Modules¶
- class sage.categories.modules.Modules(base, name=None)[source]¶
Bases:
Category_module
The category of all modules over a base ring \(R\).
An \(R\)-module \(M\) is a left and right \(R\)-module over a commutative ring \(R\) such that:
\[r*(x*s) = (r*x)*s \qquad \forall r,s \in R \text{ and } x \in M\]INPUT:
base_ring
– a ring \(R\) or subcategory ofRings()
dispatch
– boolean (for internal use; default:True
)
When the base ring is a field, the category of vector spaces is returned instead (unless
dispatch == False
).Warning
Outside of the context of symmetric modules over a commutative ring, the specifications of this category are fuzzy and not yet set in stone (see below). The code in this category and its subcategories is therefore prone to bugs or arbitrary limitations in this case.
EXAMPLES:
sage: Modules(ZZ) Category of modules over Integer Ring sage: Modules(QQ) Category of vector spaces over Rational Field sage: Modules(Rings()) Category of modules over rings sage: Modules(FiniteFields()) Category of vector spaces over finite enumerated fields sage: Modules(Integers(9)) Category of modules over Ring of integers modulo 9 sage: Modules(Integers(9)).super_categories() [Category of bimodules over Ring of integers modulo 9 on the left and Ring of integers modulo 9 on the right] sage: Modules(ZZ).super_categories() [Category of bimodules over Integer Ring on the left and Integer Ring on the right] sage: Modules == RingModules True sage: Modules(ZZ['x']).is_abelian() # see #6081 True
>>> from sage.all import * >>> Modules(ZZ) Category of modules over Integer Ring >>> Modules(QQ) Category of vector spaces over Rational Field >>> Modules(Rings()) Category of modules over rings >>> Modules(FiniteFields()) Category of vector spaces over finite enumerated fields >>> Modules(Integers(Integer(9))) Category of modules over Ring of integers modulo 9 >>> Modules(Integers(Integer(9))).super_categories() [Category of bimodules over Ring of integers modulo 9 on the left and Ring of integers modulo 9 on the right] >>> Modules(ZZ).super_categories() [Category of bimodules over Integer Ring on the left and Integer Ring on the right] >>> Modules == RingModules True >>> Modules(ZZ['x']).is_abelian() # see #6081 True
Todo
Clarify the distinction, if any, with
BiModules(R, R)
. In particular, if \(R\) is a commutative ring (e.g. a field), some pieces of the code possibly assume that \(M\) is a symmetric `R`-`R`-bimodule:\[r*x = x*r \qquad \forall r \in R \text{ and } x \in M\]Make sure that non symmetric modules are properly supported by all the code, and advertise it.
Make sure that non commutative rings are properly supported by all the code, and advertise it.
Add support for base semirings.
Implement a
FreeModules(R)
category, when so prompted by a concrete use case: e.g. modeling a free module with several bases (usingSets.SubcategoryMethods.Realizations()
) or with an atlas of local maps (see e.g. Issue #15916).
- class CartesianProducts(category, *args)[source]¶
Bases:
CartesianProductsCategory
The category of modules constructed as Cartesian products of modules.
This construction gives the direct product of modules. The implementation is based on the following resources:
- extra_super_categories()[source]¶
A Cartesian product of modules is endowed with a natural module structure.
EXAMPLES:
sage: Modules(ZZ).CartesianProducts().extra_super_categories() [Category of modules over Integer Ring] sage: Modules(ZZ).CartesianProducts().super_categories() [Category of Cartesian products of commutative additive groups, Category of modules over Integer Ring]
>>> from sage.all import * >>> Modules(ZZ).CartesianProducts().extra_super_categories() [Category of modules over Integer Ring] >>> Modules(ZZ).CartesianProducts().super_categories() [Category of Cartesian products of commutative additive groups, Category of modules over Integer Ring]
- Filtered[source]¶
alias of
FilteredModules
- class FiniteDimensional(base_category)[source]¶
Bases:
CategoryWithAxiom_over_base_ring
- class TensorProducts(category, *args)[source]¶
Bases:
TensorProductsCategory
- extra_super_categories()[source]¶
Implement the fact that a (finite) tensor product of finite dimensional modules is a finite dimensional module.
EXAMPLES:
sage: Modules(ZZ).FiniteDimensional().TensorProducts().extra_super_categories() [Category of finite dimensional modules over Integer Ring] sage: Modules(QQ).FiniteDimensional().TensorProducts().FiniteDimensional() Category of tensor products of finite dimensional vector spaces over Rational Field
>>> from sage.all import * >>> Modules(ZZ).FiniteDimensional().TensorProducts().extra_super_categories() [Category of finite dimensional modules over Integer Ring] >>> Modules(QQ).FiniteDimensional().TensorProducts().FiniteDimensional() Category of tensor products of finite dimensional vector spaces over Rational Field
- extra_super_categories()[source]¶
Implement the fact that a finite dimensional module over a finite ring is finite.
EXAMPLES:
sage: Modules(IntegerModRing(4)).FiniteDimensional().extra_super_categories() [Category of finite sets] sage: Modules(ZZ).FiniteDimensional().extra_super_categories() [] sage: Modules(GF(5)).FiniteDimensional().is_subcategory(Sets().Finite()) True sage: Modules(ZZ).FiniteDimensional().is_subcategory(Sets().Finite()) False sage: Modules(Rings().Finite()).FiniteDimensional().is_subcategory(Sets().Finite()) True sage: Modules(Rings()).FiniteDimensional().is_subcategory(Sets().Finite()) False
>>> from sage.all import * >>> Modules(IntegerModRing(Integer(4))).FiniteDimensional().extra_super_categories() [Category of finite sets] >>> Modules(ZZ).FiniteDimensional().extra_super_categories() [] >>> Modules(GF(Integer(5))).FiniteDimensional().is_subcategory(Sets().Finite()) True >>> Modules(ZZ).FiniteDimensional().is_subcategory(Sets().Finite()) False >>> Modules(Rings().Finite()).FiniteDimensional().is_subcategory(Sets().Finite()) True >>> Modules(Rings()).FiniteDimensional().is_subcategory(Sets().Finite()) False
- class FinitelyPresented(base_category)[source]¶
Bases:
CategoryWithAxiom_over_base_ring
- extra_super_categories()[source]¶
Implement the fact that a finitely presented module over a finite ring is finite.
EXAMPLES:
sage: Modules(IntegerModRing(4)).FiniteDimensional().extra_super_categories() [Category of finite sets] sage: Modules(ZZ).FiniteDimensional().extra_super_categories() [] sage: Modules(GF(5)).FiniteDimensional().is_subcategory(Sets().Finite()) True sage: Modules(ZZ).FiniteDimensional().is_subcategory(Sets().Finite()) False sage: Modules(Rings().Finite()).FiniteDimensional().is_subcategory(Sets().Finite()) True sage: Modules(Rings()).FiniteDimensional().is_subcategory(Sets().Finite()) False
>>> from sage.all import * >>> Modules(IntegerModRing(Integer(4))).FiniteDimensional().extra_super_categories() [Category of finite sets] >>> Modules(ZZ).FiniteDimensional().extra_super_categories() [] >>> Modules(GF(Integer(5))).FiniteDimensional().is_subcategory(Sets().Finite()) True >>> Modules(ZZ).FiniteDimensional().is_subcategory(Sets().Finite()) False >>> Modules(Rings().Finite()).FiniteDimensional().is_subcategory(Sets().Finite()) True >>> Modules(Rings()).FiniteDimensional().is_subcategory(Sets().Finite()) False
- Graded[source]¶
alias of
GradedModules
- class Homsets(category, *args)[source]¶
Bases:
HomsetsCategory
The category of homomorphism sets \(\hom(X,Y)\) for \(X\), \(Y\) modules.
- class Endset(base_category)[source]¶
Bases:
CategoryWithAxiom_over_base_ring
The category of endomorphism sets \(End(X)\) for \(X\) a module (this is not used yet)
- extra_super_categories()[source]¶
Implement the fact that the endomorphism set of a module is an algebra.
See also
CategoryWithAxiom.extra_super_categories()
EXAMPLES:
sage: Modules(ZZ).Endsets().extra_super_categories() [Category of magmatic algebras over Integer Ring] sage: End(ZZ^3) in Algebras(ZZ) # needs sage.modules True
>>> from sage.all import * >>> Modules(ZZ).Endsets().extra_super_categories() [Category of magmatic algebras over Integer Ring] >>> End(ZZ**Integer(3)) in Algebras(ZZ) # needs sage.modules True
- class ParentMethods[source]¶
Bases:
object
- base_ring()[source]¶
Return the base ring of
self
.EXAMPLES:
sage: # needs sage.modules sage: E = CombinatorialFreeModule(ZZ, [1,2,3]) sage: F = CombinatorialFreeModule(ZZ, [2,3,4]) sage: H = Hom(E, F) sage: H.base_ring() Integer Ring
>>> from sage.all import * >>> # needs sage.modules >>> E = CombinatorialFreeModule(ZZ, [Integer(1),Integer(2),Integer(3)]) >>> F = CombinatorialFreeModule(ZZ, [Integer(2),Integer(3),Integer(4)]) >>> H = Hom(E, F) >>> H.base_ring() Integer Ring
This
base_ring
method is actually overridden bysage.structure.category_object.CategoryObject.base_ring()
:sage: H.base_ring.__module__ # needs sage.modules
>>> from sage.all import * >>> H.base_ring.__module__ # needs sage.modules
Here we call it directly:
sage: method = H.category().parent_class.base_ring # needs sage.modules sage: method.__get__(H)() # needs sage.modules Integer Ring
>>> from sage.all import * >>> method = H.category().parent_class.base_ring # needs sage.modules >>> method.__get__(H)() # needs sage.modules Integer Ring
- zero()[source]¶
EXAMPLES:
sage: # needs sage.modules sage: E = CombinatorialFreeModule(ZZ, [1,2,3]) sage: F = CombinatorialFreeModule(ZZ, [2,3,4]) sage: H = Hom(E, F) sage: f = H.zero() sage: f Generic morphism: From: Free module generated by {1, 2, 3} over Integer Ring To: Free module generated by {2, 3, 4} over Integer Ring sage: f(E.monomial(2)) 0 sage: f(E.monomial(3)) == F.zero() True
>>> from sage.all import * >>> # needs sage.modules >>> E = CombinatorialFreeModule(ZZ, [Integer(1),Integer(2),Integer(3)]) >>> F = CombinatorialFreeModule(ZZ, [Integer(2),Integer(3),Integer(4)]) >>> H = Hom(E, F) >>> f = H.zero() >>> f Generic morphism: From: Free module generated by {1, 2, 3} over Integer Ring To: Free module generated by {2, 3, 4} over Integer Ring >>> f(E.monomial(Integer(2))) 0 >>> f(E.monomial(Integer(3))) == F.zero() True
- class ParentMethods[source]¶
Bases:
object
- linear_combination(iter_of_elements_coeff, factor_on_left=True)[source]¶
Return the linear combination \(\lambda_1 v_1 + \cdots + \lambda_k v_k\) (resp. the linear combination \(v_1 \lambda_1 + \cdots + v_k \lambda_k\)) where
iter_of_elements_coeff
iterates through the sequence \(((\lambda_1, v_1), ..., (\lambda_k, v_k))\).INPUT:
iter_of_elements_coeff
– iterator of pairs(element, coeff)
withelement
inself
andcoeff
inself.base_ring()
factor_on_left
– (optional) ifTrue
, the coefficients are multiplied from the left; ifFalse
, the coefficients are multiplied from the right
EXAMPLES:
sage: m = matrix([[0,1], [1,1]]) # needs sage.modules sage: J.<a,b,c> = JordanAlgebra(m) # needs sage.combinat sage.modules sage: J.linear_combination(((a+b, 1), (-2*b + c, -1))) # needs sage.combinat sage.modules 1 + (3, -1)
>>> from sage.all import * >>> m = matrix([[Integer(0),Integer(1)], [Integer(1),Integer(1)]]) # needs sage.modules >>> J = JordanAlgebra(m, names=('a', 'b', 'c',)); (a, b, c,) = J._first_ngens(3)# needs sage.combinat sage.modules >>> J.linear_combination(((a+b, Integer(1)), (-Integer(2)*b + c, -Integer(1)))) # needs sage.combinat sage.modules 1 + (3, -1)
- module_morphism(function, category, codomain, **keywords)[source]¶
Construct a module morphism from
self
tocodomain
.Let
self
be a module \(X\) over a ring \(R\). This constructs a morphism \(f: X \to Y\).INPUT:
self
– a parent \(X\) inModules(R)
function
– a function \(f\) from \(X\) to \(Y\)codomain
– the codomain \(Y\) of the morphism (default:f.codomain()
if it’s defined; otherwise it must be specified)category
– a category orNone
(default:None
)
EXAMPLES:
sage: # needs sage.modules sage: V = FiniteRankFreeModule(QQ, 2) sage: e = V.basis('e'); e Basis (e_0,e_1) on the 2-dimensional vector space over the Rational Field sage: neg = V.module_morphism(function=operator.neg, codomain=V); neg Generic endomorphism of 2-dimensional vector space over the Rational Field sage: neg(e[0]) Element -e_0 of the 2-dimensional vector space over the Rational Field
>>> from sage.all import * >>> # needs sage.modules >>> V = FiniteRankFreeModule(QQ, Integer(2)) >>> e = V.basis('e'); e Basis (e_0,e_1) on the 2-dimensional vector space over the Rational Field >>> neg = V.module_morphism(function=operator.neg, codomain=V); neg Generic endomorphism of 2-dimensional vector space over the Rational Field >>> neg(e[Integer(0)]) Element -e_0 of the 2-dimensional vector space over the Rational Field
- quotient(submodule, check=True, **kwds)[source]¶
Construct the quotient module
self
/submodule
.INPUT:
submodule
– a submodule with basis ofself
, or something that can be turned into one viaself.submodule(submodule)
check
, other keyword arguments – passed on toquotient_module()
.
This method just delegates to
quotient_module()
. Classes implementing modules should override that method.Parents in categories with additional structure may override
quotient()
. For example, in algebras,quotient()
will be the same asquotient_ring()
.EXAMPLES:
sage: C = CombinatorialFreeModule(QQ, ['a','b','c']) # needs sage.modules sage: TA = TensorAlgebra(C) # needs sage.combinat sage.modules sage: TA.quotient # needs sage.combinat sage.modules <bound method Rings.ParentMethods.quotient of Tensor Algebra of Free module generated by {'a', 'b', 'c'} over Rational Field>
>>> from sage.all import * >>> C = CombinatorialFreeModule(QQ, ['a','b','c']) # needs sage.modules >>> TA = TensorAlgebra(C) # needs sage.combinat sage.modules >>> TA.quotient # needs sage.combinat sage.modules <bound method Rings.ParentMethods.quotient of Tensor Algebra of Free module generated by {'a', 'b', 'c'} over Rational Field>
- tensor_square()[source]¶
Return the tensor square of
self
.EXAMPLES:
sage: A = HopfAlgebrasWithBasis(QQ).example() # needs sage.groups sage.modules sage: A.tensor_square() # needs sage.groups sage.modules An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field # An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
>>> from sage.all import * >>> A = HopfAlgebrasWithBasis(QQ).example() # needs sage.groups sage.modules >>> A.tensor_square() # needs sage.groups sage.modules An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field # An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
- class SubcategoryMethods[source]¶
Bases:
object
- DualObjects()[source]¶
Return the category of spaces constructed as duals of spaces of
self
.The dual of a vector space \(V\) is the space consisting of all linear functionals on \(V\) (see Wikipedia article Dual_space). Additional structure on \(V\) can endow its dual with additional structure; for example, if \(V\) is a finite dimensional algebra, then its dual is a coalgebra.
This returns the category of spaces constructed as dual of spaces in
self
, endowed with the appropriate additional structure.Warning
This semantic of
dual
andDualObject
is imposed on all subcategories, in particular to makedual
a covariant functorial construction.A subcategory that defines a different notion of dual needs to use a different name.
Typically, the category of graded modules should define a separate
graded_dual
construction (see Issue #15647). For now the two constructions are not distinguished which is an oversimplified model.
EXAMPLES:
sage: VectorSpaces(QQ).DualObjects() Category of duals of vector spaces over Rational Field
>>> from sage.all import * >>> VectorSpaces(QQ).DualObjects() Category of duals of vector spaces over Rational Field
The dual of a vector space is a vector space:
sage: VectorSpaces(QQ).DualObjects().super_categories() [Category of vector spaces over Rational Field]
>>> from sage.all import * >>> VectorSpaces(QQ).DualObjects().super_categories() [Category of vector spaces over Rational Field]
The dual of an algebra is a coalgebra:
sage: sorted(Algebras(QQ).DualObjects().super_categories(), key=str) [Category of coalgebras over Rational Field, Category of duals of vector spaces over Rational Field]
>>> from sage.all import * >>> sorted(Algebras(QQ).DualObjects().super_categories(), key=str) [Category of coalgebras over Rational Field, Category of duals of vector spaces over Rational Field]
The dual of a coalgebra is an algebra:
sage: sorted(Coalgebras(QQ).DualObjects().super_categories(), key=str) [Category of algebras over Rational Field, Category of duals of vector spaces over Rational Field]
>>> from sage.all import * >>> sorted(Coalgebras(QQ).DualObjects().super_categories(), key=str) [Category of algebras over Rational Field, Category of duals of vector spaces over Rational Field]
As a shorthand, this category can be accessed with the
dual()
method:sage: VectorSpaces(QQ).dual() Category of duals of vector spaces over Rational Field
>>> from sage.all import * >>> VectorSpaces(QQ).dual() Category of duals of vector spaces over Rational Field
- Filtered(base_ring=None)[source]¶
Return the subcategory of the filtered objects of
self
.INPUT:
base_ring
– this is ignored
EXAMPLES:
sage: Modules(ZZ).Filtered() Category of filtered modules over Integer Ring sage: Coalgebras(QQ).Filtered() Category of filtered coalgebras over Rational Field sage: AlgebrasWithBasis(QQ).Filtered() Category of filtered algebras with basis over Rational Field
>>> from sage.all import * >>> Modules(ZZ).Filtered() Category of filtered modules over Integer Ring >>> Coalgebras(QQ).Filtered() Category of filtered coalgebras over Rational Field >>> AlgebrasWithBasis(QQ).Filtered() Category of filtered algebras with basis over Rational Field
Todo
Explain why this does not commute with
WithBasis()
Improve the support for covariant functorial constructions categories over a base ring so as to get rid of the
base_ring
argument.
- FiniteDimensional()[source]¶
Return the full subcategory of the finite dimensional objects of
self
.EXAMPLES:
sage: Modules(ZZ).FiniteDimensional() Category of finite dimensional modules over Integer Ring sage: Coalgebras(QQ).FiniteDimensional() Category of finite dimensional coalgebras over Rational Field sage: AlgebrasWithBasis(QQ).FiniteDimensional() Category of finite dimensional algebras with basis over Rational Field
>>> from sage.all import * >>> Modules(ZZ).FiniteDimensional() Category of finite dimensional modules over Integer Ring >>> Coalgebras(QQ).FiniteDimensional() Category of finite dimensional coalgebras over Rational Field >>> AlgebrasWithBasis(QQ).FiniteDimensional() Category of finite dimensional algebras with basis over Rational Field
- FinitelyPresented()[source]¶
Return the full subcategory of the finitely presented objects of
self
.EXAMPLES:
sage: Modules(ZZ).FinitelyPresented() Category of finitely presented modules over Integer Ring sage: A = SteenrodAlgebra(2) # needs sage.combinat sage.modules sage: from sage.modules.fp_graded.module import FPModule # needs sage.combinat sage.modules sage: FPModule(A, [0, 1], [[Sq(2), Sq(1)]]).category() # needs sage.combinat sage.modules Category of finitely presented graded modules over mod 2 Steenrod algebra, milnor basis
>>> from sage.all import * >>> Modules(ZZ).FinitelyPresented() Category of finitely presented modules over Integer Ring >>> A = SteenrodAlgebra(Integer(2)) # needs sage.combinat sage.modules >>> from sage.modules.fp_graded.module import FPModule # needs sage.combinat sage.modules >>> FPModule(A, [Integer(0), Integer(1)], [[Sq(Integer(2)), Sq(Integer(1))]]).category() # needs sage.combinat sage.modules Category of finitely presented graded modules over mod 2 Steenrod algebra, milnor basis
- Graded(base_ring=None)[source]¶
Return the subcategory of the graded objects of
self
.INPUT:
base_ring
– this is ignored
EXAMPLES:
sage: Modules(ZZ).Graded() Category of graded modules over Integer Ring sage: Coalgebras(QQ).Graded() Category of graded coalgebras over Rational Field sage: AlgebrasWithBasis(QQ).Graded() Category of graded algebras with basis over Rational Field
>>> from sage.all import * >>> Modules(ZZ).Graded() Category of graded modules over Integer Ring >>> Coalgebras(QQ).Graded() Category of graded coalgebras over Rational Field >>> AlgebrasWithBasis(QQ).Graded() Category of graded algebras with basis over Rational Field
Todo
Explain why this does not commute with
WithBasis()
Improve the support for covariant functorial constructions categories over a base ring so as to get rid of the
base_ring
argument.
- Super(base_ring=None)[source]¶
Return the super-analogue category of
self
.INPUT:
base_ring
– this is ignored
EXAMPLES:
sage: Modules(ZZ).Super() Category of super modules over Integer Ring sage: Coalgebras(QQ).Super() Category of super coalgebras over Rational Field sage: AlgebrasWithBasis(QQ).Super() Category of super algebras with basis over Rational Field
>>> from sage.all import * >>> Modules(ZZ).Super() Category of super modules over Integer Ring >>> Coalgebras(QQ).Super() Category of super coalgebras over Rational Field >>> AlgebrasWithBasis(QQ).Super() Category of super algebras with basis over Rational Field
Todo
Explain why this does not commute with
WithBasis()
Improve the support for covariant functorial constructions categories over a base ring so as to get rid of the
base_ring
argument.
- TensorProducts()[source]¶
Return the full subcategory of objects of
self
constructed as tensor products.See also
RegressiveCovariantFunctorialConstruction
.
EXAMPLES:
sage: ModulesWithBasis(QQ).TensorProducts() Category of tensor products of vector spaces with basis over Rational Field
>>> from sage.all import * >>> ModulesWithBasis(QQ).TensorProducts() Category of tensor products of vector spaces with basis over Rational Field
- WithBasis()[source]¶
Return the full subcategory of the objects of
self
with a distinguished basis.EXAMPLES:
sage: Modules(ZZ).WithBasis() Category of modules with basis over Integer Ring sage: Coalgebras(QQ).WithBasis() Category of coalgebras with basis over Rational Field sage: AlgebrasWithBasis(QQ).WithBasis() Category of algebras with basis over Rational Field
>>> from sage.all import * >>> Modules(ZZ).WithBasis() Category of modules with basis over Integer Ring >>> Coalgebras(QQ).WithBasis() Category of coalgebras with basis over Rational Field >>> AlgebrasWithBasis(QQ).WithBasis() Category of algebras with basis over Rational Field
- base_ring()[source]¶
Return the base ring (category) for
self
.This implements a
base_ring
method for all subcategories ofModules(K)
.EXAMPLES:
sage: C = Modules(QQ) & Semigroups(); C Join of Category of semigroups and Category of vector spaces over Rational Field sage: C.base_ring() Rational Field sage: C.base_ring.__module__ 'sage.categories.modules' sage: C2 = Modules(Rings()) & Semigroups(); C2 Join of Category of semigroups and Category of modules over rings sage: C2.base_ring() Category of rings sage: C2.base_ring.__module__ 'sage.categories.modules' sage: # needs sage.combinat sage.groups sage.modules sage: C3 = DescentAlgebra(QQ,3).B().category() sage: C3.base_ring.__module__ 'sage.categories.modules' sage: C3.base_ring() Rational Field sage: # needs sage.combinat sage.modules sage: C4 = QuasiSymmetricFunctions(QQ).F().category() sage: C4.base_ring.__module__ 'sage.categories.modules' sage: C4.base_ring() Rational Field
>>> from sage.all import * >>> C = Modules(QQ) & Semigroups(); C Join of Category of semigroups and Category of vector spaces over Rational Field >>> C.base_ring() Rational Field >>> C.base_ring.__module__ 'sage.categories.modules' >>> C2 = Modules(Rings()) & Semigroups(); C2 Join of Category of semigroups and Category of modules over rings >>> C2.base_ring() Category of rings >>> C2.base_ring.__module__ 'sage.categories.modules' >>> # needs sage.combinat sage.groups sage.modules >>> C3 = DescentAlgebra(QQ,Integer(3)).B().category() >>> C3.base_ring.__module__ 'sage.categories.modules' >>> C3.base_ring() Rational Field >>> # needs sage.combinat sage.modules >>> C4 = QuasiSymmetricFunctions(QQ).F().category() >>> C4.base_ring.__module__ 'sage.categories.modules' >>> C4.base_ring() Rational Field
- dual()[source]¶
Return the category of spaces constructed as duals of spaces of
self
.The dual of a vector space \(V\) is the space consisting of all linear functionals on \(V\) (see Wikipedia article Dual_space). Additional structure on \(V\) can endow its dual with additional structure; for example, if \(V\) is a finite dimensional algebra, then its dual is a coalgebra.
This returns the category of spaces constructed as dual of spaces in
self
, endowed with the appropriate additional structure.Warning
This semantic of
dual
andDualObject
is imposed on all subcategories, in particular to makedual
a covariant functorial construction.A subcategory that defines a different notion of dual needs to use a different name.
Typically, the category of graded modules should define a separate
graded_dual
construction (see Issue #15647). For now the two constructions are not distinguished which is an oversimplified model.
EXAMPLES:
sage: VectorSpaces(QQ).DualObjects() Category of duals of vector spaces over Rational Field
>>> from sage.all import * >>> VectorSpaces(QQ).DualObjects() Category of duals of vector spaces over Rational Field
The dual of a vector space is a vector space:
sage: VectorSpaces(QQ).DualObjects().super_categories() [Category of vector spaces over Rational Field]
>>> from sage.all import * >>> VectorSpaces(QQ).DualObjects().super_categories() [Category of vector spaces over Rational Field]
The dual of an algebra is a coalgebra:
sage: sorted(Algebras(QQ).DualObjects().super_categories(), key=str) [Category of coalgebras over Rational Field, Category of duals of vector spaces over Rational Field]
>>> from sage.all import * >>> sorted(Algebras(QQ).DualObjects().super_categories(), key=str) [Category of coalgebras over Rational Field, Category of duals of vector spaces over Rational Field]
The dual of a coalgebra is an algebra:
sage: sorted(Coalgebras(QQ).DualObjects().super_categories(), key=str) [Category of algebras over Rational Field, Category of duals of vector spaces over Rational Field]
>>> from sage.all import * >>> sorted(Coalgebras(QQ).DualObjects().super_categories(), key=str) [Category of algebras over Rational Field, Category of duals of vector spaces over Rational Field]
As a shorthand, this category can be accessed with the
dual()
method:sage: VectorSpaces(QQ).dual() Category of duals of vector spaces over Rational Field
>>> from sage.all import * >>> VectorSpaces(QQ).dual() Category of duals of vector spaces over Rational Field
- Super[source]¶
alias of
SuperModules
- class TensorProducts(category, *args)[source]¶
Bases:
TensorProductsCategory
The category of modules constructed by tensor product of modules.
- class ParentMethods[source]¶
Bases:
object
Implement operations on tensor products of modules.
- construction()[source]¶
Return the construction of
self
.EXAMPLES:
sage: A = algebras.Free(QQ, 2) # needs sage.combinat sage.modules sage: T = A.tensor(A) # needs sage.combinat sage.modules sage: T.construction() # needs sage.combinat sage.modules (The tensor functorial construction, (Free Algebra on 2 generators (None0, None1) over Rational Field, Free Algebra on 2 generators (None0, None1) over Rational Field))
>>> from sage.all import * >>> A = algebras.Free(QQ, Integer(2)) # needs sage.combinat sage.modules >>> T = A.tensor(A) # needs sage.combinat sage.modules >>> T.construction() # needs sage.combinat sage.modules (The tensor functorial construction, (Free Algebra on 2 generators (None0, None1) over Rational Field, Free Algebra on 2 generators (None0, None1) over Rational Field))
- tensor_factors()[source]¶
Return the tensor factors of this tensor product.
EXAMPLES:
sage: # needs sage.modules sage: F = CombinatorialFreeModule(ZZ, [1,2]) sage: F.rename('F') sage: G = CombinatorialFreeModule(ZZ, [3,4]) sage: G.rename('G') sage: T = tensor([F, G]); T F # G sage: T.tensor_factors() (F, G)
>>> from sage.all import * >>> # needs sage.modules >>> F = CombinatorialFreeModule(ZZ, [Integer(1),Integer(2)]) >>> F.rename('F') >>> G = CombinatorialFreeModule(ZZ, [Integer(3),Integer(4)]) >>> G.rename('G') >>> T = tensor([F, G]); T F # G >>> T.tensor_factors() (F, G)
- extra_super_categories()[source]¶
EXAMPLES:
sage: Modules(ZZ).TensorProducts().extra_super_categories() [Category of modules over Integer Ring] sage: Modules(ZZ).TensorProducts().super_categories() [Category of modules over Integer Ring]
>>> from sage.all import * >>> Modules(ZZ).TensorProducts().extra_super_categories() [Category of modules over Integer Ring] >>> Modules(ZZ).TensorProducts().super_categories() [Category of modules over Integer Ring]
- WithBasis[source]¶
alias of
ModulesWithBasis
- additional_structure()[source]¶
Return
None
.Indeed, the category of modules defines no additional structure: a bimodule morphism between two modules is a module morphism.
See also
Todo
Should this category be a
CategoryWithAxiom
?EXAMPLES:
sage: Modules(ZZ).additional_structure()
>>> from sage.all import * >>> Modules(ZZ).additional_structure()
- super_categories()[source]¶
EXAMPLES:
sage: Modules(ZZ).super_categories() [Category of bimodules over Integer Ring on the left and Integer Ring on the right]
>>> from sage.all import * >>> Modules(ZZ).super_categories() [Category of bimodules over Integer Ring on the left and Integer Ring on the right]
Nota bene:
sage: Modules(QQ) Category of vector spaces over Rational Field sage: Modules(QQ).super_categories() [Category of modules over Rational Field]
>>> from sage.all import * >>> Modules(QQ) Category of vector spaces over Rational Field >>> Modules(QQ).super_categories() [Category of modules over Rational Field]