow c\), \(\beta: b \rightarrow c\). The output is true
if there exists an isomorphism \(\iota: a \rightarrow b\) such that \(\beta \circ \iota \sim_{a,c} \alpha\), otherwise the output is false
.
‣ AddIsEqualAsSubobjects ( 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 IsEqualAsSubobjects
. \(F: (\alpha, \beta) \mapsto \mathtt{IsEqualAsSubobjects}(\alpha,\beta)\).
‣ IsEqualAsFactorobjects ( alpha, beta ) | ( operation ) |
Returns: a boolean
The arguments are two factorobjects \(\alpha: c \rightarrow a\), \(\beta: c \rightarrow b\). The output is true
if there exists an isomorphism \(\iota: b \rightarrow a\) such that \(\iota \circ \beta \sim_{c,a} \alpha\), otherwise the output is false
.
‣ AddIsEqualAsFactorobjects ( 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 IsEqualAsFactorobjects
. \(F: (\alpha, \beta) \mapsto \mathtt{IsEqualAsFactorobjects}(\alpha,\beta)\).
‣ IsDominating ( alpha, beta ) | ( operation ) |
Returns: a boolean
In short: Returns true
iff \(\alpha\) is smaller than \(\beta\). Full description: The arguments are two subobjects \(\alpha: a \rightarrow c\), \(\beta: b \rightarrow c\). The output is true
if there exists a morphism \(\iota: a \rightarrow b\) such that \(\beta \circ \iota \sim_{a,c} \alpha\), otherwise the output is false
.
‣ AddIsDominating ( 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 IsDominating
. \(F: (\alpha, \beta) \mapsto \mathtt{IsDominating}(\alpha,\beta)\).
‣ IsCodominating ( alpha, beta ) | ( operation ) |
Returns: a boolean
In short: Returns true
iff \(\alpha\) is smaller than \(\beta\). Full description: The arguments are two factorobjects \(\alpha: c \rightarrow a\), \(\beta: c \rightarrow b\). The output is true
if there exists a morphism \(\iota: b \rightarrow a\) such that \(\iota \circ \beta \sim_{c,a} \alpha\), otherwise the output is false
.
‣ AddIsCodominating ( 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 IsCodominating
. \(F: (\alpha, \beta) \mapsto \mathtt{IsCodominating}(\alpha,\beta)\).
‣ IdentityMorphism ( a ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(a,a)\)
The argument is an object \(a\). The output is its identity morphism \(\mathrm{id}_a\).
‣ AddIdentityMorphism ( 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 IdentityMorphism
. \(F: a \mapsto \mathrm{id}_a\).
‣ PreCompose ( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a, c )\)
The arguments are two morphisms \(\alpha: a \rightarrow b\), \(\beta: b \rightarrow c\). The output is the composition \(\beta \circ \alpha: a \rightarrow c\).
‣ PreCompose ( L ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a_1, a_{n+1})\)
This is a convenience method. The argument is a list of morphisms \(L = ( \alpha_1: a_1 \rightarrow a_2, \alpha_2: a_2 \rightarrow a_3, \dots, \alpha_n: a_n \rightarrow a_{n+1} )\). The output is the composition \(\alpha_{n} \circ ( \alpha_{n-1} \circ ( \dots ( \alpha_2 \circ \alpha_1 ) ) )\).
‣ AddPreCompose ( 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 PreCompose
. \(F: (\alpha, \beta) \mapsto \beta \circ \alpha\).
‣ PostCompose ( beta, alpha ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}( a, c )\)
The arguments are two morphisms \(\beta: b \rightarrow c\), \(\alpha: a \rightarrow b\). The output is the composition \(\beta \circ \alpha: a \rightarrow c\).
‣ PostCompose ( L ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a_1, a_{n+1})\)
This is a convenience method. The argument is a list of morphisms \(L = ( \alpha_n: a_n \rightarrow a_{n+1}, \alpha_{n-1}: a_{n-1} \rightarrow a_n, \dots, \alpha_1: a_1 \rightarrow a_2 )\). The output is the composition \(((\alpha_{n} \circ \alpha_{n-1}) \circ \dots \alpha_2) \circ \alpha_1\).
‣ AddPostCompose ( 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 PostCompose
. \(F: (\alpha, \beta) \mapsto \alpha \circ \beta\).
‣ IsWellDefinedForMorphisms ( alpha ) | ( operation ) |
Returns: a boolean
The argument is a morphism \(\alpha\). The output is true
if \(\alpha\) is well-defined, otherwise the output is false
.
‣ AddIsWellDefinedForMorphisms ( 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 IsWellDefinedForMorphisms
. \(F: \alpha \mapsto \mathtt{IsWellDefinedForMorphisms}( \alpha )\).
For any pair of morphisms \(\alpha: a \rightarrow c\), \(\beta: b \rightarrow c\), we call each morphism \(\alpha / \beta: a \rightarrow b\) such that \(\beta \circ (\alpha / \beta) \sim_{a,c} \alpha\) a lift of \(\alpha\) along \(\beta\).
For any pair of morphisms \(\alpha: a \rightarrow c\), \(\beta: a \rightarrow b\), we call each morphism \(\alpha \backslash \beta: c \rightarrow b\) such that \((\alpha \backslash \beta) \circ \alpha \sim_{a,b} \beta\) a colift of \(\beta\) along \(\alpha\).
Note that such lifts (or colifts) do not have to be unique. So in general, we do not expect that algorithms computing lifts (or colifts) do this in a functorial way. Thus the operations \(\mathtt{Lift}\) and \(\mathtt{Colift}\) are not regarded as categorical operations, but only as set-theoretic operations.
‣ LiftAlongMonomorphism ( iota, tau ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(t,k)\)
The arguments are a monomorphism \(\iota: k \hookrightarrow a\) and a morphism \(\tau: t \rightarrow a\) such that there is a morphism \(u: t \rightarrow k\) with \(\iota \circ u \sim_{t,a} \tau\). The output is such a \(u\).
‣ AddLiftAlongMonomorphism ( 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 LiftAlongMonomorphism
. The function \(F\) maps a pair \((\iota, \tau)\) to a lift \(u\).
‣ ColiftAlongEpimorphism ( epsilon, tau ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(c,t)\)
The arguments are an epimorphism \(\epsilon: a \rightarrow c\) and a morphism \(\tau: a \rightarrow t\) such that there is a morphism \(u: c \rightarrow t\) with \(u \circ \epsilon \sim_{a,t} \tau\). The output is such a \(u\).
‣ AddColiftAlongEpimorphism ( 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 ColiftAlongEpimorphism
. The function \(F\) maps a pair \((\epsilon, \tau)\) to a lift \(u\).
‣ IsLiftableAlongMonomorphism ( iota, tau ) | ( operation ) |
Returns: a boolean
The arguments are a monomorphism \(\iota: k \hookrightarrow a\) and a morphism \(\tau: t \rightarrow a\). The output is true
if there exists a morphism \(u: t \rightarrow k\) with \(\iota \circ u \sim_{t,a} \tau\). Otherwise, the output is false
.
‣ AddIsLiftableAlongMonomorphism ( 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 IsLiftableAlongMonomorphism
. \(F: (\iota, \tau) \mapsto \texttt{IsLiftableAlongMonomorphism}( \iota, \tau )\).
‣ IsColiftableAlongEpimorphism ( epsilon, tau ) | ( operation ) |
Returns: a boolean
The arguments are an epimorphism \(\epsilon: a \rightarrow c\) and a morphism \(\tau: a \rightarrow t\). The output is true
if there exists a morphism \(u: c \rightarrow t\) with \(u \circ \epsilon \sim_{a,t} \tau\). Otherwise, the output is false
.
‣ AddIsColiftableAlongEpimorphism ( 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 IsColiftableAlongEpimorphism
. \(F: (\epsilon, \tau) \mapsto \texttt{IsColiftableAlongEpimorphism}( \epsilon, \tau )\).
‣ Lift ( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(a,b) + \{ \mathtt{fail} \}\)
The arguments are two morphisms \(\alpha: a \rightarrow c\), \(\beta: b \rightarrow c\). The output is a lift \(\alpha / \beta: a \rightarrow b\) of \(\alpha\) along \(\beta\) if such a lift exists or \(\mathtt{fail}\) if it doesn't. Recall that a lift \(\alpha / \beta: a \rightarrow b\) of \(\alpha\) along \(\beta\) is a morphism such that \(\beta \circ (\alpha / \beta) \sim_{a,c} \alpha\).
‣ AddLift ( 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 Lift
. The function \(F\) maps a pair \((\alpha, \beta)\) to a lift \(\alpha / \beta\) if it exists, and to fail
otherwise.
‣ Colift ( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(c,b) + \{ \mathtt{fail} \}\)
The arguments are two morphisms \(\alpha: a \rightarrow c\), \(\beta: a \rightarrow b\). The output is a colift \(\alpha \backslash \beta: c \rightarrow b\) of \(\beta\) along \(\alpha\) if such a colift exists or \(\mathtt{fail}\) if it doesn't. Recall that a colift \(\alpha \backslash \beta: c \rightarrow b\) of \(\beta\) along \(\alpha\) is a morphism such that \((\alpha \backslash \beta) \circ \alpha \sim_{a,b} \beta\).
‣ AddColift ( 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 Colift
. The function \(F\) maps a pair \((\alpha, \beta)\) to a colift \(\alpha \backslash \beta\) if it exists, and to fail
otherwise.
‣ IsLiftable ( alpha, beta ) | ( operation ) |
Returns: a boolean
The arguments are two morphisms \(\alpha: a \rightarrow c\), \(\beta: b \rightarrow c\). The output is true
if there exists a lift \(\alpha / \beta: a \rightarrow b\) of \(\alpha\) along \(\beta\), i.e., a morphism such that \(\beta \circ (\alpha / \beta) \sim_{a,c} \alpha\). Otherwise, the output is false
.
‣ AddIsLiftable ( 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 IsLiftable
. \(F: ( \alpha, \beta ) \mapsto \mathtt{IsLiftable}( \alpha, \beta )\).
‣ IsColiftable ( alpha, beta ) | ( operation ) |
Returns: a boolean
The arguments are two morphisms \(\alpha: a \rightarrow c\), \(\beta: a \rightarrow b\). The output is true
if there exists a colift \(\alpha \backslash \beta: c \rightarrow b\) of \(\beta\) along \(\alpha\)., i.e., a morphism such that \((\alpha \backslash \beta) \circ \alpha \sim_{a,b} \beta\). Otherwise, the output is false
.
‣ AddIsColiftable ( 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 IsColiftable
. \(F: ( \alpha, \beta ) \mapsto \mathtt{IsColiftable}( \alpha, \beta )\).
Let \(\alpha: a \rightarrow b\) be a morphism. An inverse of \(\alpha\) is a morphism \(\alpha^{-1}: b \rightarrow a\) such that \(\alpha \circ \alpha^{-1} \sim_{b,b} \mathrm{id}_b\) and \(\alpha^{-1} \circ \alpha \sim_{a,a} \mathrm{id}_a\).
‣ Inverse ( alpha ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(b,a)\)
The argument is an isomorphism \(\alpha: a \rightarrow b\). The output is its inverse \(\alpha^{-1}: b \rightarrow a\).
‣ AddInverse ( 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 Inverse
. \(F: \alpha \mapsto \alpha^{-1}\).
‣ IsEqualForCacheForMorphisms ( phi, psi ) | ( operation ) |
Returns: true or false
Compares two objects in the cache
‣ AddIsEqualForCacheForMorphisms ( c, F ) | ( operation ) |
Returns: northing
By default, CAP uses caches to store the values of Categorical operations. To get a value out of the cache, one needs to compare the input of a basic operation with its previous input. To compare morphisms in the category, IsEqualForCacheForMorphisms is used. By default, IsEqualForCacheForMorphisms falls back to IsEqualForCache (see ToolsForHomalg), which in turn defaults to recursive comparison for lists and IsIdenticalObj
in all other cases. If you add a function, this function used instead. A function \(F: a,b \mapsto bool\) is expected here. The output has to be true or false. Fail is not allowed in this context.
‣ IsHomSetInhabited ( A, B ) | ( operation ) |
Returns: a boolean
The arguments are two objects A and B. The output is true
if there exists a morphism from A to B, otherwise the output is false
.
‣ AddIsHomSetInhabited ( 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 IsHomSetInhabited
. \(F: A, B \mapsto \mathrm{IsHomSetInhabited}(A, B)\).
Homomorphism structures are way to "oversee" the homomorphisms between two given objects. Let \(C\), \(D\) be categories. A \(D\)-homomorphism structure for \(C\) consists of the following data:
a functor \(H: C^{\mathrm{op}} \times C \rightarrow D\) (when \(C\) and \(D\) are Ab-categories, \(H\) is assumed to be bilinear).
an object \(1 \in D\), called the distinguished object,
a bijection \(\nu: \mathrm{Hom}_{C}(a,b) \simeq \mathrm{Hom}_{D}(1, H(a,b))\) natural in \(a,b \in C\).
‣ HomomorphismStructureOnObjects ( a, b ) | ( operation ) |
Returns: an object in \(D\)
The arguments are two objects \(a, b\) in \(C\). The output is the value of the homomorphism structure on objects \(H(a,b)\).
‣ AddHomomorphismStructureOnObjects ( 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 HomomorphismStructureOnObjects
. \(F: (a,b) \mapsto H(a,b)\).
‣ HomomorphismStructureOnMorphisms ( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}_{D}(H(a',b), H(a,b'))\)
The arguments are two morphisms \(\alpha: a \rightarrow a', \beta: b \rightarrow b'\) in \(C\). The output is the value of the homomorphism structure on morphisms \(H(\alpha, \beta )\).
‣ HomomorphismStructureOnMorphismsWithGivenObjects ( s, alpha, beta, r ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}_{D}(H(a',b), H(a,b'))\)
The arguments are an object \(s = H(a',b)\) in \(D\), two morphisms \(\alpha: a \rightarrow a', \beta: b \rightarrow b'\) in \(C\), and an object \(r = H(a,b')\) in \(D\). The output is the value of the homomorphism structure on morphisms \(H(\alpha, \beta )\).
‣ AddHomomorphismStructureOnMorphismsWithGivenObjects ( 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 HomomorphismStructureOnMorphismsWithGivenObjects
. \(F: ( s, \alpha: a \rightarrow a', \beta: b \rightarrow b', r ) \mapsto H( \alpha, \beta )\).
‣ DistinguishedObjectOfHomomorphismStructure ( C ) | ( attribute ) |
Returns: an object in \(D\)
The argument is a category \(C\). The output is the distinguished object \(1\) in \(D\) of the homomorphism structure.
‣ AddDistinguishedObjectOfHomomorphismStructure ( 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 DistinguishedObjectOfHomomorphismStructure
. \(F: ( ) \mapsto 1\).
‣ InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure ( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}_{D}(1, H(a,a'))\)
The argument is a morphism \(\alpha: a \rightarrow a'\) in \(C\). The output is the corresponding morphism \(\nu( \alpha ): 1 \rightarrow H(a,a')\) in \(D\) of the homomorphism structure.
‣ AddInterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure ( 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 InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure
. \(F: (\alpha: a \rightarrow a') \mapsto (\nu(\alpha):1 \rightarrow H(a,a'))\).
‣ InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism ( a, a', iota ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}_{C}(a,a')\)
The arguments are objects \(a,a'\) in \(C\) and a morphism \(\iota: 1 \rightarrow H(a,a')\) in \(D\). The output is the corresponding morphism \(\nu^{-1}(\iota): a \rightarrow a'\) in \(C\) of the homomorphism structure.
‣ AddInterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism ( 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 InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism
. \(F: (a,a',\iota: 1 \rightarrow H(a,a')) \mapsto (\nu^{-1}(\iota): a \rightarrow a')\).
‣ SolveLinearSystemInAbCategory ( alpha, beta, gamma ) | ( operation ) |
Returns: a list of morphisms \([X_1, \dots, X_n]\)
The arguments are three lists \(\alpha\), \(\beta\), and \(\gamma\). The first list \(\alpha\) (the left coefficients) is a list of list of morphisms \(\alpha_{ij}: A_i \rightarrow B_j\), where \(i = 1 \dots m\) and \(j = 1 \dots n\) for integers \(m,n \geq 1\). The second list \(\beta\) (the right coefficients) is a list of list of morphisms \(\beta_{ij}: C_j \rightarrow D_i\), where \(i = 1 \dots m\) and \(j = 1 \dots n\). The third list \(\gamma\) (the right side) is a list of morphisms \(\gamma_i: A_i \rightarrow D_i\), where \(i = 1, \dots, m\). The output is either a list of morphisms \(X_j: B_j \rightarrow C_j\) for \(j=1\dots n\) solving the linear system defined by \(\alpha\), \(\beta\), \(\gamma\), i.e., \(\sum_{j = 1}^n \alpha_{ij}\cdot X_j \cdot \beta_{ij} = \gamma_i\) for all \(i = 1 \dots m\), or \(\texttt{fail}\) if no such solution exists.
‣ MereExistenceOfSolutionOfLinearSystemInAbCategory ( alpha, beta, gamma ) | ( operation ) |
Returns: a boolean
Like SolveLinearSystemInAbCategory
, but the output is simply true
if a solution exists, false
otherwise.
‣ HomStructure ( alpha, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}_{D}(H(a',b), H(a,b'))\)
This is a convenience method. The arguments are two morphisms \(\alpha: a \rightarrow a', \beta: b \rightarrow b'\) in \(C\). The output is HomomorphismStructureOnMorphisms
called on \(\alpha\), \(\beta\).
‣ HomStructure ( alpha, b ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}_{D}(H(a',b), H(a,b))\)
This is a convenience method. The arguments are a morphism \(\alpha: a \rightarrow a'\) and an object \(b\) in \(C\). The output is HomomorphismStructureOnMorphisms
called on \(\alpha\), \(\mathrm{id}_b\).
‣ HomStructure ( a, beta ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}_{D}(H(a,b), H(a,b'))\)
This is a convenience method. The arguments are an object \(a\) and a morphism \(\beta: b \rightarrow b'\) in \(C\). The output is HomomorphismStructureOnMorphisms
called on \(\mathrm{id}_a\), \(\beta\).
‣ HomStructure ( a, b ) | ( operation ) |
Returns: an object
This is a convenience method. The arguments are two objects \(a\) and \(b\) in \(C\). The output is HomomorphismStructureOnObjects
called on \(a,b\).
‣ HomStructure ( arg ) | ( operation ) |
This is a convenience method for InterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure
.
‣ HomStructure ( arg1, arg2, arg3 ) | ( operation ) |
This is a convenience method for InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism
.
‣ HomStructure ( arg ) | ( operation ) |
This is a convenience method for DistinguishedObjectOfHomomorphismStructure
.
‣ BasisOfExternalHom ( a, b ) | ( operation ) |
Returns: a list of morphisms in \(\mathrm{Hom}_{C}(a,b)\)
The arguments are objects \(a,b\) in a \(k\)-linear category \(C\). The output is a list \(L\) of morphisms which is a basis of \(\mathrm{Hom}_{C}(a,b)\) in the sense that any given morphism \(\alpha: a \to b\) can uniquely be written as a linear combination of \(L\) with the coefficients in CoefficientsOfMorphismWithGivenBasisOfExternalHom
(\(\alpha,L\)).
‣ AddBasisOfExternalHom ( 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 BasisOfExternalHom
.
‣ CoefficientsOfMorphismWithGivenBasisOfExternalHom ( alpha, L ) | ( operation ) |
Returns: a list of elements in \(k\)
The arguments are a morphism \(\alpha: a \to b\) in a \(k\)-linear category \(C\) and a list L=BasisOfExternalHom
(\(a,b\)). The output is a list of coefficients of \(\alpha\) with respect to \(L\).
‣ AddCoefficientsOfMorphismWithGivenBasisOfExternalHom ( 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 CoefficientsOfMorphismWithGivenBasisOfExternalHom
.
‣ CoefficientsOfMorphism ( alpha ) | ( attribute ) |
Returns: a list of elements in \(k\)
This is a convenience method. The argument is a morphism \(\alpha: a \to b\) in a \(k\)-linear category \(C\). The output is a list of coefficients of \(\alpha\) with respect to the list BasisOfExternalHom
(a,b).
Let \(\phi: A \rightarrow B\) be a morphism. There are several different natural ways to look at \(\phi\) as an object in an ambient category:
\(\mathrm{Hom}( A, B )\), the set of homomorphisms with the equivalence relation \(\mathtt{IsCongruentForMorphisms}\) regarded as a category,
\(\sum_{A}\mathrm{Hom}( A, B )\), the category of morphisms where the range is fixed,
\(\sum_{B}\mathrm{Hom}( A, B )\), the category of morphisms where the source is fixed,
\(\sum_{A,B}\mathrm{Hom}( A, B )\), the category of morphisms where neither source nor range is fixed,
and furthermore, if \(\phi\) happens to be an endomorphism \(A \rightarrow A\), we also have
\(\sum_{A}\mathrm{Hom}(A,A)\), the category of endomorphisms.
Let \(\mathbf{C}\) be one of the categories above in which \(\phi\) may reside as an object, and let \(i\) be a non-negative integer or \(\infty\). CAP provides commands for passing from \(\phi\) to \(\phi_i\), where \(\phi_i\) is isomorphic to \(\phi\) in \(\mathbf{C}\), but "simpler". The idea is that the greater the \(i\), the "simpler" the \(\phi_i\) (but this could mean the harder the computation), with \(\infty\) as a possible value. The case \(i = 0\) defaults to the identity operator for all simplifications. For the Add-operatations, only the cases \(i \geq 1\) have to be given as functions.
\(\ \)
If we regard \(\phi\) as an object in the category \(\mathrm{Hom}( A, B )\), \(\phi_i\) is again in \(\mathrm{Hom}( A, B )\) such that \(\phi \sim_{A,B} \phi_i\). This case is handled by the following commands:
‣ SimplifyMorphism ( phi, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(A,B)\)
The arguments are a morphism \(\phi: A \rightarrow B\) and a non-negative integer \(i\) or infinity
. The output is a simplified morphism \(\phi_i\).
‣ AddSimplifyMorphism ( 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 SimplifyMorphism
. The function \(F\) maps \((\phi,i \geq 1)\) to \(\phi_i\).
\(\ \)
If we regard \(\phi\) as an object in the category \(\sum_{A}\mathrm{Hom}( A, B )\), then \(\phi_i\) is a morphism of type \(A_i \rightarrow B\) and there is an isomorphism \(\sigma_i: A \rightarrow A_i\) such that \(\phi_i \circ \sigma_i \sim_{A,B} \phi\). This case is handled by the following commands:
‣ SimplifySource ( phi, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(A_i,B)\)
The arguments are a morphism \(\phi: A \rightarrow B\) and a non-negative integer \(i\) or infinity
. The output is a simplified morphism with simplified source \(\phi_i: A_i \rightarrow B\).
‣ AddSimplifySource ( 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 SimplifySource
. The function \(F\) maps \((\phi,i \geq 1)\) to \(\phi_i\).
‣ SimplifySource_IsoToInputObject ( phi, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(A_i,A)\)
The arguments are a morphism \(\phi: A \rightarrow B\) and a non-negative integer \(i\) or infinity
. The output is the isomorphism \((\sigma_i)^{-1}: A_i \rightarrow A\).
‣ AddSimplifySource_IsoToInputObject ( 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 SimplifySource_IsoToInputObject
. The function \(F\) maps \((\phi,i)\) to \((\sigma_i)^{-1}\).
‣ SimplifySource_IsoFromInputObject ( phi, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(A,A_i)\)
The arguments are a morphism \(\phi: A \rightarrow B\) and a non-negative integer \(i\) or infinity
. The output is the isomorphism \(\sigma_i: A \rightarrow A_i\).
‣ AddSimplifySource_IsoFromInputObject ( 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 SimplifySource_IsoFromInputObject
. The function \(F\) maps \((\phi,i)\) to \((\sigma_i)\).
\(\ \)
If we regard \(\phi\) as an object in the category \(\sum_{B}\mathrm{Hom}( A, B )\), then \(\phi_i\) is a morphism of type \(A \rightarrow B_i\) and there is an isomorphism \(\rho_i: B \rightarrow B_i\) such that \( \rho_i^{-1} \circ \phi_i\sim_{A,B} \phi\). This case is handled by the following commands:
‣ SimplifyRange ( phi, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(A,B_i)\)
The arguments are a morphism \(\phi: A \rightarrow B\) and a non-negative integer \(i\) or infinity
. The output is a simplified morphism with simplified range \(\phi_i: A \rightarrow B_i\).
‣ AddSimplifyRange ( 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 SimplifyRange
. The function \(F\) maps \((\phi,i \geq 1)\) to \(\phi_i\).
‣ SimplifyRange_IsoToInputObject ( phi, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(B_i,B)\)
The arguments are a morphism \(\phi: A \rightarrow B\) and a non-negative integer \(i\) or infinity
. The output is the isomorphism \((\rho_i)^{-1}: B_i \rightarrow B\).
‣ AddSimplifyRange_IsoToInputObject ( 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 SimplifyRange_IsoToInputObject
. The function \(F\) maps \((\phi,i)\) to \((\rho_i)^{-1}\).
‣ SimplifyRange_IsoFromInputObject ( phi, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(B,B_i)\)
The arguments are a morphism \(\phi: A \rightarrow B\) and a non-negative integer \(i\) or infinity
. The output is the isomorphism \(\rho_i: B \rightarrow B_i\).
‣ AddSimplifyRange_IsoFromInputObject ( 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 SimplifyRange_IsoFromInputObject
. The function \(F\) maps \((\phi,i)\) to \(\rho_i\).
\(\ \)
If we regard \(\phi\) as an object in the category \(\sum_{A, B}\mathrm{Hom}( A, B )\), then \(\phi_i\) is a morphism of type \(A_i \rightarrow B_i\) and there is are isomorphisms \(\sigma_i: A \rightarrow A_i\) and \(\rho_i: B \rightarrow B_i\) such that \( \rho_i^{-1} \circ \phi_i \circ \sigma_i \sim_{A,B} \phi\). This case is handled by the following commands:
‣ SimplifySourceAndRange ( phi, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(A_i,B_i)\)
The arguments are a morphism \(\phi: A \rightarrow B\) and a non-negative integer \(i\) or infinity
. The output is a simplified morphism with simplified source and range \(\phi_i: A_i \rightarrow B_i\).
‣ AddSimplifySourceAndRange ( 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 SimplifySourceAndRange
. The function \(F\) maps \((\phi,i \geq 1)\) to \(\phi_i\).
‣ SimplifySourceAndRange_IsoToInputRange ( phi, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(B_i,B)\)
The arguments are a morphism \(\phi: A \rightarrow B\) and a non-negative integer \(i\) or infinity
. The output is the isomorphism \((\rho_i)^{-1}: B_i \rightarrow B\).
‣ AddSimplifySourceAndRange_IsoToInputRange ( 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 SimplifySourceAndRange_IsoToInputRange
. The function \(F\) maps \((\phi,i)\) to \((\rho_i)^{-1}\).
‣ SimplifySourceAndRange_IsoFromInputRange ( phi, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(B,B_i)\)
The arguments are a morphism \(\phi: A \rightarrow B\) and a non-negative integer \(i\) or infinity
. The output is the isomorphism \(\rho_i: B \rightarrow B_i\).
‣ AddSimplifySourceAndRange_IsoFromInputRange ( 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 SimplifySourceAndRange_IsoFromInputRange
. The function \(F\) maps \((\phi,i)\) to \(\rho_i\).
‣ SimplifySourceAndRange_IsoToInputSource ( phi, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(A_i,A)\)
The arguments are a morphism \(\phi: A \rightarrow B\) and a non-negative integer \(i\) or infinity
. The output is the isomorphism \((\sigma_i)^{-1}: A_i \rightarrow A\).
‣ AddSimplifySourceAndRange_IsoToInputSource ( 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 SimplifySourceAndRange_IsoToInputSource
. The function \(F\) maps \((\phi,i)\) to \((\sigma_i)^{-1}\).
‣ SimplifySourceAndRange_IsoFromInputSource ( phi, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(A,A_i)\)
The arguments are a morphism \(\phi: A \rightarrow B\) and a non-negative integer \(i\) or infinity
. The output is the isomorphism \(\sigma_i: A \rightarrow A_i\).
‣ AddSimplifySourceAndRange_IsoFromInputSource ( 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 SimplifySourceAndRange_IsoFromInputSource
. The function \(F\) maps \((\phi,i)\) to \((\sigma_i)\).
\(\ \)
If \(\phi:A \rightarrow A\) is an endomorphism, we may regard it as an object in the category \(\sum_{A}\mathrm{Hom}( A, A )\). In this case \(\phi_i\) is a morphism of type \(A_i \rightarrow A_i\) and there is an isomorphism \(\sigma_i: A \rightarrow A_i\) such that \( \sigma_i^{-1} \circ \phi_i \circ \sigma_i \sim_{A,A} \phi\). This case is handled by the following commands:
‣ SimplifyEndo ( phi, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(A_i,A_i)\)
The arguments are an endomorphism \(\phi: A \rightarrow A\) and a non-negative integer \(i\) or infinity
. The output is a simplified endomorphism \(\phi_i: A_i \rightarrow A_i\).
‣ AddSimplifyEndo ( 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 SimplifyEndo
. The function \(F\) maps \((\phi,i \geq 1)\) to \(\phi_i\).
‣ SimplifyEndo_IsoToInputObject ( phi, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(A_i,A)\)
The arguments are an endomorphism \(\phi: A \rightarrow A\) and a non-negative integer \(i\) or infinity
. The output is the isomorphism \((\sigma_i)^{-1}: A_i \rightarrow A\).
‣ AddSimplifyEndo_IsoToInputObject ( 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 SimplifyEndo_IsoToInputObject
. The function \(F\) maps \((\phi,i)\) to \((\sigma_i)^{-1}\).
‣ SimplifyEndo_IsoFromInputObject ( phi, i ) | ( operation ) |
Returns: a morphism in \(\mathrm{Hom}(A,A_i)\)
The arguments are an endomorphism \(\phi: A \rightarrow A\) and a non-negative integer \(i\) or infinity
. The output is the isomorphism \(\sigma_i: A \rightarrow A_i\).
‣ AddSimplifyEndo_IsoFromInputObject ( 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 SimplifyEndo_IsoFromInputObject
. The function \(F\) maps \((\phi,i)\) to \((\sigma_i)\).
‣ Simplify ( phi ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(A_{\infty},B_{\infty})\)
This is a convenient method. The argument is a morphism \(\phi: A \rightarrow B\). The output is a "simplified" version of \(\phi\) that may change the source and range of \(\phi\) (up to isomorphism). To be precise, the output is an \(\infty\)-th simplified morphism of \((\iota_A^{\infty})^{-1}\circ \phi \circ \iota_A^{\infty}\).
Let \(\alpha: A \rightarrow B\) be a morphism in an additive category. Suppose we are given direct sum decompositions of \(A \simeq A' \oplus A''\) and \(B \simeq B' \oplus B''\) such that If \(\alpha''\) is a split epimorphism, then we call \(\alpha': A' \rightarrow B'\) some reduction of \(\alpha\) by split epi summands. The inclusions/projections of the decompositions into direct sums induce commutative diagrams and
‣ SomeReductionBySplitEpiSummand ( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(A',B')\)
The argument is a morphism \(\alpha: A \rightarrow B\). The output is some reduction of \(\alpha\) by split epi summands \(\alpha': A' \rightarrow B'\).
‣ AddSomeReductionBySplitEpiSummand ( 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 SomeReductionBySplitEpiSummand
. The function \(F\) maps \(\alpha\) to \(\alpha'\).
‣ SomeReductionBySplitEpiSummand_MorphismToInputRange ( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(B',B)\)
The argument is a morphism \(\alpha: A \rightarrow B\). The output is the morphism \(\beta': B' \rightarrow B\) linking \(\alpha\) with some reduction by split epi summands.
‣ AddSomeReductionBySplitEpiSummand_MorphismToInputRange ( 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 SomeReductionBySplitEpiSummand_MorphismToInputRange
. The function \(F\) maps \(\alpha\) to \(\beta'\).
‣ SomeReductionBySplitEpiSummand_MorphismFromInputRange ( alpha ) | ( attribute ) |
Returns: a morphism in \(\mathrm{Hom}(B,B')\)
The argument is a morphism \(\alpha: A \rightarrow B\). The output is the morphism \(\beta: B \rightarrow B'\) linking \(\alpha\) with some reduction by split epi summands.
‣ AddSomeReductionBySplitEpiSummand_MorphismFromInputRange ( 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 SomeReductionBySplitEpiSummand_MorphismFromInputRange
. The function \(F\) maps \(\alpha\) to \(\beta\).
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