AUTHORS: David Kohel and William Stein
Every SAGE object lies in a category. Categories in SAGE are modeled on
the mathematical idea of category, and are distinct from Python
classes, which are a programming construct.
In most cases, typing x.category()
returns the category to
which
belongs. If
is a category and
is any object,
tries to make an object in
from
.
We create a couple of categories.
sage: Sets()
Category of sets
sage: GSets()
Category of G-sets
sage: Semigroups()
Category of semigroups
sage: VectorSpaces(FiniteField(11))
Category of vector spaces over Finite field of size 11
sage: Ideals(IntegerRing())
Category of ring ideals in Integer Ring
The default category for elements
of an objects
is the
category of all objects of
. For example,
sage: V = VectorSpace(RationalField(), 3)
sage: x = V.gen(1)
sage: x.category()
Category of elements of Vector space of dimension 3 over Rational Field
The module categories.categories defines the following methods:
-
Returns True if x is a category.
The module categories.categories defines the following classes:
- class AbelianCategory
- class AbelianGroups
-
The category of all abelian groups.
- class AbelianMonoids
-
The category of all monoids.
- class AbelianSemigroups
-
The category of all abelian semigroups.
- class AlgebraIdeals
-
The category of ideals in a fixed algebra
.
- class AlgebraModules
-
The category of modules over a fixed algebra
.
- class Algebras
-
The category of algebras over a fixed base ring.
- class Category_ideal
- class CommutativeAlgebraIdeals
-
The category of ideals in a fixed commutative algebra
.
- class CommutativeAlgebras
-
The category of commutative algebras over a given base ring.
- class CommutativeRingIdeals
-
The category of ideals in a fixed commutative ring.
- class CommutativeRings
-
The category of commutative rings.
- class Elements
-
The category of all elements of a given object.
- class Fields
-
The category of fields.
- class FiniteFields
-
The category of all finite fields.
- class FreeModules
-
The category of free modules over a base ring.
- class GroupAlgebras
-
EXAMPLES:
sage: GroupAlgebras(IntegerRing())
Category of group algebras over Integer Ring
- class Groups
-
The category of groups.
- class GSets
-
The category of
-sets, for a group
.
- class HeckeModules
-
The category of Hecke modules.
- class Ideals
-
The category of all ideals in a fixed ring.
- class MatrixAlgebras
-
The category of matrix algebras over a field.
- class Modules
-
The category of all modules over a base ring.
- class MonoidAlgebras
-
The category of all monoid algebras over a given base ring.
- class Monoids
-
The category of monoids.
- class NumberFields
-
The category of number fields.
- class Objects
-
The category of all SAGE objects.
- class RingIdeals
-
The category of all ideals in a fixed ring.
- class RingModules
-
The category of all modules over a base ring.
- class Rings
-
The category of all rings.
- class Semigroups
-
The category of all semigroups.
- class Sets
-
The category of sets.
- class VectorSpaces
-
The category of vector spaces over a specified field,
with an emedding in an ambient vector space.
Release 0.7.7, documentation updated on October 4, 2005.
See About this document... for information on suggesting changes.