Super modules¶
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sage.categories.super_modules.
SuperModules
¶ The category of super modules.
An \(R\)-super module (where \(R\) is a ring) is an \(R\)-module \(M\) equipped with a decomposition \(M = M_0 \oplus M_1\) into two \(R\)-submodules \(M_0\) and \(M_1\) (called the even part and the odd part of \(M\), respectively).
Thus, an \(R\)-super module automatically becomes a \(\ZZ / 2 \ZZ\)-graded \(R\)-module, with \(M_0\) being the degree-\(0\) component and \(M_1\) being the degree-\(1\) component.
EXAMPLES:
sage: Modules(ZZ).Super() Category of super modules over Integer Ring sage: Modules(ZZ).Super().super_categories() [Category of graded modules over Integer Ring]
The category of super modules defines the super structure which shall be preserved by morphisms:
sage: Modules(ZZ).Super().additional_structure() Category of super modules over Integer Ring
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class
sage.categories.super_modules.
SuperModulesCategory
(base_category)¶ Bases:
sage.categories.covariant_functorial_construction.CovariantConstructionCategory
,sage.categories.category_types.Category_over_base_ring
EXAMPLES:
sage: C = Algebras(QQ).Super() sage: C Category of super algebras over Rational Field sage: C.base_category() Category of algebras over Rational Field sage: sorted(C.super_categories(), key=str) [Category of graded algebras over Rational Field, Category of super modules over Rational Field] sage: AlgebrasWithBasis(QQ).Super().base_ring() Rational Field sage: HopfAlgebrasWithBasis(QQ).Super().base_ring() Rational Field
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classmethod
default_super_categories
(category, *args)¶ Return the default super categories of \(F_{Cat}(A,B,...)\) for \(A,B,...\) parents in \(Cat\).
INPUT:
cls
– the category class for the functor \(F\)category
– a category \(Cat\)*args
– further arguments for the functor
OUTPUT:
A join category.
This implements the property that subcategories constructed by the set of whitelisted axioms is a subcategory.
EXAMPLES:
sage: HopfAlgebras(ZZ).WithBasis().FiniteDimensional().Super() # indirect doctest Category of finite dimensional super hopf algebras with basis over Integer Ring
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classmethod