# Elements, parents, and categories in Sage: a (draft of) primer¶

## Abstract¶

The purpose of categories in Sage is to translate the mathematical concept of categories (category of groups, of vector spaces, ...) into a concrete software engineering design pattern for:

• organizing and promoting generic code
• fostering consistency across the Sage library (naming conventions, doc, tests)
• embedding more mathematical knowledge into the system

This design pattern is largely inspired from Axiom and its followers (Aldor, Fricas, MuPAD, ...). It differs from those by:

• blending in the Magma inspired concept of Parent/Element
• being built on top of (and not into) the standard Python object oriented and class hierarchy mechanism. This did not require changing the language, and could in principle be implemented in any language supporting the creation of new classes dynamically.

The general philosophy is that Building mathematical information into the system yields more expressive, more conceptual and, at the end, easier to maintain and faster code (within a programming realm; this would not necessarily apply to specialized libraries like gmp!).

### One line pitch for mathematicians¶

Categories in Sage provide a library of interrelated bookshelves, with each bookshelf containing algorithms, tests, documentation, or some mathematical facts about the objects of a given category (e.g. groups).

### One line pitch for programmers¶

Categories in Sage provide a large hierarchy of abstract classes for mathematical objects. To keep it maintainable, the inheritance information between the classes is not hardcoded but instead reconstructed dynamically from duplication free semantic information.

## Introduction: Sage as a library of objects and algorithms¶

The Sage library, with more than one million lines of code, documentation, and tests, implements:

• Thousands of different kinds of objects (classes):

Integers, polynomials, matrices, groups, number fields, elliptic curves, permutations, morphisms, languages, ... and a few racoons ...

• Tens of thousands methods and functions:

Arithmetic, integer and polynomial factorization, pattern matching on words, ...

### Some challenges¶

• How to organize this library?

One needs some bookshelves to group together related objects and algorithms.

• How to ensure consistency?

Similar objects should behave similarly:

sage: Permutations(5).cardinality()
120

sage: GL(2,2).cardinality()
6

sage: A=random_matrix(ZZ,6,3,x=7)
sage: L=LatticePolytope(A.rows())
sage: L.npoints()                # oops!   # random
37

• How to ensure robustness?

• How to reduce duplication?

Example: binary powering:

sage: m = 3
sage: m^8 == m*m*m*m*m*m*m*m == ((m^2)^2)^2
True

sage: m=random_matrix(QQ, 4, algorithm='echelonizable', rank=3, upper_bound=60)
sage: m^8 == m*m*m*m*m*m*m*m == ((m^2)^2)^2
True


We want to implement binary powering only once, as generic code that will apply in all cases.

## A bit of help from abstract algebra¶

### The hierarchy of categories¶

What makes binary powering work in the above examples? In both cases, we have a set endowed with a multiplicative binary operation which is associative. Such a set is called a semigroup, and binary powering works generally for any semigroup.

sage: Semigroups()
Category of semigroups


and sure enough, binary powering is defined there:

sage: m._pow_.__module__
'sage.categories.semigroups'


That’s our bookshelf! And it’s used in many places:

sage: GL(2,ZZ) in Semigroups()
True
sage: NN in Semigroups()
True


For a less trivial bookshelf we can consider euclidean rings: once we know how to do euclidean division in some set $$R$$, we can compute gcd’s in $$R$$ generically using the Euclidean algorithm.

We are in fact very lucky: abstract algebra provides us right away with a large and robust set of bookshelves which is the result of centuries of work of mathematicians to identify the important concepts. This includes for example:

sage: Sets()
Category of sets

sage: Groups()
Category of groups

sage: Rings()
Category of rings

sage: Fields()
Category of fields

sage: HopfAlgebras(QQ)
Category of hopf algebras over Rational Field


Each of the above is called a category. It typically specifies what are the operations on the elements, as well as the axioms satisfied by those operations. For example the category of groups specifies that a group is a set endowed with a binary operation (the multiplication) which is associative and admits a unit and inverses.

Each set in Sage knows which bookshelf of generic algorithms it can use, that is to which category it belongs:

sage: G = GL(2,ZZ)
sage: G.category()
Category of groups


sage: Groups().is_subcategory(Semigroups())
True
sage: G in Semigroups()
True


Altogether, our group gets algorithms from a bunch of bookshelves:

sage: G.categories()
[Category of groups, Category of monoids, Category of semigroups,
...,
Category of magmas,
Category of sets, ...]


Those can be viewed graphically:

sage: g = Groups().category_graph()
sage: g.set_latex_options(format="dot2tex")
sage: view(g, tightpage=True)                 # not tested


In case dot2tex is not available, you can use instead:

sage: g.show(vertex_shape=None, figsize=20)


Here is an overview of all categories in Sage:

sage: g = sage.categories.category.category_graph()
sage: g.set_latex_options(format="dot2tex")
sage: view(g, tightpage=True)                 # not tested


Wrap-up: generic algorithms in Sage are organized in a hierarchy of bookshelves modelled upon the usual hierarchy of categories provided by abstract algebra.

### Elements, Parents, Categories¶

Parent

A parent is a Python instance modelling a set of mathematical elements together with its additional (algebraic) structure.

Examples include the ring of integers, the group $$S_3$$, the set of prime numbers, the set of linear maps between two given vector spaces, and a given finite semigroup.

These sets are often equipped with additional structure: the set of all integers forms a ring. The main way of encoding this information is specifying which categories a parent belongs to.

It is completely possible to have different Python instances modelling the same set of elements. For example, one might want to consider the ring of integers, or the poset of integers under their standard order, or the poset of integers under divisibility, or the semiring of integers under the operations of maximum and addition. Each of these would be a different instance, belonging to different categories.

For a given model, there should be a unique instance in Sage representing that parent:

sage: IntegerRing() is IntegerRing()
True


Element

An element is a Python instance modelling a mathematical element of a set.

Examples of element include $$5$$ in the integer ring, $$x^3 - x$$ in the polynomial ring in $$x$$ over the rationals, $$4 + O(3^3)$$ in the 3-adics, the transposition $$(1 2)$$ in $$S_3$$, and the identity morphism in the set of linear maps from $$\QQ^3$$ to $$\QQ^3$$.

Every element in Sage has a parent. The standard idiom in Sage for creating elements is to create their parent, and then provide enough data to define the element:

sage: R = PolynomialRing(ZZ, name='x')
sage: R([1,2,3])
3*x^2 + 2*x + 1


One can also create elements using various methods on the parent and arithmetic of elements:

sage: x = R.gen()
sage: 1 + 2*x + 3*x^2
3*x^2 + 2*x + 1


Unlike parents, elements in Sage are not necessarily unique:

sage: ZZ(5040) is ZZ(5040)
False


Many parents model algebraic structures, and their elements support arithmetic operations. One often further wants to do arithmetic by combining elements from different parents: adding together integers and rationals for example. Sage supports this feature using coercion (see sage.structure.coerce for more details).

It is possible for a parent to also have simultaneously the structure of an element. Consider for example the monoid of all finite groups, endowed with the cartesian product operation. Then, every finite group (which is a parent) is also an element of this monoid. This is not yet implemented, and the design details are not yet fixed but experiments are underway in this direction.

Todo

Give a concrete example, typically using ElementWrapper.

Category

A category is a Python instance modelling a mathematical category.

Examples of categories include the category of finite semigroups, the category of all (Python) objects, the category of $$\ZZ$$-algebras, and the category of cartesian products of $$\ZZ$$-algebras:

sage: FiniteSemigroups()
Category of finite semigroups
sage: Objects()
Category of objects
sage: Algebras(ZZ)
Category of algebras over Integer Ring
sage: Algebras(ZZ).CartesianProducts()
Category of Cartesian products of algebras over Integer Ring


Mind the ‘s’ in the names of the categories above; GroupAlgebra and GroupAlgebras are distinct things.

Every parent belongs to a collection of categories. Moreover, categories are interrelated by the super categories relation. For example, the category of rings is a super category of the category of fields, because every field is also a ring.

A category serves two roles:

• to provide a model for the mathematical concept of a category and the associated structures: homsets, morphisms, functorial constructions, axioms.
• to organize and promote generic code, naming conventions, documentation, and tests across similar mathematical structures.

CategoryObject

Objects of a mathematical category are not necessarily parents. Parent has a superclass that provides a means of modeling such.

For example, the category of schemes does not have a faithful forgetful functor to the category of sets, so it does not make sense to talk about schemes as parents.

Morphisms, Homsets

As category theorists will expect, Morphisms and Homsets will play an ever more important role, as support for them will improve.

Much of the mathematical information in Sage is encoded as relations between elements and their parents, parents and their categories, and categories and their super categories:

sage: 1.parent()
Integer Ring

sage: ZZ
Integer Ring

sage: ZZ.category()
Category of euclidean domains

sage: ZZ.categories()
[Category of euclidean domains, Category of principal ideal domains,
Category of unique factorization domains, Category of gcd domains,
Category of integral domains, Category of domains,
Category of commutative rings, Category of rings, ...
Category of magmas and additive magmas, ...
Category of monoids, Category of semigroups,
Category of commutative magmas, Category of unital magmas, Category of magmas,
Category of sets,
Category of sets with partial maps,
Category of objects]

sage: g = EuclideanDomains().category_graph()
sage: g.set_latex_options(format="dot2tex")
sage: view(g, tightpage=True)                 # not tested


## A bit of help from computer science¶

### Hierarchy of classes¶

How are the bookshelves implemented in practice?

Sage uses the classical design paradigm of Object Oriented Programming (OOP). Its fundamental principle is that any object that a program is to manipulate should be modelled by an instance of a class. The class implements:

• a data structure: which describes how the object is stored,
• methods: which describe the operations on the object.

The instance itself contains the data for the given object, according to the specified data structure.

Hence, all the objects mentioned above should be instances of some classes. For example, an integer in Sage is an instance of the class Integer (and it knows about it!):

sage: i = 12
sage: type(i)
<type 'sage.rings.integer.Integer'>


Applying an operation is generally done by calling a method:

sage: i.factor()
2^2 * 3

sage: x = var('x')
sage: p = 6*x^2 + 12*x + 6
sage: type(p)
<type 'sage.symbolic.expression.Expression'>
sage: p.factor()
6*(x + 1)^2

sage: R.<x> = PolynomialRing(QQ, sparse=True)
sage: pQ = R ( p )
sage: type(pQ)
<class 'sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_sparse_field'>
sage: pQ.factor()
(6) * (x + 1)^2

sage: pZ = ZZ[x] ( p )
sage: type(pZ)
<type 'sage.rings.polynomial.polynomial_integer_dense_flint.Polynomial_integer_dense_flint'>
sage: pZ.factor()
2 * 3 * (x + 1)^2


Factoring integers, expressions, or polynomials are distinct tasks, with completely different algorithms. Yet, from a user (or caller) point of view, all those objects can be manipulated alike. This illustrates the OOP concepts of polymorphism, data abstraction, and encapsulation.

Let us be curious, and see where some methods are defined. This can be done by introspection:

sage: i._mul_??                   # not tested

For plain Python methods, one can also just ask in which module they are implemented:

sage: i._pow_.__module__
'sage.categories.semigroups'

sage: pQ._mul_.__module__
'sage.rings.polynomial.polynomial_element_generic'
sage: pQ._pow_.__module__
'sage.categories.semigroups'


We see that integers and polynomials have each their own multiplication method: the multiplication algorithms are indeed unrelated and deeply tied to their respective datastructures. On the other hand, as we have seen above, they share the same powering method because the set $$\ZZ$$ of integers, and the set $$\QQ[x]$$ of polynomials are both semigroups. Namely, the class for integers and the class for polynomials both derive from an abstract class for semigroup elements, which factors out the generic methods like _pow_. This illustrates the use of hierarchy of classes to share common code between classes having common behaviour.

OOP design is all about isolating the objects that one wants to model together with their operations, and designing an appropriate hierarchy of classes for organizing the code. As we have seen above, the design of the class hierarchy is easy since it can be modelled upon the hierarchy of categories (bookshelves). Here is for example a piece of the hierarchy of classes for an element of a group of matrices:

sage: G = GL(2,ZZ)
sage: m = G.an_element()
sage: for cls in m.__class__.mro(): print cls
<class 'sage.groups.matrix_gps.group_element.LinearMatrixGroup_gap_with_category.element_class'>
<class 'sage.groups.matrix_gps.group_element.MatrixGroupElement_gap'>
...
<class 'sage.categories.groups.Groups.element_class'>
<class 'sage.categories.monoids.Monoids.element_class'>
<class 'sage.categories.semigroups.Semigroups.element_class'>
...


On the top, we see concrete classes that describe the data structure for matrices and provide the operations that are tied to this data structure. Then follow abstract classes that are attached to the hierarchy of categories and provide generic algorithms.

The full hierarchy is best viewed graphically:

sage: g = class_graph(m.__class__)
sage: g.set_latex_options(format="dot2tex")
sage: view(g, tightpage=True)                 # not tested


### Parallel hierarchy of classes for parents¶

Let us recall that we do not just want to compute with elements of mathematical sets, but with the sets themselves:

sage: ZZ.one()
1

sage: R = QQ['x,y']
sage: R.krull_dimension()
2
sage: A = R.quotient( R.ideal(x^2 - 2) )
sage: A.krull_dimension() # todo: not implemented


Here are some typical operations that one may want to carry on various kinds of sets:

• The set of permutations of 5, the set of rational points of an elliptic curve: counting, listing, random generation
• A language (set of words): rationality testing, counting elements, generating series
• A finite semigroup: left/right ideals, center, representation theory
• A vector space, an algebra: cartesian product, tensor product, quotient

Hence, following the OOP fundamental principle, parents should also be modelled by instances of some (hierarchy of) classes. For example, our group $$G$$ is an instance of the following class:

sage: G = GL(2,ZZ)
sage: type(G)
<class 'sage.groups.matrix_gps.linear.LinearMatrixGroup_gap_with_category'>


Here is a piece of the hierarchy of classes above it:

sage: for cls in G.__class__.mro(): print cls
<class 'sage.groups.matrix_gps.linear.LinearMatrixGroup_gap_with_category'>
...
<class 'sage.categories.groups.Groups.parent_class'>
<class 'sage.categories.monoids.Monoids.parent_class'>
<class 'sage.categories.semigroups.Semigroups.parent_class'>
...


Note that the hierarchy of abstract classes is again attached to categories and parallel to that we had seen for the elements. This is best viewed graphically:

sage: g = class_graph(m.__class__)
sage: g.relabel(lambda x: x.replace("_","\_"))
sage: g.set_latex_options(format="dot2tex")
sage: view(g, tightpage=True)                 # not tested


Note

This is a progress upon systems like Axiom or MuPAD where a parent is modelled by the class of its elements; this oversimplification leads to confusion between methods on parents and elements, and makes parents special; in particular it prevents potentially interesting constructions like “groups of groups”.

## Sage categories¶

Why this business of categories? And to start with, why don’t we just have a good old hierarchy of classes Group, Semigroup, Magma, ... ?

### Dynamic hierarchy of classes¶

As we have just seen, when we manipulate groups, we actually manipulate several kinds of objects:

• groups
• group elements
• morphisms between groups
• and even the category of groups itself!

Thus, on the group bookshelf, we want to put generic code for each of the above. We therefore need three, parallel hierarchies of abstract classes:

• Group, Monoid, Semigroup, Magma, ...
• GroupElement, MonoidElement, SemigroupElement, MagmaElement, ...
• GroupMorphism, SemigroupElement, SemigroupMorphism, MagmaMorphism, ...

(and in fact many more as we will see).

We could implement the above hierarchies as usual:

class Group(Monoid):
# generic methods that apply to all groups

class GroupElement(MonoidElement):
# generic methods that apply to all group elements

class GroupMorphism(MonoidMorphism):
# generic methods that apply to all group morphisms

And indeed that’s how it was done in Sage before 2009, and there are still many traces of this. The drawback of this approach is duplication: the fact that a group is a monoid is repeated three times above!

Instead, Sage now uses the following syntax, where the Groups bookshelf is structured into units with nested classes:

class Groups(Category):

def super_categories(self):
return [Monoids(), ...]

class ParentMethods:
# generic methods that apply to all groups

class ElementMethods:
# generic methods that apply to all group elements

class MorphismMethods:
# generic methods that apply to all group morphisms (not yet implemented)

class SubcategoryMethods:
# generic methods that apply to all subcategories of Groups()

With this syntax, the information that a group is a monoid is specified only once, in the Category.super_categories() method. And indeed, when the category of inverse unital magmas was introduced, there was a single point of truth to update in order to reflect the fact that a group is an inverse unital magma:

sage: Groups().super_categories()
[Category of monoids, Category of inverse unital magmas]


The price to pay (there is no free lunch) is that some magic is required to construct the actual hierarchy of classes for parents, elements, and morphisms. Namely, Groups.ElementMethods should be seen as just a bag of methods, and the actual class Groups().element_class is constructed from it by adding the appropriate super classes according to Groups().super_categories():

sage: Groups().element_class
<class 'sage.categories.groups.Groups.element_class'>

sage: Groups().element_class.__bases__
(<class 'sage.categories.monoids.Monoids.element_class'>,
<class 'sage.categories.magmas.Magmas.Unital.Inverse.element_class'>)


We now see that the hierarchy of classes for parents and elements is parallel to the hierarchy of categories:

sage: Groups().all_super_categories()
[Category of groups,
Category of monoids,
Category of semigroups,
...
Category of magmas,
Category of sets,
...]

sage: for cls in Groups().element_class.mro(): print cls
<class 'sage.categories.groups.Groups.element_class'>
<class 'sage.categories.monoids.Monoids.element_class'>
<class 'sage.categories.semigroups.Semigroups.element_class'>
...
<class 'sage.categories.magmas.Magmas.element_class'>
...
sage: for cls in Groups().parent_class.mro(): print cls
<class 'sage.categories.groups.Groups.parent_class'>
<class 'sage.categories.monoids.Monoids.parent_class'>
<class 'sage.categories.semigroups.Semigroups.parent_class'>
...
<class 'sage.categories.magmas.Magmas.parent_class'>
...


Another advantage of building the hierarchy of classes dynamically is that, for parametrized categories, the hierarchy may depend on the parameters. For example an algebra over $$\QQ$$ is a $$\QQ$$-vector space, but an algebra over $$\ZZ$$ is not (it is just a $$\ZZ$$-module)!

Note

At this point this whole infrastructure may feel like overdesigning, right? We felt like this too! But we will see later that, once one gets used to it, this approach scales very naturally.

From a computer science point of view, this infrastructure implements, on top of standard multiple inheritance, a dynamic composition mechanism of mixin classes (Wikipedia article Mixin), governed by mathematical properties.

For implementation details on how the hierarchy of classes for parents and elements is constructed, see Category.

### On the category hierarchy: subcategories and super categories¶

We have seen above that, for example, the category of sets is a super category of the category of groups. This models the fact that a group can be unambiguously considered as a set by forgetting its group operation. In object-oriented parlance, we want the relation “a group is a set”, so that groups can directly inherit code implemented on sets.

Formally, a category Cs() is a super category of a category Ds() if Sage considers any object of Ds() to be an object of Cs(), up to an implicit application of a canonical functor from Ds() to Cs(). This functor is normally an inclusion of categories or a forgetful functor. Reciprocally, Ds() is said to be a subcategory of Cs().

Warning

This terminology deviates from the usual mathematical definition of subcategory and is subject to change. Indeed, the forgetful functor from the category of groups to the category of sets is not an inclusion of categories, as it is not injective: a given set may admit more than one group structure. See trac ticket #16183 for more details. The name supercategory is also used with a different meaning in certain areas of mathematics.

### Categories are instances and have operations¶

Note that categories themselves are naturally modelled by instances because they can have operations of their own. An important one is:

sage: Groups().example()
General Linear Group of degree 4 over Rational Field


which gives an example of object of the category. Besides illustrating the category, the example provides a minimal template for implementing a new object in the category:

sage: S = Semigroups().example(); S
An example of a semigroup: the left zero semigroup


Its source code can be obtained by introspection:

sage: S??                                     # not tested

This example is also typically used for testing generic methods. See Category.example() for more.

Other operations on categories include querying the super categories or the axioms satisfied by the operations of a category:

sage: Groups().super_categories()
[Category of monoids, Category of inverse unital magmas]
sage: Groups().axioms()
frozenset(['Inverse', 'Associative', 'Unital'])


or constructing the intersection of two categories, or the smallest category containing them:

sage: Groups() & FiniteSets()
Category of finite groups
sage: Algebras(QQ) | Groups()
Category of monoids


### Specifications and generic documentation¶

Categories do not only contain code but also the specifications of the operations. In particular a list of mandatory and optional methods to be implemented can be found by introspection with:

sage: Groups().required_methods()
{'parent': {'required': ['__contains__'], 'optional': []},
'element': {'required': [], 'optional': ['_mul_']}}


Documentation about those methods can be obtained with:

sage: G = Groups()
sage: G.element_class._mul_?        # not tested
sage: G.parent_class.one?           # not tested

Warning

Well, more precisely, that’s how things should be, but there is still some work to do in this direction. For example, the inverse operation is not specified above. Also, we are still missing a good programmatic syntax to specify the input and output types of the methods. Finally, in many cases the implementer must provide at least one of two methods, each having a default implementation using the other one (e.g. listing or iterating for a finite enumerated set); there is currently no good programmatic way to specify this.

### Generic tests¶

Another feature that parents and elements receive from categories is generic tests; their purpose is to check (at least to some extent) that the parent satisfies the required mathematical properties (is my semigroup indeed associative?) and is implemented according to the specifications (does the method an_element indeed return an element of the parent?):

sage: S = FiniteSemigroups().example(alphabet=('a', 'b'))
sage: TestSuite(S).run(verbose = True)
running ._test_an_element() . . . pass
running ._test_associativity() . . . pass
running ._test_category() . . . pass
running ._test_elements() . . .
Running the test suite of self.an_element()
running ._test_category() . . . pass
running ._test_eq() . . . pass
running ._test_not_implemented_methods() . . . pass
running ._test_pickling() . . . pass
pass
running ._test_elements_eq_reflexive() . . . pass
running ._test_elements_eq_symmetric() . . . pass
running ._test_elements_eq_transitive() . . . pass
running ._test_elements_neq() . . . pass
running ._test_enumerated_set_contains() . . . pass
running ._test_enumerated_set_iter_cardinality() . . . pass
running ._test_enumerated_set_iter_list() . . . pass
running ._test_eq() . . . pass
running ._test_not_implemented_methods() . . . pass
running ._test_pickling() . . . pass
running ._test_some_elements() . . . pass


Tests can be run individually:

sage: S._test_associativity()


Here is how to access the code of this test:

sage: S._test_associativity?? # not tested

Here is how to run the test on all elements:

sage: L = S.list()
sage: S._test_associativity(elements=L)


Let us see what happens when a test fails. Here we redefine the product of $$S$$ to something definitely not associative:

sage: S.product = lambda x, y: S("("+x.value +y.value+")")


And rerun the test:

sage: S._test_associativity(elements=L)
Traceback (most recent call last):
...
File ".../sage/categories/semigroups.py", line ..., in _test_associativity
tester.assert_((x * y) * z == x * (y * z))
...
AssertionError: False is not true

We can recover instantly the actual values of x, y, z, that is, a counterexample to the associativity of our broken semigroup, using post mortem introspection with the Python debugger pdb (this does not work yet in the notebook):

sage: import pdb
sage: pdb.pm()                       # not tested
> /opt/sage-5.11.rc1/local/lib/python/unittest/case.py(424)assertTrue()
-> raise self.failureException(msg)
(Pdb) u
> /opt/sage-5.11.rc1/local/lib/python2.7/site-packages/sage/categories/semigroups.py(145)_test_associativity()
-> tester.assert_((x * y) * z == x * (y * z))
(Pdb) p x, y, z
('a', 'a', 'a')
(Pdb) p (x * y) * z
'((aa)a)'
(Pdb) p x * (y * z)
'(a(aa))'


### Wrap-up¶

• Categories provide a natural hierarchy of bookshelves to organize not only code, but also specifications and testing tools.
• Everything about, say, algebras with a distinguished basis is gathered in AlgebrasWithBasis or its super categories. This includes properties and algorithms for elements, parents, morphisms, but also, as we will see, for constructions like cartesian products or quotients.
• The mathematical relations between elements, parents, and categories translate dynamically into a traditional hierarchy of classes.
• This design enforces robustness and consistency, which is particularly welcome given that Python is an interpreted language without static type checking.

## Case study¶

In this section, we study an existing parent in detail; a good followup is to go through the sage.categories.tutorial or the thematic tutorial on coercion and categories (“How to implement new algebraic structures in Sage”) to learn how to implement a new one!

We consider the example of finite semigroup provided by the category:

sage: S = FiniteSemigroups().example(); S
An example of a finite semigroup: the left regular band generated by ('a', 'b', 'c', 'd')
sage: S?                    # not tested

Where do all the operations on S and its elements come from?

sage: x = S('a')


_repr_ is a technical method which comes with the data structure (ElementWrapper); since it’s implemented in Cython, we need to use Sage’s introspection tools to recover where it’s implemented:

sage: x._repr_.__module__
sage: sage.misc.sageinspect.sage_getfile(x._repr_)
'.../sage/structure/element_wrapper.pyx'


__pow__ is a generic method for all finite semigroups:

sage: x.__pow__.__module__
'sage.categories.semigroups'


__mul__ is a default implementation from the Magmas category (a magma is a set with an inner law $$*$$, not necessarily associative):

sage: x.__mul__.__module__
'sage.categories.magmas'


It delegates the work to the parent (following the advice: if you do not know what to do, ask your parent):

sage: x.__mul__??                             # not tested

product is a mathematical method implemented by the parent:

sage: S.product.__module__
'sage.categories.examples.finite_semigroups'


cayley_graph is a generic method on the parent, provided by the FiniteSemigroups category:

sage: S.cayley_graph.__module__
'sage.categories.semigroups'


multiplication_table is a generic method on the parent, provided by the Magmas category (it does not require associativity):

sage: S.multiplication_table.__module__
'sage.categories.magmas'


Consider now the implementation of the semigroup:

sage: S??                                     # not tested

This implementation specifies a data structure for the parents and the elements, and makes a promise: the implemented parent is a finite semigroup. Then it fulfills the promise by implementing the basic operation product. It also implements the optional method semigroup_generators. In exchange, $$S$$ and its elements receive generic implementations of all the other operations. $$S$$ may override any of those by more efficient ones. It may typically implement the element method is_idempotent to always return True.

A (not yet complete) list of mandatory and optional methods to be implemented can be found by introspection with:

sage: FiniteSemigroups().required_methods()
{'parent': {'required': ['__contains__'], 'optional': []},
'element': {'required': [], 'optional': ['_mul_']}}


product does not appear in the list because a default implementation is provided in term of the method _mul_ on elements. Of course, at least one of them should be implemented. On the other hand, a default implementation for __contains__ is provided by Parent.

Documentation about those methods can be obtained with:

sage: C = FiniteSemigroups().element_class
sage: C._mul_?                                # not tested

Here is the code for the finite semigroups category:

sage: FiniteSemigroups??                      # not tested

## Specifying the category of a parent¶

Some parent constructors (not enough!) allow to specify the desired category for the parent. This can typically be used to specify additional properties of the parent that we know to hold a priori. For example, permutation groups are by default in the category of finite permutation groups (no surprise):

sage: P = PermutationGroup([[(1,2,3)]]); P
Permutation Group with generators [(1,2,3)]
sage: P.category()
Category of finite permutation groups


In this case, the group is commutative, so we can specify this:

sage: P = PermutationGroup([[(1,2,3)]], category=PermutationGroups().Finite().Commutative()); P
Permutation Group with generators [(1,2,3)]
sage: P.category()
Category of finite commutative permutation groups


This feature can even be used, typically in experimental code, to add more structure to existing parents, and in particular to add methods for the parents or the elements, without touching the code base:

sage: class Foos(Category):
....:     def super_categories(self):
....:          return [PermutationGroups().Finite().Commutative()]
....:     class ParentMethods:
....:         def foo(self): print "foo"
....:     class ElementMethods:
....:         def bar(self): print "bar"

sage: P = PermutationGroup([[(1,2,3)]], category=Foos())
sage: P.foo()
foo
sage: p = P.an_element()
sage: p.bar()
bar


In the long run, it would be thinkable to use this idiom to implement forgetful functors; for example the above group could be constructed as a plain set with:

sage: P = PermutationGroup([[(1,2,3)]], category=Sets()) # todo: not implemented


At this stage though, this is still to be explored for robustness and practicality. For now, most parents that accept a category argument only accept a subcategory of the default one.

## Scaling further: functorial constructions, axioms, ...¶

In this section, we explore more advanced features of categories. Along the way, we illustrate that a large hierarchy of categories is desirable to model complicated mathematics, and that scaling to support such a large hierarchy is the driving motivation for the design of the category infrastructure.

### Functorial constructions¶

Sage has support for a certain number of so-called covariant functorial constructions which can be used to construct new parents from existing ones while carrying over as much as possible of their algebraic structure. This includes:

• Cartesian products: See cartesian_product.
• Tensor products: See tensor.
• Subquotients / quotients / subobjects / isomorphic objects: See:
• Dual objects: See Modules().DualObjects.
• Algebras, as in group algebras, monoid algebras, ...: See: Sets.ParentMethods.algebras().

Let for example $$A$$ and $$B$$ be two parents, and let us construct the cartesian product $$A \times B \times B$$:

sage: A = AlgebrasWithBasis(QQ).example();     A.rename("A")
sage: B = HopfAlgebrasWithBasis(QQ).example(); B.rename("B")
sage: C = cartesian_product([A, B, B]); C
A (+) B (+) B


In which category should this new parent be? Since $$A$$ and $$B$$ are vector spaces, the result is, as a vector space, the direct sum $$A \oplus B \oplus B$$, hence the notation. Also, since both $$A$$ and $$B$$ are monoids, $$A \times B \times B$$ is naturally endowed with a monoid structure for pointwise multiplication:

sage: C in Monoids()
True


the unit being the cartesian product of the units of the operands:

sage: C.one()
B[(0, word: )] + B[(1, ())] + B[(2, ())]
sage: cartesian_product([A.one(), B.one(), B.one()])
B[(0, word: )] + B[(1, ())] + B[(2, ())]


The pointwise product can be implemented generically for all magmas (i.e. sets endowed with a multiplicative operation) that are constructed as cartesian products. It’s thus implemented in the Magmas category:

sage: C.product.__module__
'sage.categories.magmas'


More specifically, keeping on using nested classes to structure the code, the product method is put in the nested class Magmas.CartesianProducts.ParentMethods:

class Magmas(Category):
class ParentMethods:
# methods for magmas
class ElementMethods:
# methods for elements of magmas
class CartesianProduct(CartesianProductCategory):
class ParentMethods:
# methods for magmas that are constructed as cartesian products
def product(self, x, y):
# ...
class ElementMethods:
# ...

Note

The support for nested classes in Python is relatively recent. Their intensive use for the category infrastructure did reveal some glitches in their implementation, in particular around class naming and introspection. Sage currently works around the more annoying ones but some remain visible. See e.g. sage.misc.nested_class_test.

Let us now look at the categories of C:

sage: C.categories()
[Category of Cartesian products of algebras with basis over Rational Field, ...
Category of Cartesian products of semigroups, Category of semigroups, ...
Category of Cartesian products of magmas, ..., Category of magmas, ...
Category of Cartesian products of additive magmas, ..., Category of additive magmas,
Category of Cartesian products of sets, Category of sets, ...]


This reveals the parallel hierarchy of categories for cartesian products of semigroups magmas, ... We are thus glad that Sage uses its knowledge that a monoid is a semigroup to automatically deduce that a cartesian product of monoids is a cartesian product of semigroups, and build the hierarchy of classes for parents and elements accordingly.

In general, the cartesian product of $$A$$ and $$B$$ can potentially be an algebra, a coalgebra, a differential module, and be finite dimensional, or graded, or .... This can only be decided at runtime, by introspection into the properties of $$A$$ and $$B$$; furthermore, the number of possible combinations (e.g. finite dimensional differential algebra) grows exponentially with the number of properties.

### Axioms¶

#### First examples¶

We have seen that Sage is aware of the axioms satisfied by, for example, groups:

sage: Groups().axioms()
frozenset(['Inverse', 'Associative', 'Unital'])


In fact, the category of groups can be defined by stating that a group is a magma, that is a set endowed with an internal binary multiplication, which satisfies the above axioms. Accordingly, we can construct the category of groups from the category of magmas:

sage: Magmas().Associative().Unital().Inverse()
Category of groups


In general, we can construct new categories in Sage by specifying the axioms that are satisfied by the operations of the super categories. For example, starting from the category of magmas, we can construct all the following categories just by specifying the axioms satisfied by the multiplication:

sage: Magmas()
Category of magmas
sage: Magmas().Unital()
Category of unital magmas

sage: Magmas().Commutative().Unital()
Category of commutative unital magmas
sage: Magmas().Unital().Commutative()
Category of commutative unital magmas

sage: Magmas().Associative()
Category of semigroups

sage: Magmas().Associative().Unital()
Category of monoids

sage: Magmas().Associative().Unital().Commutative()
Category of commutative monoids

sage: Magmas().Associative().Unital().Inverse()
Category of groups


#### Axioms and categories with axioms¶

Here, Associative, Unital, Commutative are axioms. In general, any category Cs in Sage can declare a new axiom A. Then, the category with axiom Cs.A() models the subcategory of the objects of Cs satisfying the axiom A. Similarly, for any subcategory Ds of Cs, Ds.A() models the subcategory of the objects of Ds satisfying the axiom A. In most cases, it’s a full subcategory (see Wikipedia article Subcategory).

For example, the category of sets defines the Finite axiom, and this axiom is available in the subcategory of groups:

sage: Sets().Finite()
Category of finite sets
sage: Groups().Finite()
Category of finite groups


The meaning of each axiom is described in the documentation of the corresponding method, which can be obtained as usual by instrospection:

sage: C = Groups()
sage: C.Finite?              # not tested

The purpose of categories with axioms is no different from other categories: to provide bookshelves of code, documentation, mathematical knowledge, tests, for their objects. The extra feature is that, when intersecting categories, axioms are automatically combined together:

sage: C = Magmas().Associative() & Magmas().Unital().Inverse() & Sets().Finite(); C
Category of finite groups
sage: sorted(C.axioms())
['Associative', 'Finite', 'Inverse', 'Unital']


For a more advanced example, Sage knows that a ring is a set $$C$$ endowed with a multiplication which distributes over addition, such that $$(C, +)$$ is a commutative additive group and $$(C, *)$$ is a monoid:

sage: C = (CommutativeAdditiveGroups() & Monoids()).Distributive(); C
Category of rings

sage: sorted(C.axioms())


The infrastructure allows for specifying further deduction rules, in order to encode mathematical facts like Wedderburn’s theorem:

sage: DivisionRings() & Sets().Finite()
Category of finite fields


Note

When an axiom specifies the properties of some operations in Sage, the notations for those operations are tied to this axiom. For example, as we have seen above, we need two distinct axioms for associativity: the axiom “AdditiveAssociative” is about the properties of the addition $$+$$, whereas the axiom “Associative” is about the properties of the multiplication $$*$$.

We are touching here an inherent limitation of the current infrastructure. There is indeed no support for providing generic code that is independent of the notations. In particular, the category hierarchy about additive structures (additive monoids, additive groups, ...) is completely duplicated by that for multiplicative structures (monoids, groups, ...).

As far as we know, none of the existing computer algebra systems has a good solution for this problem. The difficulty is that this is not only about a single notation but a bunch of operators and methods: +, -, zero, summation, sum, ... in one case, *, /, one, product, prod, factor, ... in the other. Sharing something between the two hierarchies of categories would only be useful if one could write generic code that applies in both cases; for that one needs to somehow automatically substitute the right operations in the right spots in the code. That’s kind of what we are doing manually between e.g. AdditiveMagmas.ParentMethods.addition_table() and Magmas.ParentMethods.multiplication_table(), but doing this systematically is a different beast from what we have been doing so far with just usual inheritance.

#### Single entry point and name space usage¶

A nice feature of the notation Cs.A() is that, from a single entry point (say the category Magmas as above), one can explore a whole range of related categories, typically with the help of introspection to discover which axioms are available, and without having to import new Python modules. This feature will be used in trac ticket #15741 to unclutter the global name space from, for example, the many variants of the category of algebras like:

sage: FiniteDimensionalAlgebrasWithBasis(QQ)
Category of finite dimensional algebras with basis over Rational Field


There will of course be a deprecation step, but it’s recommended to prefer right away the more flexible notation:

sage: Algebras(QQ).WithBasis().FiniteDimensional()
Category of finite dimensional algebras with basis over Rational Field


Design discussion

How far should this be pushed? Fields should definitely stay, but should FiniteGroups or DivisionRings be removed from the global namespace? Do we want to further completely deprecate the notation FiniteGroups() in favor of Groups().Finite()?

#### On the potential combinatorial explosion of categories with axioms¶

Even for a very simple category like Magmas, there are about $$2^5$$ potential combinations of the axioms! Think about what this becomes for a category with two operations $$+$$ and $$*$$:

sage: C = (Magmas() & AdditiveMagmas()).Distributive(); C
Category of distributive magmas and additive magmas

Category of rngs

Category of semirings

Category of rings

sage: Rings().Division()
Category of division rings

sage: Rings().Division().Commutative()
Category of fields

sage: Rings().Division().Finite()
Category of finite fields


sage: g = HopfAlgebras(QQ).WithBasis().Graded().Connected().category_graph()
sage: g.set_latex_options(format="dot2tex")
sage: view(g, tightpage=True)                 # not tested


#### Difference between axioms and regressive covariant functorial constructions¶

Our running examples here will be the axiom FiniteDimensional and the regressive covariant functorial construction Graded. Let Cs be some subcategory of Modules, say the category of modules itself:

sage: Cs = Modules(QQ)


Then, Cs.FiniteDimensional() (respectively Cs.Graded()) is the subcategory of the objects O of Cs which are finite dimensional (respectively graded).

Let also Ds be a subcategory of Cs, say:

sage: Ds = Algebras(QQ)


A finite dimensional algebra is also a finite dimensional module:

sage: Algebras(QQ).FiniteDimensional().is_subcategory( Modules(QQ).FiniteDimensional() )
True


sage: Algebras(QQ).Graded().is_subcategory( Modules(QQ).Graded() )
True


This is the covariance property: for A an axiom or a covariant functorial construction, if Ds is a subcategory of Cs, then Ds.A() is a subcategory of Cs.A().

What happens if we consider reciprocally an object of Cs.A() which is also in Ds? A finite dimensional module which is also an algebra is a finite dimensional algebra:

sage: Modules(QQ).FiniteDimensional() & Algebras(QQ)
Category of finite dimensional algebras over Rational Field


On the other hand, a graded module $$O$$ which is also an algebra is not necessarily a graded algebra! Indeed, the grading on $$O$$ may not be compatible with the product on $$O$$:

sage: Modules(QQ).Graded() & Algebras(QQ)
Join of Category of algebras over Rational Field and Category of graded modules over Rational Field


The relevant difference between FiniteDimensional and Graded is that FiniteDimensional is a statement about the properties of O seen as a module (and thus does not depend on the given category), whereas Graded is a statement about the properties of O and all its operations in the given category.

In general, if a category satisfies a given axiom, any subcategory also satisfies that axiom. Another formulation is that, for an axiom A defined in a super category Cs of Ds, Ds.A() is the intersection of the categories Ds and Cs.A():

sage: As = Algebras(QQ).FiniteDimensional(); As
Category of finite dimensional algebras over Rational Field
sage: Bs = Algebras(QQ) & Modules(QQ).FiniteDimensional(); As
Category of finite dimensional algebras over Rational Field
sage: As is Bs
True


An immediate consequence is that, as we have already noticed, axioms commute:

sage: As = Algebras(QQ).FiniteDimensional().WithBasis(); As
Category of finite dimensional algebras with basis over Rational Field
sage: Bs = Algebras(QQ).WithBasis().FiniteDimensional(); Bs
Category of finite dimensional algebras with basis over Rational Field
sage: As is Bs
True


On the other hand, axioms do not necessarily commute with functorial constructions, even if the current printout may missuggest so:

sage: As = Algebras(QQ).Graded().WithBasis(); As
Category of graded algebras with basis over Rational Field
Category of graded algebras with basis over Rational Field
sage: As is Bs
False


This is because Bs is the category of algebras endowed with basis, which are further graded; in particular the basis must respect the grading (i.e. be made of homogeneous elements). On the other hand, As is the category of graded algebras, which are further endowed with some basis; that basis need not respect the grading. In fact As is really a join category:

sage: type(As)
<class 'sage.categories.category.JoinCategory_with_category'>
sage: As._repr_(as_join=True)
'Join of Category of algebras with basis over Rational Field and Category of graded algebras over Rational Field'


Todo

Improve the printing of functorial constructions and joins to raise this potentially dangerous ambiguity.

We refer to sage.categories.category_with_axiom for how to implement axioms.

### Wrap-up¶

As we have seen, there is a combinatorial explosion of possible classes. Constructing by hand the full class hierarchy would not scale unless one would restrict to a very rigid subset. Even if it was possible to construct automatically the full hierarchy, this would not scale with respect to system resources.

When designing software systems with large hierarchies of abstract classes for business objects, the difficulty is usually to identify a proper set of key concepts. Here we are lucky, as the key concepts have been long identified and are relatively few:

• Operations ($$+$$, $$*$$, ...)
• Axioms on those operations (associativity, ...)
• Constructions (cartesian products, ...)

Better, those concepts are sufficiently well known so that a user can reasonably be expected to be familiar with the concepts that are involved for his own needs.

Instead, the difficulty is concentrated in the huge number of possible combinations, an unpredictable large subset of which being potentially of interest; at the same time, only a small – but moving – subset has code naturally attached to it.

This has led to the current design, where one focuses on writing the relatively few classes for which there is actual code or mathematical information, and lets Sage compose dynamically and lazily those building blocks to construct the minimal hierarchy of classes needed for the computation at hand. This allows for the infrastructure to scale smoothly as bookshelves are added, extended, or reorganized.

## Writing a new category¶

Each category $$C$$ must be provided with a method C.super_categories() and can be provided with a method C._subcategory_hook_(D). Also, it may be needed to insert $$C$$ into the output of the super_categories() method of some other category. This determines the position of $$C$$ in the category graph.

A category may provide methods that can be used by all its objects, respectively by all elements of its objects.

Each category should come with a good example, in sage.categories.examples.

### Inserting the new category into the category graph¶

C.super_categories() must return a list of categories, namely the immediate super categories of $$C$$. Of course, if you know that your new category $$C$$ is an immediate super category of some existing category $$D$$, then you should also update the method D.super_categories to include $$C$$.

The immediate super categories of $$C$$ should not be join categories. Furthermore, one always should have:

Cs().is_subcategory( Category.join(Cs().super_categories()) )

Cs()._cmp_key  >  other._cmp_key  for other in Cs().super_categories()

This is checked by _test_category().

In several cases, the category $$C$$ is directly provided with a generic implementation of super_categories; a typical example is when $$C$$ implements an axiom or a functorial construction; in such a case, $$C$$ may implement C.extra_super_categories() to complement the super categories discovered by the generic implementation. This method needs not return immediate super categories; instead it’s usually best to specify the largest super category providing the desired mathematical information. For example, the category Magmas.Commutative.Algebras just states that the algebra of a commutative magma is a commutative magma. This is sufficient to let Sage deduce that it’s in fact a commutative algebra.

### Methods for objects and elements¶

Different objects of the same category share some algebraic features, and very often these features can be encoded in a method, in a generic way. For example, for every commutative additive monoid, it makes sense to ask for the sum of a list of elements. Sage’s category framework allows to provide a generic implementation for all objects of a category.

If you want to provide your new category with generic methods for objects (or elements of objects), then you simply add a nested class called ParentMethods (or ElementMethods). The methods of that class will automatically become methods of the objects (or the elements). For instance:

sage: P.<x,y> = ZZ[]
sage: P.prod([x,y,2])
2*x*y
sage: P.prod.__module__
'sage.categories.monoids'
sage: P.prod.__func__ is Monoids().ParentMethods.prod.__func__
True


We recommend to study the code of one example:

sage: C = CommutativeAdditiveMonoids()
sage: C??                               # not tested

### On the order of super categories¶

The generic method C.all_super_categories() determines recursively the list of all super categories of $$C$$.

The order of the categories in this list does influence the inheritance of methods for parents and elements. Namely, if $$P$$ is an object in the category $$C$$ and if $$C_1$$ and $$C_2$$ are both super categories of $$C$$ defining some method foo in ParentMethods, then $$P$$ will use $$C_1$$‘s version of foo if and only if $$C_1$$ appears in C.all_super_categories() before $$C_2$$.

However this must be considered as an implementation detail: if $$C_1$$ and $$C_2$$ are incomparable categories, then the order in which they appear must be mathematically irrelevant: in particular, the methods foo in $$C_1$$ and $$C_2$$ must have the same semantic. Code should not rely on any specific order, as it is subject to later change. Whenever one of the implementations is preferred in some common subcategory of $$C_1$$ and $$C_2$$, for example for efficiency reasons, the ambiguity should be resolved explicitly by definining a method foo in this category. See the method some_elements in the code of the category FiniteCoxeterGroups for an example.

Since trac ticket #11943, C.all_super_categories() is computed by the so-called C3 algorithm used by Python to compute Method Resolution Order of new-style classes. Thus the order in C.all_super_categories(), C.parent_class.mro() and C.element_class.mro() are guaranteed to be consistent.

Since trac ticket #13589, the C3 algorithm is put under control of some total order on categories. This order is not necessarily meaningful, but it guarantees that C3 always finds a consistent Method Resolution Order. For background, see sage.misc.c3_controlled. A visible effect is that the order in which categories are specified in C.super_categories(), or in a join category, no longer influences the result of C.all_super_categories().

### Subcategory hook (advanced optimization feature)¶

The default implementation of the method C.is_subcategory(D) is to look up whether $$D$$ appears in C.all_super_categories(). However, building the list of all the super categories of $$C$$ is an expensive operation that is sometimes best avoided. For example, if both $$C$$ and $$D$$ are categories defined over a base, but the bases differ, then one knows right away that they can not be subcategories of each other.

When such a short-path is known, one can implement a method _subcategory_hook_. Then, C.is_subcategory(D) first calls D._subcategory_hook_(C). If this returns Unknown, then C.is_subcategory(D) tries to find D in C.all_super_categories(). Otherwise, C.is_subcategory(D) returns the result of D._subcategory_hook_(C).

By default, D._subcategory_hook_(C) tests whether issubclass(C.parent_class,D.parent_class), which is very often giving the right answer:

sage: Rings()._subcategory_hook_(Algebras(QQ))
True
sage: HopfAlgebras(QQ)._subcategory_hook_(Algebras(QQ))
False
sage: Algebras(QQ)._subcategory_hook_(HopfAlgebras(QQ))
True
`