Axioms¶
This documentation covers how to implement axioms and proceeds with an overview of the implementation of the axiom infrastructure. It assumes that the reader is familiar with the category primer, and in particular its section about axioms.
Implementing axioms¶
Simple case involving a single predefined axiom¶
Suppose that one wants to provide code (and documentation, tests, …)
for the objects of some existing category Cs()
that satisfy some
predefined axiom A
.
The first step is to open the hood and check whether there already
exists a class implementing the category Cs().A()
. For example,
taking Cs=Semigroups
and the Finite
axiom, there already
exists a class for the category of finite semigroups:
sage: Semigroups().Finite()
Category of finite semigroups
sage: type(Semigroups().Finite())
<class 'sage.categories.finite_semigroups.FiniteSemigroups_with_category'>
In this case, we say that the category of semigroups implements the
axiom Finite
, and code about finite semigroups should go in the
class FiniteSemigroups
(or, as usual, in its nested classes
ParentMethods
, ElementMethods
, and so on).
On the other hand, there is no class for the category of infinite semigroups:
sage: Semigroups().Infinite()
Category of infinite semigroups
sage: type(Semigroups().Infinite())
<class 'sage.categories.category.JoinCategory_with_category'>
This category is indeed just constructed as the intersection of the categories of semigroups and of infinite sets respectively:
sage: Semigroups().Infinite().super_categories()
[Category of semigroups, Category of infinite sets]
In this case, one needs to create a new class to implement the axiom
Infinite
for this category. This boils down to adding a nested
class Semigroups.Infinite
inheriting from CategoryWithAxiom
.
In the following example, we implement a category Cs
, with a
subcategory for the objects satisfying the Finite
axiom defined in
the super category Sets
(we will see later on how to define new
axioms):
sage: from sage.categories.category_with_axiom import CategoryWithAxiom
sage: class Cs(Category):
....: def super_categories(self):
....: return [Sets()]
....: class Finite(CategoryWithAxiom):
....: class ParentMethods:
....: def foo(self):
....: print("I am a method on finite C's")
sage: Cs().Finite()
Category of finite cs
sage: Cs().Finite().super_categories()
[Category of finite sets, Category of cs]
sage: Cs().Finite().all_super_categories()
[Category of finite cs, Category of finite sets,
Category of cs, Category of sets, ...]
sage: Cs().Finite().axioms()
frozenset({'Finite'})
Now a parent declared in the category Cs().Finite()
inherits from
all the methods of finite sets and of finite \(C\)’s, as desired:
sage: P = Parent(category=Cs().Finite())
sage: P.is_finite() # Provided by Sets.Finite.ParentMethods
True
sage: P.foo() # Provided by Cs.Finite.ParentMethods
I am a method on finite C's
Note
This follows the same idiom as for Covariant Functorial Constructions.
From an object oriented point of view, any subcategory
Cs()
ofSets
inherits aFinite
method. UsuallyCs
could complement this method by overriding it with a methodCs.Finite
which would make a super call toSets.Finite
and then do extra stuff.In the above example,
Cs
also wants to complementSets.Finite
, though not by doing more stuff, but by providing it with an additional mixin class containing the code for finiteCs
. To keep the analogy, this mixin class is to be put inCs.Finite
.By defining the axiom
Finite
,Sets
fixes the semantic ofCs.Finite()
for all its subcategoriesCs
: namely “the category ofCs
which are finite as sets”. Hence, for example,Modules.Free.Finite
cannot be used to model the category of free modules of finite rank, even though their traditional name “finite free modules” might suggest it.It may come as a surprise that we can actually use the same name
Finite
for the mixin class and for the method defining the axiom; indeed, by default a class does not have a binding behavior and would completely override the method. See the section Defining a new axiom for details and the rationale behind it.An alternative would have been to give another name to the mixin class, like
FiniteCategory
. However this would have resulted in more namespace pollution, whereas usingFinite
is already clear, explicit, and easier to remember.Under the hood, the category
Cs().Finite()
is aware that it has been constructed from the categoryCs()
by adding the axiomFinite
:sage: Cs().Finite().base_category() Category of cs sage: Cs().Finite()._axiom 'Finite'
Over time, the nested class Cs.Finite
may become large and too
cumbersome to keep as a nested subclass of Cs
. Or the category with
axiom may have a name of its own in the literature, like semigroups
rather than associative magmas, or fields rather than commutative
division rings. In this case, the category with axiom can be put
elsewhere, typically in a separate file, with just a link from
Cs
:
sage: class Cs(Category):
....: def super_categories(self):
....: return [Sets()]
sage: class FiniteCs(CategoryWithAxiom):
....: class ParentMethods:
....: def foo(self):
....: print("I am a method on finite C's")
sage: Cs.Finite = FiniteCs
sage: Cs().Finite()
Category of finite cs
For a real example, see the code of the class FiniteGroups
and the
link to it in Groups
. Note that the link is implemented using
LazyImport
; this is highly recommended: it
makes sure that FiniteGroups
is imported after Groups
it
depends upon, and makes it explicit that the class Groups
can be
imported and is fully functional without importing FiniteGroups
.
Note
Some categories with axioms are created upon Sage’s startup. In such a
case, one needs to pass the at_startup=True
option to
LazyImport
, in order to quiet the warning
about that lazy import being resolved upon startup. See for example
Sets.Finite
.
This is undoubtedly a code smell. Nevertheless, it is preferable
to stick to lazy imports, first to resolve the import order
properly, and more importantly as a reminder that the category
would be best not constructed upon Sage’s startup. This is to spur
developers to reduce the number of parents (and therefore
categories) that are constructed upon startup. Each
at_startup=True
that will be removed will be a measure of
progress in this direction.
Note
In principle, due to a limitation of
LazyImport
with nested classes (see
trac ticket #15648), one should pass the option as_name
to
LazyImport
:
Finite = LazyImport('sage.categories.finite_groups', 'FiniteGroups', as_name='Finite')
in order to prevent Groups.Finite
to keep on reimporting
FiniteGroups
.
Given that passing this option introduces some redundancy and is
error prone, the axiom infrastructure includes a little workaround
which makes the as_name
unnecessary in this case.
Making the category with axiom directly callable¶
If desired, a category with axiom can be constructed directly through its class rather than through its base category:
sage: Semigroups()
Category of semigroups
sage: Semigroups() is Magmas().Associative()
True
sage: FiniteGroups()
Category of finite groups
sage: FiniteGroups() is Groups().Finite()
True
For this notation to work, the class Semigroups
needs to be
aware of the base category class (here, Magmas
) and of the
axiom (here, Associative
):
sage: Semigroups._base_category_class_and_axiom
(<class 'sage.categories.magmas.Magmas'>, 'Associative')
sage: Fields._base_category_class_and_axiom
(<class 'sage.categories.division_rings.DivisionRings'>, 'Commutative')
sage: FiniteGroups._base_category_class_and_axiom
(<class 'sage.categories.groups.Groups'>, 'Finite')
sage: FiniteDimensionalAlgebrasWithBasis._base_category_class_and_axiom
(<class 'sage.categories.algebras_with_basis.AlgebrasWithBasis'>, 'FiniteDimensional')
In our example, the attribute _base_category_class_and_axiom
was
set upon calling Cs().Finite()
, which makes the notation seemingly
work:
sage: FiniteCs()
Category of finite cs
sage: FiniteCs._base_category_class_and_axiom
(<class '__main__.Cs'>, 'Finite')
sage: FiniteCs._base_category_class_and_axiom_origin
'set by __classget__'
But calling FiniteCs()
right after defining the class would have
failed (try it!). In general, one needs to set the attribute explicitly:
sage: class FiniteCs(CategoryWithAxiom):
....: _base_category_class_and_axiom = (Cs, 'Finite')
....: class ParentMethods:
....: def foo(self):
....: print("I am a method on finite C's")
Having to set explicitly this link back from FiniteCs
to Cs
introduces redundancy in the code. It would therefore be desirable to
have the infrastructure set the link automatically instead (a
difficulty is to achieve this while supporting lazy imported
categories with axiom).
As a first step, the link is set automatically upon accessing the class from the base category class:
sage: Algebras.WithBasis._base_category_class_and_axiom
(<class 'sage.categories.algebras.Algebras'>, 'WithBasis')
sage: Algebras.WithBasis._base_category_class_and_axiom_origin
'set by __classget__'
Hence, for whatever this notation is worth, one can currently do:
sage: Algebras.WithBasis(QQ)
Category of algebras with basis over Rational Field
We don’t recommend using syntax like Algebras.WithBasis(QQ)
, as it
may eventually be deprecated.
As a second step, Sage tries some obvious heuristics to deduce the link
from the name of the category with axiom (see
base_category_class_and_axiom()
for the details). This typically
covers the following examples:
sage: FiniteCoxeterGroups()
Category of finite coxeter groups
sage: FiniteCoxeterGroups() is CoxeterGroups().Finite()
True
sage: FiniteCoxeterGroups._base_category_class_and_axiom_origin
'deduced by base_category_class_and_axiom'
sage: FiniteDimensionalAlgebrasWithBasis(QQ)
Category of finite dimensional algebras with basis over Rational Field
sage: FiniteDimensionalAlgebrasWithBasis(QQ) is Algebras(QQ).FiniteDimensional().WithBasis()
True
If the heuristic succeeds, the result is guaranteed to be correct. If
it fails, typically because the category has a name of its own like
Fields
, the attribute _base_category_class_and_axiom
should be set explicitly. For more examples, see the code of the
classes Semigroups
or Fields
.
Note
When printing out a category with axiom, the heuristic determines
whether a category has a name of its own by checking out how
_base_category_class_and_axiom
was set:
sage: Fields._base_category_class_and_axiom_origin
'hardcoded'
See CategoryWithAxiom._without_axioms()
,
CategoryWithAxiom._repr_object_names_static()
.
In our running example FiniteCs
, Sage failed to deduce
automatically the base category class and axiom because the class
Cs
is not in the standard location sage.categories.cs
.
Design discussion
The above deduction, based on names, is undoubtedly inelegant. But
it’s safe (either the result is guaranteed to be correct, or an
error is raised), it saves on some redundant information, and it
is only used for the simple shorthands like FiniteGroups()
for
Groups().Finite()
. Finally, most if not all of these
shorthands are likely to eventually disappear (see trac ticket #15741
and the related discussion in the primer).
Defining a new axiom¶
We describe now how to define a new axiom. The first step is to figure
out the largest category where the axiom makes sense. For example
Sets
for Finite
, Magmas
for Associative
, or
Modules
for FiniteDimensional
. Here we define the axiom
Green
for the category Cs
and its subcategories:
sage: from sage.categories.category_with_axiom import CategoryWithAxiom
sage: class Cs(Category):
....: def super_categories(self):
....: return [Sets()]
....: class SubcategoryMethods:
....: def Green(self):
....: '<documentation of the axiom Green>'
....: return self._with_axiom("Green")
....: class Green(CategoryWithAxiom):
....: class ParentMethods:
....: def foo(self):
....: print("I am a method on green C's")
With the current implementation, the name of the axiom must also be added to a global container:
sage: all_axioms = sage.categories.category_with_axiom.all_axioms
sage: all_axioms += ("Green",)
We can now use the axiom as usual:
sage: Cs().Green()
Category of green cs
sage: P = Parent(category=Cs().Green())
sage: P.foo()
I am a method on green C's
Compared with our first example, the only newcomer is the method
.Green()
that can be used by any subcategory Ds()
of Cs()
to add the axiom Green
. Note that the expression Ds().Green
always evaluates to this method, regardless of whether Ds
has a
nested class Ds.Green
or not (an implementation detail):
sage: Cs().Green
<bound method Cs_with_category.Green of Category of cs>
Thanks to this feature (implemented in CategoryWithAxiom.__classget__()
),
the user is systematically referred to the documentation of this
method when doing introspection on Ds().Green
:
sage: C = Cs()
sage: C.Green? # not tested
sage: Cs().Green.__doc__
'<documentation of the axiom Green>'
It is therefore the natural spot for the documentation of the axiom.
Note
The presence of the nested class Green
in Cs
is currently
mandatory even if it is empty.
Todo
Specify whether or not one should systematically use @cached_method in the definition of the axiom. And make sure all the definition of axioms in Sage are consistent in this respect!
Todo
We could possibly define an @axiom decorator? This could hide two little implementation details: whether or not to make the method a cached method, and the call to _with_axiom(…) under the hood. It could do possibly do some more magic. The gain is not obvious though.
Note
all_axioms
is only used marginally, for sanity checks and when
trying to derive automatically the base category class. The order
of the axioms in this tuple also controls the order in which they
appear when printing out categories with axioms (see
CategoryWithAxiom._repr_object_names_static()
).
During a Sage session, new axioms should only be added at the end
of all_axioms
, as above, so as to not break the cache of
axioms_rank()
. Otherwise, they can be inserted statically
anywhere in the tuple. For axioms defined within the Sage library,
the name is best inserted by editing directly the definition of
all_axioms
in sage.categories.category_with_axiom
.
Design note
Let us state again that, unlike what the existence of
all_axioms
might suggest, the definition of an axiom is local
to a category and its subcategories. In particular, two
independent categories Cs()
and Ds()
can very well define
axioms with the same name and different semantics. As long as the
two hierarchies of subcategories don’t intersect, this is not a
problem. And if they do intersect naturally (that is if one is
likely to create a parent belonging to both categories), this
probably means that the categories Cs
and Ds
are about
related enough areas of mathematics that one should clear the
ambiguity by having either the same semantic or different names.
This caveat is no different from that of name clashes in hierarchy of classes involving multiple inheritance.
Todo
Explore ways to get rid of this global all_axioms
tuple,
and/or have automatic registration there, and/or having a
register_axiom(…) method.
Special case: defining an axiom depending on several categories¶
In some cases, the largest category where the axiom makes sense is the
intersection of two categories. This is typically the case for axioms
specifying compatibility conditions between two otherwise unrelated
operations, like Distributive
which specifies a compatibility
between \(*\) and \(+\). Ideally, we would want the Distributive
axiom
to be defined by:
sage: Magmas() & AdditiveMagmas()
Join of Category of magmas and Category of additive magmas
The current infrastructure does not support this perfectly: indeed,
defining an axiom for a category \(C\) requires \(C\) to have a class of
its own; hence a JoinCategory
as above won’t do;
we need to implement a new class like
MagmasAndAdditiveMagmas
;
furthermore, we cannot yet model the fact that MagmasAndAdditiveMagmas()
is the intersection of Magmas()
and AdditiveMagmas()
rather than a
mere subcategory:
sage: from sage.categories.magmas_and_additive_magmas import MagmasAndAdditiveMagmas
sage: Magmas() & AdditiveMagmas() is MagmasAndAdditiveMagmas()
False
sage: Magmas() & AdditiveMagmas() # todo: not implemented
Category of magmas and additive magmas
Still, there is a workaround to get the natural notations:
sage: (Magmas() & AdditiveMagmas()).Distributive()
Category of distributive magmas and additive magmas
sage: (Monoids() & CommutativeAdditiveGroups()).Distributive()
Category of rings
The trick is to define Distributive
as usual in
MagmasAndAdditiveMagmas
, and to
add a method Magmas.SubcategoryMethods.Distributive()
which
checks that self
is a subcategory of both Magmas()
and
AdditiveMagmas()
, complains if not, and otherwise takes the
intersection of self
with MagmasAndAdditiveMagmas()
before
calling Distributive
.
The downsides of this workaround are:
Creation of an otherwise empty class
MagmasAndAdditiveMagmas
.Pollution of the namespace of
Magmas()
(and subcategories likeGroups()
) with a method that is irrelevant (but safely complains if called).C._with_axiom('Distributive')
is not strictly equivalent toC.Distributive()
, which can be unpleasantly surprising:sage: (Monoids() & CommutativeAdditiveGroups()).Distributive() Category of rings sage: (Monoids() & CommutativeAdditiveGroups())._with_axiom('Distributive') Join of Category of monoids and Category of commutative additive groups
Todo
Other categories that would be better implemented via an axiom depending on a join category include:
Algebras
: defining an associative unital algebra as a ring and a module satisfying the suitable compatibility axiom between inner multiplication and multiplication by scalars (bilinearity). Of course this should be implemented at the level ofMagmaticAlgebras
, if not higher.Bialgebras
: defining an bialgebra as an algebra and coalgebra where the coproduct is a morphism for the product.Bimodules
: defining a bimodule as a left and right module where the two actions commute.
Todo
 Design and implement an idiom for the definition of an axiom by a join category.
 Or support more advanced joins, through some hook or registration process to specify that a given category is the intersection of two (or more) categories.
 Or at least improve the above workaround to avoid the last issue; this
possibly could be achieved using a class
Magmas.Distributive
with a bit of__classcall__
magic.
Handling multiple axioms, arborescence structure of the code¶
Prelude¶
Let us consider the category of magmas, together with two of its
axioms, namely Associative
and Unital
. An associative magma is
a semigroup and a unital semigroup is a monoid. We have also seen
that axioms commute:
sage: Magmas().Unital()
Category of unital magmas
sage: Magmas().Associative()
Category of semigroups
sage: Magmas().Associative().Unital()
Category of monoids
sage: Magmas().Unital().Associative()
Category of monoids
At the level of the classes implementing these categories, the following comes as a general naturalization of the previous section:
sage: Magmas.Unital
<class 'sage.categories.magmas.Magmas.Unital'>
sage: Magmas.Associative
<class 'sage.categories.semigroups.Semigroups'>
sage: Magmas.Associative.Unital
<class 'sage.categories.monoids.Monoids'>
However, the following may look suspicious at first:
sage: Magmas.Unital.Associative
Traceback (most recent call last):
...
AttributeError: type object 'Magmas.Unital' has no attribute 'Associative'
The purpose of this section is to explain the design of the code layout and the rationale for this mismatch.
Abstract model¶
As we have seen in the Primer,
the objects of a category Cs()
can usually satisfy, or not, many
different axioms. Out of all combinations of axioms, only a small
number are relevant in practice, in the sense that we actually want to
provide features for the objects satisfying these axioms.
Therefore, in the context of the category class Cs
, we want to
provide the system with a collection \((D_S)_{S\in \mathcal S}\) where
each \(S\) is a subset of the axioms and the corresponding \(D_S\) is a
class for the subcategory of the objects of Cs()
satisfying the
axioms in \(S\). For example, if Cs()
is the category of magmas, the
pairs \((S, D_S)\) would include:
{Associative} : Semigroups
{Associative, Unital} : Monoids
{Associative, Unital, Inverse}: Groups
{Associative, Commutative} : Commutative Semigroups
{Unital, Inverse} : Loops
Then, given a subset \(T\) of axioms, we want the system to be able to
select automatically the relevant classes
\((D_S)_{S\in \mathcal S, S\subset T}\),
and build from them a category for the objects of Cs
satisfying
the axioms in \(T\), together with its hierarchy of super categories. If
\(T\) is in the indexing set \(\mathcal S\), then the class of the
resulting category is directly \(D_T\):
sage: C = Magmas().Unital().Inverse().Associative(); C
Category of groups
sage: type(C)
<class 'sage.categories.groups.Groups_with_category'>
Otherwise, we get a join category:
sage: C = Magmas().Infinite().Unital().Associative(); C
Category of infinite monoids
sage: type(C)
<class 'sage.categories.category.JoinCategory_with_category'>
sage: C.super_categories()
[Category of monoids, Category of infinite sets]
Concrete model as an arborescence of nested classes¶
We further want the construction to be efficient and amenable to
laziness. This led us to the following design decision: the collection
\((D_S)_{S\in \mathcal S}\) of classes should be structured as an
arborescence (or equivalently a rooted forest). The root is Cs
,
corresponding to \(S=\emptyset\). Any other class \(D_S\) should be the
child of a single class \(D_{S'}\) where \(S'\) is obtained from \(S\) by
removing a single axiom \(A\). Of course, \(D_{S'}\) and \(A\) are
respectively the base category class and axiom of the category with
axiom \(D_S\) that we have met in the first section.
At this point, we urge the reader to explore the code of
Magmas
and
DistributiveMagmasAndAdditiveMagmas
and see how the arborescence structure on the categories with axioms
is reflected by the nesting of category classes.
Discussion of the design¶
Performance¶
Thanks to the arborescence structure on subsets of axioms, constructing the hierarchy of categories and computing intersections can be made efficient with, roughly speaking, a linear/quadratic complexity in the size of the involved category hierarchy multiplied by the number of axioms (see Section Algorithms). This is to be put in perspective with the manipulation of arbitrary collections of subsets (aka boolean functions) which can easily raise NPhard problems.
Furthermore, thanks to its locality, the algorithms can be made suitably lazy: in particular, only the involved category classes need to be imported.
Flexibility¶
This design also brings in quite some flexibility, with the possibility to support features such as defining new axioms depending on other axioms and deduction rules. See below.
Asymmetry¶
As we have seen at the beginning of this section, this design
introduces an asymmetry. It’s not so bad in practice, since in most
practical cases, we want to work incrementally. It’s for example more
natural to describe FiniteFields
as Fields
with the
axiom Finite
rather than Magmas
and
AdditiveMagmas
with all (or at least sufficiently many) of
the following axioms:
sage: sorted(Fields().axioms())
['AdditiveAssociative', 'AdditiveCommutative', 'AdditiveInverse',
'AdditiveUnital', 'Associative', 'Commutative', 'Distributive',
'Division', 'NoZeroDivisors', 'Unital']
The main limitation is that the infrastructure currently imposes to be incremental by steps of a single axiom.
In practice, among the roughly 60 categories with axioms that are currently implemented in Sage, most admitted a (rather) natural choice of a base category and single axiom to add. For example, one usually thinks more naturally of a monoid as a semigroup which is unital rather than as a unital magma which is associative. Modeling this asymmetry in the code actually brings a bonus: it is used for printing out categories in a (heuristically) mathematicianfriendly way:
sage: Magmas().Commutative().Associative()
Category of commutative semigroups
Only in a few cases is a choice made that feels mathematically
arbitrary. This is essentially in the chain of nested classes
distributive_magmas_and_additive_magmas.DistributiveMagmasAndAdditiveMagmas.AdditiveAssociative.AdditiveCommutative.AdditiveUnital.Associative
.
Placeholder classes¶
Given that we can only add a single axiom at a time when implementing
a CategoryWithAxiom
, we need to create a few category classes
that are just placeholders. For the worst example, see the chain of
nested classes
distributive_magmas_and_additive_magmas.DistributiveMagmasAndAdditiveMagmas.AdditiveAssociative.AdditiveCommutative.AdditiveUnital.Associative
.
This is suboptimal, but fits within the scope of the axiom infrastructure which is to reduce a potentially exponential number of placeholder category classes to just a couple.
Note also that, in the above example, it’s likely that some of the intermediate classes will grow to non placeholder ones, as people will explore more weaker variants of rings.
Mismatch between the arborescence of nested classes and the hierarchy of categories¶
The fact that the hierarchy relation between categories is not reflected directly as a relation between the classes may sound suspicious at first! However, as mentioned in the primer, this is actually a big selling point of the axioms infrastructure: by calculating automatically the hierarchy relation between categories with axioms one avoids the nightmare of maintaining it by hand. Instead, only a rather minimal number of links needs to be maintainted in the code (one per category with axiom).
Besides, with the flexibility introduced by runtime deduction rules (see below), the hierarchy of categories may depend on the parameters of the categories and not just their class. So it’s fine to make it clear from the onset that the two relations do not match.
Evolutivity¶
At this point, the arborescence structure has to be hardcoded by hand with the annoyances we have seen. This does not preclude, in a future iteration, to design and implement some idiom for categories with axioms that adds several axioms at once to a base category; maybe some variation around:
class DistributiveMagmasAndAdditiveMagmas:
...
@category_with_axiom(
AdditiveAssociative,
AdditiveCommutative,
AdditiveUnital,
AdditiveInverse,
Associative)
def _(): return LazyImport('sage.categories.rngs', 'Rngs', at_startup=True)
or:
register_axiom_category(DistributiveMagmasAndAdditiveMagmas,
{AdditiveAssociative,
AdditiveCommutative,
AdditiveUnital,
AdditiveInverse,
Associative},
'sage.categories.rngs', 'Rngs', at_startup=True)
The infrastructure would then be in charge of building the appropriate arborescence under the hood. Or rely on some database (see discussion on trac ticket #10963, in particular at the end of comment 332).
Axioms defined upon other axioms¶
Sometimes an axiom can only be defined when some other axiom
holds. For example, the axiom NoZeroDivisors
only makes sense if
there is a zero, that is if the axiom AdditiveUnital
holds. Hence,
for the category
MagmasAndAdditiveMagmas
, we
consider in the abstract model only those subsets of axioms where the
presence of NoZeroDivisors
implies that of AdditiveUnital
. We
also want the axiom to be only available if meaningful:
sage: Rings().NoZeroDivisors()
Category of domains
sage: Rings().Commutative().NoZeroDivisors()
Category of integral domains
sage: Semirings().NoZeroDivisors()
Traceback (most recent call last):
...
AttributeError: 'Semirings_with_category' object has no attribute 'NoZeroDivisors'
Concretely, this is to be implemented by defining the new axiom in the
(SubcategoryMethods
nested class of the) appropriate category with
axiom. For example the axiom NoZeroDivisors
would be naturally
defined in
magmas_and_additive_magmas.MagmasAndAdditiveMagmas.Distributive.AdditiveUnital
.
Note
The axiom NoZeroDivisors
is currently defined in
Rings
, by simple lack of need for the feature; it should
be lifted up as soon as relevant, that is when some code will be
available for parents with no zero divisors that are not
necessarily rings.
Deduction rules¶
A similar situation is when an axiom A
of a category Cs
implies some other axiom B
, with the same consequence as above on
the subsets of axioms appearing in the abstract model. For example, a
division ring necessarily has no zero divisors:
sage: 'NoZeroDivisors' in Rings().Division().axioms()
True
sage: 'NoZeroDivisors' in Rings().axioms()
False
This deduction rule is implemented by the method
Rings.Division.extra_super_categories()
:
sage: Rings().Division().extra_super_categories()
(Category of domains,)
In general, this is to be implemented by a method
Cs.A.extra_super_categories
returning a tuple (Cs().B(),)
, or
preferably (Ds().B(),)
where Ds
is the category defining the
axiom B
.
This follows the same idiom as for deduction rules about functorial
constructions (see covariant_functorial_construction.CovariantConstructionCategory.extra_super_categories()
).
For example, the fact that a Cartesian product of associative magmas
(i.e. of semigroups) is an associative magma is implemented in
Semigroups.CartesianProducts.extra_super_categories()
:
sage: Magmas().Associative()
Category of semigroups
sage: Magmas().Associative().CartesianProducts().extra_super_categories()
[Category of semigroups]
Similarly, the fact that the algebra of a commutative magma is
commutative is implemented in
Magmas.Commutative.Algebras.extra_super_categories()
:
sage: Magmas().Commutative().Algebras(QQ).extra_super_categories()
[Category of commutative magmas]
Warning
In some situations this idiom is inapplicable as it would require to implement two classes for the same category. This is the purpose of the next section.
Special case¶
In the previous examples, the deduction rule only had an influence on
the super categories of the category with axiom being constructed. For
example, when constructing Rings().Division()
, the rule
Rings.Division.extra_super_categories()
simply adds
Rings().NoZeroDivisors()
as a super category thereof.
In some situations this idiom is inapplicable because a class for the
category with axiom under construction already exists elsewhere. Take
for example Wedderburn’s theorem: any finite division ring is
commutative, i.e. is a finite field. In other words,
DivisionRings().Finite()
coincides with Fields().Finite()
:
sage: DivisionRings().Finite()
Category of finite enumerated fields
sage: DivisionRings().Finite() is Fields().Finite()
True
Therefore we cannot create a class DivisionRings.Finite
to hold
the desired extra_super_categories
method, because there is
already a class for this category with axiom, namely
Fields.Finite
.
A natural idiom would be to have DivisionRings.Finite
be a link to
Fields.Finite
(locally introducing an undirected cycle in the
arborescence of nested classes). It would be a bit tricky to implement
though, since one would need to detect, upon constructing
DivisionRings().Finite()
, that DivisionRings.Finite
is
actually Fields.Finite
, in order to construct appropriately
Fields().Finite()
; and reciprocally, upon computing the super
categories of Fields().Finite()
, to not try to add
DivisionRings().Finite()
as a super category.
Instead the current idiom is to have a method
DivisionRings.Finite_extra_super_categories
which mimicks the
behavior of the wouldbe
DivisionRings.Finite.extra_super_categories
:
sage: DivisionRings().Finite_extra_super_categories()
(Category of commutative magmas,)
This idiom is admittedly rudimentary, but consistent with how
mathematical facts specifying non trivial inclusion relations between
categories are implemented elsewhere in the various
extra_super_categories
methods of axiom categories and covariant
functorial constructions. Besides, it gives a natural spot (the
docstring of the method) to document and test the modeling of the
mathematical fact. Finally, Wedderburn’s theorem is arguably a theorem
about division rings (in the context of division rings, finiteness
implies commutativity) and therefore lives naturally in
DivisionRings
.
An alternative would be to implement the category of finite division
rings (i.e. finite fields) in a class DivisionRings.Finite
rather
than Fields.Finite
:
sage: from sage.categories.category_with_axiom import CategoryWithAxiom
sage: class MyDivisionRings(Category):
....: def super_categories(self):
....: return [Rings()]
sage: class MyFields(Category):
....: def super_categories(self):
....: return [MyDivisionRings()]
sage: class MyFiniteFields(CategoryWithAxiom):
....: _base_category_class_and_axiom = (MyDivisionRings, "Finite")
....: def extra_super_categories(self): # Wedderburn's theorem
....: return [MyFields()]
sage: MyDivisionRings.Finite = MyFiniteFields
sage: MyDivisionRings().Finite()
Category of my finite fields
sage: MyFields().Finite() is MyDivisionRings().Finite()
True
In general, if several categories C1s()
, C2s()
, … are mapped to
the same category when applying some axiom A
(that is C1s().A()
== C2s().A() == ...
), then one should be careful to implement this
category in a single class Cs.A
, and set up methods
extra_super_categories
or A_extra_super_categories
methods as
appropriate. Each such method should return something like
[C2s()]
and not [C2s().A()]
for the latter would likely lead
to an infinite recursion.
Design discussion
Supporting similar deduction rules will be an important feature in the future, with quite a few occurrences already implemented in upcoming tickets. For the time being though there is a single occurrence of this idiom outside of the tests. So this would be an easy thing to refactor after trac ticket #10963 if a better idiom is found.
Larger synthetic examples¶
We now consider some larger synthetic examples to check that the
machinery works as expected. Let us start with a category defining a
bunch of axioms, using axiom()
for conciseness (don’t do it for
real axioms; they deserve a full documentation!):
sage: from sage.categories.category_singleton import Category_singleton
sage: from sage.categories.category_with_axiom import axiom
sage: import sage.categories.category_with_axiom
sage: all_axioms = sage.categories.category_with_axiom.all_axioms
sage: all_axioms += ("B","C","D","E","F")
sage: class As(Category_singleton):
....: def super_categories(self):
....: return [Objects()]
....:
....: class SubcategoryMethods:
....: B = axiom("B")
....: C = axiom("C")
....: D = axiom("D")
....: E = axiom("E")
....: F = axiom("F")
....:
....: class B(CategoryWithAxiom):
....: pass
....: class C(CategoryWithAxiom):
....: pass
....: class D(CategoryWithAxiom):
....: pass
....: class E(CategoryWithAxiom):
....: pass
....: class F(CategoryWithAxiom):
....: pass
Now we construct a subcategory where, by some theorem of William,
axioms B
and C
together are equivalent to E
and F
together:
sage: class A1s(Category_singleton):
....: def super_categories(self):
....: return [As()]
....:
....: class B(CategoryWithAxiom):
....: def C_extra_super_categories(self):
....: return [As().E(), As().F()]
....:
....: class E(CategoryWithAxiom):
....: def F_extra_super_categories(self):
....: return [As().B(), As().C()]
sage: A1s().B().C()
Category of e f a1s
The axioms B
and C
do not show up in the name of the obtained
category because, for concision, the printing uses some heuristics to
not show axioms that are implied by others. But they are satisfied:
sage: sorted(A1s().B().C().axioms())
['B', 'C', 'E', 'F']
Note also that this is a join category:
sage: type(A1s().B().C())
<class 'sage.categories.category.JoinCategory_with_category'>
sage: A1s().B().C().super_categories()
[Category of e a1s,
Category of f as,
Category of b a1s,
Category of c as]
As desired, William’s theorem holds:
sage: A1s().B().C() is A1s().E().F()
True
and propagates appropriately to subcategories:
sage: C = A1s().E().F().D().B().C()
sage: C is A1s().B().C().E().F().D() # commutativity
True
sage: C is A1s().E().F().E().F().D() # William's theorem
True
sage: C is A1s().E().E().F().F().D() # commutativity
True
sage: C is A1s().E().F().D() # idempotency
True
sage: C is A1s().D().E().F()
True
In this quick variant, we actually implement the category of b c
a2s
, and choose to do so in A2s.B.C
:
sage: class A2s(Category_singleton):
....: def super_categories(self):
....: return [As()]
....:
....: class B(CategoryWithAxiom):
....: class C(CategoryWithAxiom):
....: def extra_super_categories(self):
....: return [As().E(), As().F()]
....:
....: class E(CategoryWithAxiom):
....: def F_extra_super_categories(self):
....: return [As().B(), As().C()]
sage: A2s().B().C()
Category of e f a2s
sage: sorted(A2s().B().C().axioms())
['B', 'C', 'E', 'F']
sage: type(A2s().B().C())
<class '__main__.A2s.B.C_with_category'>
As desired, William’s theorem and its consequences hold:
sage: A2s().B().C() is A2s().E().F()
True
sage: C = A2s().E().F().D().B().C()
sage: C is A2s().B().C().E().F().D() # commutativity
True
sage: C is A2s().E().F().E().F().D() # William's theorem
True
sage: C is A2s().E().E().F().F().D() # commutativity
True
sage: C is A2s().E().F().D() # idempotency
True
sage: C is A2s().D().E().F()
True
Finally, we “accidentally” implement the category of b c a1s
, both
in A3s.B.C
and A3s.E.F
:
sage: class A3s(Category_singleton):
....: def super_categories(self):
....: return [As()]
....:
....: class B(CategoryWithAxiom):
....: class C(CategoryWithAxiom):
....: def extra_super_categories(self):
....: return [As().E(), As().F()]
....:
....: class E(CategoryWithAxiom):
....: class F(CategoryWithAxiom):
....: def extra_super_categories(self):
....: return [As().B(), As().C()]
We can still construct, say:
sage: A3s().B()
Category of b a3s
sage: A3s().C()
Category of c a3s
However,
sage: A3s().B().C() # not tested
runs into an infinite recursion loop, as A3s().B().C()
wants to
have A3s().E().F()
as super category and reciprocally.
Todo
The above example violates the specifications (a category should be modelled by at most one class), so it’s appropriate that it fails. Yet, the error message could be usefully complemented by some hint at what the source of the problem is (a category implemented in two distinct classes). Leaving a large enough piece of the backtrace would be useful though, so that one can explore where the issue comes from (e.g. with post mortem debugging).
Specifications¶
After fixing some vocabulary, we summarize here some specifications about categories and axioms.
The lattice of constructible categories¶
A mathematical category \(C\) is implemented if there is a class in
Sage modelling it; it is constructible if it is either implemented,
or is the intersection of implemented categories; in the latter case
it is modelled by a JoinCategory
. The comparison of two
constructible categories with the Category.is_subcategory()
method is supposed to model the comparison of the corresponding
mathematical categories for inclusion of the objects (see
On the category hierarchy: subcategories and super categories for details). For example:
sage: Fields().is_subcategory(Rings())
True
However this modelling may be incomplete. It can happen that a mathematical fact implying that a category \(A\) is a subcategory of a category \(B\) is not implemented. Still, the comparison should endow the set of constructible categories with a poset structure and in fact a lattice structure.
In this lattice, the join of two categories (Category.join()
) is
supposed to model their intersection. Given that we compare categories
for inclusion, it would be more natural to call this operation the
meet; blames go to me (Nicolas) for originally comparing categories
by amount of structure rather than by inclusion. In practice, the
join of two categories may be a strict super category of their
intersection; first because this intersection might not be
constructible; second because Sage might miss some mathematical
information to recover the smallest constructible super category of
the intersection.
Axioms¶
We say that an axiom A
is defined by a category Cs()
if
Cs
defines an appropriate method Cs.SubcategoryMethods.A
, with
the semantic of the axiom specified in the documentation; for any
subcategory Ds()
, Ds().A()
models the subcategory of the
objects of Ds()
satisfying A
. In this case, we say that the
axiom A
is defined for the category Ds()
. Furthermore,
Ds
implements the axiom A
if Ds
has a category with
axiom as nested class Ds.A
. The category Ds()
satisfies the
axiom if Ds()
is a subcategory of Cs().A()
(meaning that all
the objects of Ds()
are known to satisfy the axiom A
).
A digression on the structure of fibers when adding an axiom¶
Consider the application \(\phi_A\) which maps a category to its
category of objects satisfying \(A\). Equivalently, \(\phi_A\) is
computing the intersection with the defining category with axiom of
\(A\). It follows immediately from the latter that \(\phi_A\) is a
regressive endomorphism of the lattice of categories. It restricts
to a regressive endomorphism Cs() > Cs().A()
on the lattice of constructible categories.
This endomorphism may have non trivial fibers, as in our favorite
example: DivisionRings()
and Fields()
are in the same fiber
for the axiom Finite
:
sage: DivisionRings().Finite() is Fields().Finite()
True
Consider the intersection \(S\) of such a fiber of \(\phi_A\) with the
upper set \(I_A\) of categories that do not satisfy A
. The fiber
itself is a sublattice. However \(I_A\) is not guaranteed to be stable
under intersection (though exceptions should be rare). Therefore,
there is a priori no guarantee that \(S\) would be stable under
intersection. Also it’s presumably finite, in fact small, but this is
not guaranteed either.
Specifications¶
Any constructible category
C
should admit a finite number of larger constructible categories.The methods
super_categories
,extra_super_categories
, and friends should always return strict supercategories.For example, to specify that a finite division ring is a finite field,
DivisionRings.Finite_extra_super_categories
should not returnFields().Finite()
! It could possibly returnFields()
; but it’s preferable to return the largest category that contains the relevant information, in this caseMagmas().Commutative()
, and to let the infrastructure apply the derivations.The base category of a
CategoryWithAxiom
should be an implemented category (i.e. not aJoinCategory
). This is checked byCategoryWithAxiom._test_category_with_axiom()
.Arborescent structure: Let
Cs()
be a category, and \(S\) be some set of axioms defined in some super categories ofCs()
but not satisfied byCs()
. Suppose we want to provide a category with axiom for the elements ofCs()
satisfying the axioms in \(S\). Then, there should be a single enumerationA1, A2, ..., Ak
without repetition of axioms in \(S\) such thatCs.A1.A2....Ak
is an implemented category. Furthermore, every intermediate stepCs.A1.A2....Ai
with \(i\leq k\) should be a category with axiom havingAi
as axiom andCs.A1.A2....Ai1
as base category class; this base category class should not satisfyAi
. In particular, when some axioms of \(S\) can be deduced from previous ones by deduction rules, they should not appear in the enumerationA1, A2, ..., Ak
.In particular, if
Cs()
is a category that satisfies some axiomA
(e.g. from one of its super categories), then it should not implement that axiom. For example, a category classCs
can never have a nested classCs.A.A
. Similarly, applying the specification recursively, a category satisfyingA
cannot have a nested classCs.A1.A2.A3.A
whereA1
,A2
,A3
are axioms.A category can only implement an axiom if this axiom is defined by some super category. The code has not been systematically checked to support having two super categories defining the same axiom (which should of course have the same semantic). You are welcome to try, at your own risk. :)
When a category defines an axiom or functorial construction
A
, this fixes the semantic ofA
for all the subcategories. In particular, if two categories defineA
, then these categories should be independent, and either the semantic ofA
should be the same, or there should be no natural intersection between the two hierarchies of subcategories.Any super category of a
CategoryWithParameters
should either be aCategoryWithParameters
or aCategory_singleton
.A
CategoryWithAxiom
having aCategory_singleton
as base category should be aCategoryWithAxiom_singleton
. This is handled automatically byCategoryWithAxiom.__init__()
and checked inCategoryWithAxiom._test_category_with_axiom()
.A
CategoryWithAxiom
having aCategory_over_base_ring
as base category should be aCategory_over_base_ring
. This currently has to be handled by hand, usingCategoryWithAxiom_over_base_ring
. This is checked inCategoryWithAxiom._test_category_with_axiom()
.
Todo
The following specifications would be desirable but are not yet implemented:
A functorial construction category (Graded, CartesianProducts, …) having a
Category_singleton
as base category should be aCategoryWithAxiom_singleton
.Nothing difficult to implement, but this will need to rework the current “no subclass of a concrete class” assertion test of
Category_singleton.__classcall__()
.Similarly, a covariant functorial construction category having a
Category_over_base_ring
as base category should be aCategory_over_base_ring
.
The following specification might be desirable, or not:
 A join category involving a
Category_over_base_ring
should be aCategory_over_base_ring
. In the mean time, abase_ring
method is automatically provided for most of those byModules.SubcategoryMethods.base_ring()
.
Design goals¶
As pointed out in the primer, the main design goal of the axioms infrastructure is to subdue the potential combinatorial explosion of the category hierarchy by letting the developer focus on implementing a few bookshelves for which there is actual code or mathematical information, and let Sage compose dynamically and lazily these 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.
Other design goals include:
Flexibility in the code layout: the category of, say, finite sets can be implemented either within the Sets category (in a nested class
Sets.Finite
), or in a separate file (typically in a classFiniteSets
in a lazily imported module sage.categories.finite_sets).Single point of truth: a theorem, like Wedderburn’s, should be implemented in a single spot.
Single entry point: for example, from the entry
Rings
, one can explore a whole range of related categories just by applying axioms and constructions:sage: Rings().Commutative().Finite().NoZeroDivisors() Category of finite integral domains sage: Rings().Finite().Division() Category of finite enumerated fieldsThis will allow for progressively getting rid of all the entries like
GradedHopfAlgebrasWithBasis
which are polluting the global name space.Note that this is not about precluding the existence of multiple natural ways to construct the same category:
sage: Groups().Finite() Category of finite groups sage: Monoids().Finite().Inverse() Category of finite groups sage: Sets().Finite() & Monoids().Inverse() Category of finite groupsConcise idioms for the users (adding axioms, …)
Concise idioms and well highlighted hierarchy of bookshelves for the developer (especially with code folding)
Introspection friendly (listing the axioms, recovering the mixins)
Note
The constructor for instances of this class takes as input the base category. Hence, they should in principle be constructed as:
sage: FiniteSets(Sets())
Category of finite sets
sage: Sets.Finite(Sets())
Category of finite sets
None of these idioms are really practical for the user. So instead, this object is to be constructed using any of the following idioms:
sage: Sets()._with_axiom('Finite')
Category of finite sets
sage: FiniteSets()
Category of finite sets
sage: Sets().Finite()
Category of finite sets
The later two are implemented using respectively
CategoryWithAxiom.__classcall__()
and
CategoryWithAxiom.__classget__()
.
Upcoming features¶
Todo
Implement compatibility axiom / functorial constructions. For example, one would want to have:
A.CartesianProducts() & B.CartesianProducts() = (A&B).CartesianProducts()
Once full subcategories are implemented (see trac ticket #10668), make the relevant categories with axioms be such. This can be done systematically for, e.g., the axioms
Associative
orCommutative
, but not for the axiomUnital
: a semigroup morphism between two monoids need not preserve the unit.Should all full subcategories be implemented in term of axioms?
Algorithms¶
Computing joins¶
The workhorse of the axiom infrastructure is the algorithm for
computing the join \(J\) of a set \(C_1, \ldots, C_k\) of categories (see
Category.join()
). Formally, \(J\) is defined as the largest
constructible category such that \(J \subset C_i\) for all \(i\), and
\(J \subset C.A()\) for every constructible category \(C \supset J\)
and any axiom \(A\) satisfied by \(J\).
The join \(J\) is naturally computed as a closure in the lattice of
constructible categories: it starts with the \(C_i\)’s, gathers the set
\(S\) of all the axioms satisfied by them, and repeatedly adds each
axiom \(A\) to those categories that do not yet satisfy \(A\) using
Category._with_axiom()
. Due to deduction rules or (extra) super
categories, new categories or new axioms may appear in the
process. The process stops when each remaining category has been
combined with each axiom. In practice, only the smallest categories
are kept along the way; this is correct because adding an axiom is
covariant: C.A()
is a subcategory of D.A()
whenever C
is a
subcategory of D
.
As usual in such closure computations, the result does not depend on the order of execution. Futhermore, given that adding an axiom is an idempotent and regressive operation, the process is guaranteed to stop in a number of steps which is bounded by the number of super categories of \(J\). In particular, it is a finite process.
Todo
Detail this a bit. What could typically go wrong is a situation
where, for some category C1
, C1.A()
specifies a category
C2
as super category such that C2.A()
specifies C3
as
super category such that …; this would clearly cause an infinite
execution. Note that this situation violates the specifications
since C1.A()
is supposed to be a subcategory of C2.A()
,
… so we would have an infinite increasing chain of constructible
categories.
It’s reasonable to assume that there is a finite number of axioms defined in the code. There remains to use this assumption to argue that any infinite execution of the algorithm would give rise to such an infinite sequence.
Adding an axiom¶
Let Cs
be a category and A
an axiom defined for this
category. To compute Cs().A()
, there are two cases.
Adding an axiom A
to a category Cs()
not implementing it¶
In this case, Cs().A()
returns the join of:
Cs()
Bs().A()
for every direct super categoryBs()
ofCs()
 the categories appearing in
Cs().A_extra_super_categories()
This is a highly recursive process. In fact, as such, it would run
right away into an infinite loop! Indeed, the join of Cs()
with
Bs().A()
would trigger the construction of Cs().A()
and
reciprocally. To avoid this, the Category.join()
method itself
does not use Category._with_axiom()
to add axioms, but its
sister Category._with_axiom_as_tuple()
; the latter builds a
tuple of categories that should be joined together but leaves the
computation of the join to its caller, the master join calculation.
Adding an axiom A
to a category Cs()
implementing it¶
In this case Cs().A()
simply constructs an instance \(D\) of
Cs.A
which models the desired category. The non trivial part is
the construction of the super categories of \(D\). Very much like
above, this includes:
Cs()
Bs().A()
for every super categoryBs()
ofCs()
 the categories appearing in
D.extra_super_categories()
This by itself may not be sufficient, due in particular to deduction
rules. On may for example discover a new axiom A1
satisfied by
\(D\), imposing to add A1
to all of the above categories. Therefore
the super categories are computed as the join of the above categories.
Up to one twist: as is, the computation of this join would trigger
recursively a recalculation of Cs().A()
! To avoid this,
Category.join()
is given an optional argument to specify that
the axiom A
should not be applied to Cs()
.
Sketch of proof of correctness and evaluation of complexity¶
As we have seen, this is a highly recursive process! In particular, one needs to argue that, as long as the specifications are satisfied, the algorithm won’t run in an infinite recursion, in particular in case of deduction rule.
Theorem
Consider the construction of a category \(C\) by adding an axiom to a category (or computing of a join). Let \(H\) be the hierarchy of implemented categories above \(C\). Let \(n\) and \(m\) be respectively the number of categories and the number of inheritance edges in \(H\).
Assuming that the specifications are satisfied, the construction of \(C\) involves constructing the categories in \(H\) exactly once (and no other category), and at most \(n\) join calculations. In particular, the time complexity should be, roughly speaking, bounded by \(n^2\). In particular, it’s finite.
Remark
It’s actually to be expected that the complexity is more of the order of magnitude of \(na+m\), where \(a\) is the number of axioms satisfied by \(C\). But this is to be checked in detail, in particular due to the many category inclusion tests involved.
The key argument is that Category.join
cannot call itself
recursively without going through the construction of some implemented
category. In turn, the construction of some implemented category \(C\)
only involves constructing strictly smaller categories, and possibly a
direct join calculation whose result is strictly smaller than
\(C\). This statement is obvious if \(C\) implements the
super_categories
method directly, and easy to check for functorial
construction categories. It requires a proof for categories with
axioms since there is a recursive join involved.
Lemma
Let \(C\) be a category implementing an axiom \(A\). Recall that the
construction of C.A()
involves a single direct join
calculation for computing the super categories. No other direct
join calculation occur, and the calculation involves only
implemented categories that are strictly smaller than C.A()
.
Proof
Let \(D\) be a category involved in the join calculation for the
super categories of C.A()
, and assume by induction that \(D\) is
strictly smaller than C.A()
. A category \(E\) newly constructed
from \(D\) can come from:
D.(extra_)super_categories()
In this case, the specifications impose that \(E\) should be strictly smaller than \(D\) and therefore strictly smaller than \(C\).
D.with_axiom_as_tuple('B')
orD.B_extra_super_categories()
for some axiom \(B\)In this case, the axiom \(B\) is satisfied by some subcategory of
C.A()
, and therefore must be satisfied byC.A()
itself. Since adding an axiom is a regressive construction, \(E\) must be a subcategory ofC.A()
. If there is equality, then \(E\) andC.A()
must have the same class, and therefore, \(E\) must be directly constructed asC.A()
. However the join construction explicitly prevents this call.
Note that a call to D.with_axiom_as_tuple('B')
does not trigger
a direct join calculation; but of course, if \(D\) implements \(B\),
the construction of the implemented category E = D.B()
will
involve a strictly smaller join calculation.
Conclusion¶
This is the end of the axioms documentation. Congratulations on having read that far!
Tests¶
Note
Quite a few categories with axioms are constructed early on during Sage’s startup. Therefore, when playing around with the implementation of the axiom infrastructure, it is easy to break Sage. The following sequence of tests is designed to test the infrastructure from the ground up even in a partially broken Sage. Please don’t remove the imports!
sage: Magmas()
Category of magmas
sage: Magmas().Finite()
Category of finite magmas
sage: Magmas().Unital()
Category of unital magmas
sage: Magmas().Commutative().Unital()
Category of commutative unital magmas
sage: Magmas().Associative()
Category of semigroups
sage: Magmas().Associative() & Magmas().Unital().Inverse() & Sets().Finite()
Category of finite groups
sage: _ is Groups().Finite()
True
sage: from sage.categories.semigroups import Semigroups
sage: Semigroups()
Category of semigroups
sage: Semigroups().Finite()
Category of finite semigroups
sage: from sage.categories.modules_with_basis import ModulesWithBasis
sage: ModulesWithBasis(QQ) is Modules(QQ).WithBasis()
True
sage: ModulesWithBasis(ZZ) is Modules(ZZ).WithBasis()
True
sage: Semigroups().Unital()
Category of monoids
sage: Semigroups().Unital().Commutative()
Category of commutative monoids
sage: Semigroups().Commutative()
Category of commutative semigroups
sage: Semigroups().Commutative().Unital()
Category of commutative monoids
sage: Semigroups().Commutative().Unital().super_categories()
[Category of monoids, Category of commutative magmas]
sage: AdditiveMagmas().AdditiveAssociative().AdditiveCommutative()
Category of commutative additive semigroups
sage: from sage.categories.magmas_and_additive_magmas import MagmasAndAdditiveMagmas
sage: C = CommutativeAdditiveMonoids() & Monoids() & MagmasAndAdditiveMagmas().Distributive(); C
Category of semirings
sage: C is (CommutativeAdditiveMonoids() & Monoids()).Distributive()
True
sage: C.AdditiveInverse()
Category of rings
sage: Rings().axioms()
frozenset({'AdditiveAssociative',
'AdditiveCommutative',
'AdditiveInverse',
'AdditiveUnital',
'Associative',
'Distributive',
'Unital'})
sage: sorted(Rings().axioms())
['AdditiveAssociative', 'AdditiveCommutative', 'AdditiveInverse',
'AdditiveUnital', 'Associative', 'Distributive', 'Unital']
sage: Domains().Commutative()
Category of integral domains
sage: DivisionRings().Finite() # Wedderburn's theorem
Category of finite enumerated fields
sage: FiniteMonoids().Algebras(QQ)
Join of Category of monoid algebras over Rational Field
and Category of finite dimensional algebras with basis over Rational Field
and Category of finite set algebras over Rational Field
sage: FiniteGroups().Algebras(QQ)
Category of finite group algebras over Rational Field

class
sage.categories.category_with_axiom.
Bars
(s=None)¶ Bases:
sage.categories.category_singleton.Category_singleton
A toy singleton category, for testing purposes.
See also

Unital_extra_super_categories
()¶ Return extraneous super categories for the unital objects of
self
.This method specifies that a unital bar is a test object. Thus, the categories of unital bars and of unital test objects coincide.
EXAMPLES:
sage: from sage.categories.category_with_axiom import Bars, TestObjects sage: Bars().Unital_extra_super_categories() [Category of test objects] sage: Bars().Unital() Category of unital test objects sage: TestObjects().Unital().all_super_categories() [Category of unital test objects, Category of unital blahs, Category of test objects, Category of bars, Category of blahs, Category of sets, Category of sets with partial maps, Category of objects]

super_categories
()¶


class
sage.categories.category_with_axiom.
Blahs
(s=None)¶ Bases:
sage.categories.category_singleton.Category_singleton
A toy singleton category, for testing purposes.
This is the root of a hierarchy of mathematically meaningless categories, used for testing Sage’s category framework:

Blue_extra_super_categories
()¶ Illustrates a current limitation in the way to have an axiom imply another one.
Here, we would want
Blue
to implyUnital
, and to put the class for the category of unital blue blahs inBlahs.Unital.Blue
rather thanBlahs.Blue
.This currently fails because
Blahs
is the category where the axiomBlue
is defined, and the specifications currently impose that a category defining an axiom should also implement it (here in an category with axiomBlahs.Blue
). In practice, due to this violation of the specifications, the axiom is lost during the join calculation.Todo
Decide whether we care about this feature. In such a situation, we are not really defining a new axiom, but just defining an axiom as an alias for a couple others, which might not be that useful.
Todo
Improve the infrastructure to detect and report this violation of the specifications, if this is easy. Otherwise, it’s not so bad: when defining an axiom A in a category
Cs
the first thing one is supposed to doctest is thatCs().A()
works. So the problem should not go unnoticed.

class
Commutative
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom

class
Connected
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom

class
FiniteDimensional
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom

class
Flying
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom

extra_super_categories
()¶ This illustrates a way to have an axiom imply another one.
Here, we want
Flying
to implyUnital
, and to put the class for the category of unital flying blahs inBlahs.Flying
rather thanBlahs.Unital.Flying
.


class
SubcategoryMethods
¶ 
Blue
()¶

Commutative
()¶

Connected
()¶

FiniteDimensional
()¶

Flying
()¶

Unital
()¶


class
Unital
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom

class
Blue
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom

class

super_categories
()¶


class
sage.categories.category_with_axiom.
CategoryWithAxiom
(base_category)¶ Bases:
sage.categories.category.Category
An abstract class for categories obtained by adding an axiom to a base category.
See the
category primer
, and in particular its section about axioms for an introduction to axioms, andCategoryWithAxiom
for how to implement axioms and the documentation of the axiom infrastructure.
static
__classcall__
(*args, **options)¶ Make
FoosBar(**)
an alias forFoos(**)._with_axiom("Bar")
.EXAMPLES:
sage: FiniteGroups() Category of finite groups sage: ModulesWithBasis(ZZ) Category of modules with basis over Integer Ring sage: AlgebrasWithBasis(QQ) Category of algebras with basis over Rational Field
This is relevant when e.g.
Foos(**)
does some non trivial transformations:sage: Modules(QQ) is VectorSpaces(QQ) True sage: type(Modules(QQ)) <class 'sage.categories.vector_spaces.VectorSpaces_with_category'> sage: ModulesWithBasis(QQ) is VectorSpaces(QQ).WithBasis() True sage: type(ModulesWithBasis(QQ)) <class 'sage.categories.vector_spaces.VectorSpaces.WithBasis_with_category'>

static
__classget__
(base_category, base_category_class)¶ Implement the binding behavior for categories with axioms.
This method implements a binding behavior on category with axioms so that, when a category
Cs
implements an axiomA
with a nested classCs.A
, the expressionCs().A
evaluates to the method defining the axiomA
and not the nested class. See those design notes for the rationale behind this behavior.EXAMPLES:
sage: Sets().Infinite() Category of infinite sets sage: Sets().Infinite Cached version of <function ...Infinite at ...> sage: Sets().Infinite.f == Sets.SubcategoryMethods.Infinite.f True
We check that this also works when the class is implemented in a separate file, and lazy imported:
sage: Sets().Finite Cached version of <function ...Finite at ...>
There is no binding behavior when accessing
Finite
orInfinite
from the class of the category instead of the category itself:sage: Sets.Finite <class 'sage.categories.finite_sets.FiniteSets'> sage: Sets.Infinite <class 'sage.categories.sets_cat.Sets.Infinite'>
This method also initializes the attribute
_base_category_class_and_axiom
if not already set:sage: Sets.Infinite._base_category_class_and_axiom (<class 'sage.categories.sets_cat.Sets'>, 'Infinite') sage: Sets.Infinite._base_category_class_and_axiom_origin 'set by __classget__'

__init__
(base_category)¶

_repr_object_names
()¶ The names of the objects of this category, as used by
_repr_
.See also
EXAMPLES:
sage: FiniteSets()._repr_object_names() 'finite sets' sage: AlgebrasWithBasis(QQ).FiniteDimensional()._repr_object_names() 'finite dimensional algebras with basis over Rational Field' sage: Monoids()._repr_object_names() 'monoids' sage: Semigroups().Unital().Finite()._repr_object_names() 'finite monoids' sage: Algebras(QQ).Commutative()._repr_object_names() 'commutative algebras over Rational Field'
Note
This is implemented by taking _repr_object_names from self._without_axioms(named=True), and adding the names of the relevant axioms in appropriate order.

static
_repr_object_names../_static
(category, axioms)¶ INPUT:
base_category
– a categoryaxioms
– a list or iterable of strings
EXAMPLES:
sage: from sage.categories.category_with_axiom import CategoryWithAxiom sage: CategoryWithAxiom._repr_object_names../_static(Semigroups(), ["Flying", "Blue"]) 'flying blue semigroups' sage: CategoryWithAxiom._repr_object_names../_static(Algebras(QQ), ["Flying", "WithBasis", "Blue"]) 'flying blue algebras with basis over Rational Field' sage: CategoryWithAxiom._repr_object_names../_static(Algebras(QQ), ["WithBasis"]) 'algebras with basis over Rational Field' sage: CategoryWithAxiom._repr_object_names../_static(Sets().Finite().Subquotients(), ["Finite"]) 'subquotients of finite sets' sage: CategoryWithAxiom._repr_object_names../_static(Monoids(), ["Unital"]) 'monoids' sage: CategoryWithAxiom._repr_object_names../_static(Algebras(QQ['x']['y']), ["Flying", "WithBasis", "Blue"]) 'flying blue algebras with basis over Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field'
If the axioms is a set or frozen set, then they are first sorted using
canonicalize_axioms()
:sage: CategoryWithAxiom._repr_object_names../_static(Semigroups(), set(["Finite", "Commutative", "Facade"])) 'facade finite commutative semigroups'
See also
Note
The logic here is shared between
_repr_object_names()
andcategory.JoinCategory._repr_object_names()

_test_category_with_axiom
(**options)¶ Run generic tests on this category with axioms.
See also
This check that an axiom category of a
Category_singleton
is a singleton category, and similarwise forCategory_over_base_ring
.EXAMPLES:
sage: Sets().Finite()._test_category_with_axiom() sage: Modules(ZZ).FiniteDimensional()._test_category_with_axiom()

_without_axioms
(named=False)¶ Return the category without the axioms that have been added to create it.
EXAMPLES:
sage: Sets().Finite()._without_axioms() Category of sets sage: Monoids().Finite()._without_axioms() Category of magmas
This is because:
sage: Semigroups().Unital() is Monoids() True
If
named
isTrue
, then_without_axioms
stops at the first category that has an explicit name of its own:sage: Sets().Finite()._without_axioms(named=True) Category of sets sage: Monoids().Finite()._without_axioms(named=True) Category of monoids
Technically we test this by checking if the class specifies explicitly the attribute
_base_category_class_and_axiom
by looking up_base_category_class_and_axiom_origin
.Some more examples:
sage: Algebras(QQ).Commutative()._without_axioms() Category of magmatic algebras over Rational Field sage: Algebras(QQ).Commutative()._without_axioms(named=True) Category of algebras over Rational Field

additional_structure
()¶ Return the additional structure defined by
self
.OUTPUT:
None
By default, a category with axiom defines no additional structure.
See also
EXAMPLES:
sage: Sets().Finite().additional_structure() sage: Monoids().additional_structure()

axioms
()¶ Return the axioms known to be satisfied by all the objects of
self
.See also
EXAMPLES:
sage: C = Sets.Finite(); C Category of finite sets sage: C.axioms() frozenset({'Finite'}) sage: C = Modules(GF(5)).FiniteDimensional(); C Category of finite dimensional vector spaces over Finite Field of size 5 sage: sorted(C.axioms()) ['AdditiveAssociative', 'AdditiveCommutative', 'AdditiveInverse', 'AdditiveUnital', 'Finite', 'FiniteDimensional'] sage: sorted(FiniteMonoids().Algebras(QQ).axioms()) ['AdditiveAssociative', 'AdditiveCommutative', 'AdditiveInverse', 'AdditiveUnital', 'Associative', 'Distributive', 'FiniteDimensional', 'Unital', 'WithBasis'] sage: sorted(FiniteMonoids().Algebras(GF(3)).axioms()) ['AdditiveAssociative', 'AdditiveCommutative', 'AdditiveInverse', 'AdditiveUnital', 'Associative', 'Distributive', 'Finite', 'FiniteDimensional', 'Unital', 'WithBasis'] sage: from sage.categories.magmas_and_additive_magmas import MagmasAndAdditiveMagmas sage: MagmasAndAdditiveMagmas().Distributive().Unital().axioms() frozenset({'Distributive', 'Unital'}) sage: D = MagmasAndAdditiveMagmas().Distributive() sage: X = D.AdditiveAssociative().AdditiveCommutative().Associative() sage: X.Unital().super_categories()[1] Category of monoids sage: X.Unital().super_categories()[1] is Monoids() True

base_category
()¶ Return the base category of
self
.EXAMPLES:
sage: C = Sets.Finite(); C Category of finite sets sage: C.base_category() Category of sets sage: C._without_axioms() Category of sets

extra_super_categories
()¶ Return the extra super categories of a category with axiom.
Default implementation which returns
[]
.EXAMPLES:
sage: FiniteSets().extra_super_categories() []

super_categories
()¶ Return a list of the (immediate) super categories of
self
, as perCategory.super_categories()
.This implements the property that if
As
is a subcategory ofBs
, then the intersection ofAs
withFiniteSets()
is a subcategory ofAs
and of the intersection ofBs
withFiniteSets()
.EXAMPLES:
A finite magma is both a magma and a finite set:
sage: Magmas().Finite().super_categories() [Category of magmas, Category of finite sets]
Variants:
sage: Sets().Finite().super_categories() [Category of sets] sage: Monoids().Finite().super_categories() [Category of monoids, Category of finite semigroups]
EXAMPLES:

static

class
sage.categories.category_with_axiom.
CategoryWithAxiom_over_base_ring
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom
,sage.categories.category_types.Category_over_base_ring

class
sage.categories.category_with_axiom.
CategoryWithAxiom_singleton
(base_category)¶ Bases:
sage.categories.category_singleton.Category_singleton
,sage.categories.category_with_axiom.CategoryWithAxiom

class
sage.categories.category_with_axiom.
TestObjects
(s=None)¶ Bases:
sage.categories.category_singleton.Category_singleton
A toy singleton category, for testing purposes.
See also

class
Commutative
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom

class
Facade
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom

class
Finite
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom

class
FiniteDimensional
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom

class

class
FiniteDimensional
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom

class
Finite
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom

class
Unital
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom

class
Commutative
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom

class

class

class
Unital
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom

super_categories
()¶

class

class
sage.categories.category_with_axiom.
TestObjectsOverBaseRing
(base, name=None)¶ Bases:
sage.categories.category_types.Category_over_base_ring
A toy singleton category, for testing purposes.
See also

class
Commutative
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

class
Facade
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

class
Finite
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

class
FiniteDimensional
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

class

class
FiniteDimensional
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

class
Finite
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

class
Unital
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

class
Commutative
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

class

class

class
Unital
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

super_categories
()¶

class

sage.categories.category_with_axiom.
axiom
(axiom)¶ Return a function/method
self > self._with_axiom(axiom)
.This can used as a shorthand to define axioms, in particular in the tests below. Usually one will want to attach documentation to an axiom, so the need for such a shorthand in real life might not be that clear, unless we start creating lots of axioms.
In the long run maybe this could evolve into an
@axiom
decorator.EXAMPLES:
sage: from sage.categories.category_with_axiom import axiom sage: axiom("Finite")(Semigroups()) Category of finite semigroups
Upon assigning the result to a class this becomes a method:
sage: class As: ....: def _with_axiom(self, axiom): return self, axiom ....: Finite = axiom("Finite") sage: As().Finite() (<__main__.As ... at ...>, 'Finite')

sage.categories.category_with_axiom.
axiom_of_nested_class
(cls, nested_cls)¶ Given a class and a nested axiom class, return the axiom.
EXAMPLES:
This uses some heuristics like checking if the nested_cls carries the name of the axiom, or is built by appending or prepending the name of the axiom to that of the class:
sage: from sage.categories.category_with_axiom import TestObjects, axiom_of_nested_class sage: axiom_of_nested_class(TestObjects, TestObjects.FiniteDimensional) 'FiniteDimensional' sage: axiom_of_nested_class(TestObjects.FiniteDimensional, TestObjects.FiniteDimensional.Finite) 'Finite' sage: axiom_of_nested_class(Sets, FiniteSets) 'Finite' sage: axiom_of_nested_class(Algebras, AlgebrasWithBasis) 'WithBasis'
In all other cases, the nested class should provide an attribute
_base_category_class_and_axiom
:sage: Semigroups._base_category_class_and_axiom (<class 'sage.categories.magmas.Magmas'>, 'Associative') sage: axiom_of_nested_class(Magmas, Semigroups) 'Associative'

sage.categories.category_with_axiom.
base_category_class_and_axiom
(cls)¶ Try to deduce the base category and the axiom from the name of
cls
.The heuristic is to try to decompose the name as the concatenation of the name of a category and the name of an axiom, and looking up that category in the standard location (i.e. in
sage.categories.hopf_algebras
forHopfAlgebras
, and insage.categories.sets_cat
as a special case forSets
).If the heuristic succeeds, the result is guaranteed to be correct. Otherwise, an error is raised.
EXAMPLES:
sage: from sage.categories.category_with_axiom import base_category_class_and_axiom, CategoryWithAxiom sage: base_category_class_and_axiom(FiniteSets) (<class 'sage.categories.sets_cat.Sets'>, 'Finite') sage: Sets.Finite <class 'sage.categories.finite_sets.FiniteSets'> sage: base_category_class_and_axiom(Sets.Finite) (<class 'sage.categories.sets_cat.Sets'>, 'Finite') sage: base_category_class_and_axiom(FiniteDimensionalHopfAlgebrasWithBasis) (<class 'sage.categories.hopf_algebras_with_basis.HopfAlgebrasWithBasis'>, 'FiniteDimensional') sage: base_category_class_and_axiom(HopfAlgebrasWithBasis) (<class 'sage.categories.hopf_algebras.HopfAlgebras'>, 'WithBasis')
Along the way, this does some sanity checks:
sage: class FacadeSemigroups(CategoryWithAxiom): ....: pass sage: base_category_class_and_axiom(FacadeSemigroups) Traceback (most recent call last): ... AssertionError: Missing (lazy import) link for <class 'sage.categories.semigroups.Semigroups'> to <class '__main__.FacadeSemigroups'> for axiom Facade? sage: Semigroups.Facade = FacadeSemigroups sage: base_category_class_and_axiom(FacadeSemigroups) (<class 'sage.categories.semigroups.Semigroups'>, 'Facade')
Note
In the following example, we could possibly retrieve
Sets
from the class name. However this cannot be implemented robustly until trac ticket #9107 is fixed. Anyway this feature has not been needed so far:sage: Sets.Infinite <class 'sage.categories.sets_cat.Sets.Infinite'> sage: base_category_class_and_axiom(Sets.Infinite) Traceback (most recent call last): ... TypeError: Could not retrieve the base category class and axiom for <class 'sage.categories.sets_cat.Sets.Infinite'>. ...

sage.categories.category_with_axiom.
uncamelcase
(s, separator=' ')¶ EXAMPLES:
sage: sage.categories.category_with_axiom.uncamelcase("FiniteDimensionalAlgebras") 'finite dimensional algebras' sage: sage.categories.category_with_axiom.uncamelcase("JTrivialMonoids") 'j trivial monoids' sage: sage.categories.category_with_axiom.uncamelcase("FiniteDimensionalAlgebras", "_") 'finite_dimensional_algebras'