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baseclasses.py
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baseclasses.py
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from sympy.core import (Set, Basic, FiniteSet, EmptySet, Dict, Symbol,
Dummy, Tuple)
class Class(Set):
r"""
The base class for any kind of class in the set-theoretic sense.
In axiomatic set theories, everything is a class. A class which
can be a member of another class is a set. A class which is not a
member of another class is a proper class. The class `\{1, 2\}`
is a set; the class of all sets is a proper class.
This class is essentially a synonym for :class:`sympy.core.Set`.
The goal of this class is to assure easier migration to the
eventual proper implementation of set theory.
"""
is_proper = False
def _make_symbol(name):
"""
If ``name`` is not empty, creates a :class:`Symbol` with this
name. Otherwise creates a :class:`Dummy`.
"""
if name:
return Symbol(name)
else:
return Dummy("")
class Object(Basic):
"""
The base class for any kind of object in an abstract category.
While concrete categories may have some concrete SymPy classes as
object types, in abstract categories only the name of an object is
known. Anonymous objects are not allowed.
Two objects with the same name are the same object.
Examples
========
>>> from sympy.categories import Object
>>> Object("A") == Object("A")
True
"""
def __new__(cls, name):
if not name:
raise ValueError("Anonymous Objects are not allowed.")
return Basic.__new__(cls, Symbol(name))
@property
def name(self):
"""
Returns the name of this object.
Examples
========
>>> from sympy.categories import Object
>>> A = Object("A")
>>> A.name
'A'
"""
return self._args[0].name
class Morphism(Basic):
"""
The base class for any kind of morphism in an abstract category.
In abstract categories, a morphism is an arrow between two
category objects. The object where the arrow starts is called the
domain, while the object where the arrow ends is called the
codomain.
Two simple (not composed) morphisms with the same name, domain,
and codomain are the same morphisms. A simple unnamed morphism is
not equal to any other morphism.
Two composed morphisms are equal if they have the same components,
in the same order (which guarantees the equality of domains and
codomains). The names of such composed morphisms are not taken in
consideration at comparison.
Morphisms with the same domain and codomain can be defined to be
identity morphisms. Identity morphisms with the same (co)domains
are equal. Identity morphisms are identities with respect to
composition. Identity morphisms are instances of
:class:`IdentityMorphism`.
Examples
========
>>> from sympy.categories import Object, Morphism
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = Morphism(A, B, "f")
>>> g = Morphism(B, C, "g")
>>> f == Morphism(A, B, "f")
True
>>> f == Morphism(A, B, "")
False
>>> g * f
Morphism(Object("B"), Object("C"), "g") *
Morphism(Object("A"), Object("B"), "f")
>>> id_A = Morphism(A, A, identity=True)
>>> id_A == Morphism(A, A, identity=True)
True
>>> f * id_A == f
True
"""
def __new__(cls, domain, codomain, name="", identity=False):
if identity and (domain != codomain):
raise ValueError(
"identity morphisms must have the same domain and codomain")
# The last component of self.args represents the components of
# this morphism.
if identity:
return IdentityMorphism(domain, name)
else:
return Basic.__new__(cls, domain, codomain,
_make_symbol(name), Tuple())
@property
def domain(self):
"""
Returns the domain of this morphism.
Examples
========
>>> from sympy.categories import Object, Morphism
>>> A = Object("A")
>>> B = Object("B")
>>> f = Morphism(A, B)
>>> f.domain
Object("A")
"""
return self.args[0]
@property
def codomain(self):
"""
Returns the codomain of this morphism.
Examples
========
>>> from sympy.categories import Object, Morphism
>>> A = Object("A")
>>> B = Object("B")
>>> f = Morphism(A, B)
>>> f.codomain
Object("B")
"""
return self.args[1]
@property
def name(self):
"""
Returns the name of this morphism.
Examples
========
>>> from sympy.categories import Object, Morphism
>>> A = Object("A")
>>> B = Object("B")
>>> f = Morphism(A, B, "f")
>>> f.name
'f'
"""
return self.args[2].name
@property
def identity(self):
"""
Is ``True`` if this morphism is known to be an identity
morphism.
Examples
========
>>> from sympy.categories import Object, Morphism
>>> A = Object("A")
>>> B = Object("B")
>>> f = Morphism(A, B, "f")
>>> f.identity
False
>>> id_A = Morphism(A, A, identity=True)
>>> id_A.identity
True
"""
return False
@property
def components(self):
"""
Returns the components of this morphisms.
Examples
========
>>> from sympy.categories import Object, Morphism
>>> from sympy import Tuple
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = Morphism(A, B, "f")
>>> g = Morphism(B, C, "g")
>>> (g * f).components == Tuple(f, g)
True
"""
components = self.args[3]
if not components:
return Tuple(self)
else:
return components
def compose(self, g, new_name=""):
"""
If ``self`` is a morphism from `B` to `C` and ``g`` is a
morphism from `A` to `B`, returns the morphism from `A` to `C`
which results from the composition of these morphisms.
If either ``self`` or ``g`` are morphisms resulted from some
previous composition, components in the resulting morphism
will be the concatenation of ``g.components`` and
``self.components``, in this order.
Instead of ``f.compose(g)`` it is possible to write ``f * g``.
Examples
========
>>> from sympy.categories import Object, Morphism
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = Morphism(A, B, "f")
>>> g = Morphism(B, C, "g")
>>> g.compose(f)
Morphism(Object("B"), Object("C"), "g") *
Morphism(Object("A"), Object("B"), "f")
>>> (g.compose(f, "h")).name
'h'
"""
if g.codomain != self.domain:
raise ValueError("Uncomponsable morphisms.")
if g.identity:
return self
# We don't really know whether the new morphism is an identity
# (even if g.domain == self.codomain), so let's suppose it's
# not an identity.
return Basic.__new__(Morphism, g.domain, self.codomain,
_make_symbol(new_name), g.components +
self.components)
def __mul__(self, g):
"""
Returns the result of the composition of ``self`` with the
argument, if this composition is defined.
The semantics of multiplication is as follows: ``f * g =
f.compose(g)``.
See Also
=======
compose
"""
return self.compose(g)
def flatten(self, new_name=""):
"""
If ``self`` resulted from composition of other morphisms,
returns a new morphism without any information about the
morphisms it resulted from.
Note that comparing ``self`` with the new morphism need NOT
return ``True``.
Examples
========
>>> from sympy.categories import Object, Morphism
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = Morphism(A, B, "f")
>>> g = Morphism(B, C, "g")
>>> (g * f).flatten("h")
Morphism(Object("A"), Object("C"), "h")
See Also
========
compose
"""
return Morphism(self.domain, self.codomain, new_name)
def __eq__(self, other):
if not isinstance(other, Morphism):
return False
if self.identity and other.identity:
# All identities are equal.
return self.domain == other.domain
elif self.identity or other.identity:
# One of the morphisms is an identity, but not both.
return False
if (len(self.components) == 1) and (len(other.components) == 1):
# We are comparing two simple morphisms.
if (not self.name) or (not other.name):
return False
return (self.name == other.name) and \
(self.domain == other.domain) and \
(self.codomain == other.codomain)
else:
# One of the morphisms is composed. Compare the
# components.
for (self_component, other_component) in \
zip(self.components, other.components):
if self_component != other_component:
return False
return True
def __ne__(self, other):
return not (self == other)
def __hash__(self):
return hash((self.name, self.domain, self.codomain))
class IdentityMorphism(Morphism):
"""
An identity morphism.
An identity morphism is a morphism with equal domain and codomain,
which acts as an identity with respect to composition.
Examples
========
>>> from sympy.categories import Object, Morphism, IdentityMorphism
>>> A = Object("A")
>>> B = Object("B")
>>> f = Morphism(A, B, "f")
>>> id_A = IdentityMorphism(A)
>>> f * id_A == f
True
"""
def __new__(cls, domain, name=""):
return Basic.__new__(cls, domain, _make_symbol(name))
@property
def domain(self):
"""
Returns the domain of this identity morphism.
Examples
========
>>> from sympy.categories import Object, IdentityMorphism
>>> A = Object("A")
>>> id_A = IdentityMorphism(A)
>>> id_A.domain
Object("A")
"""
return self.args[0]
@property
def codomain(self):
"""
Returns the codomain of this identity morphism.
Examples
========
>>> from sympy.categories import Object, Morphism
>>> A = Object("A")
>>> id_A = Morphism(A, A, identity=True)
>>> id_A.codomain
Object("A")
"""
return self.args[0]
@property
def name(self):
"""
Returns the name of this identity morphism.
Examples
========
>>> from sympy.categories import Object, IdentityMorphism
>>> A = Object("A")
>>> id_A = IdentityMorphism(A, "id_A")
>>> id_A.name
'id_A'
"""
return self.args[1].name
@property
def identity(self):
"""
Is ``True`` if this morphism is known to be an identity
morphism.
Examples
========
>>> from sympy.categories import Object, IdentityMorphism
>>> A = Object("A")
>>> id_A = IdentityMorphism(A)
>>> id_A.identity
True
"""
return True
@property
def components(self):
r"""
Returns the components of this morphisms.
Since this is an identity morphisms, it always has itself as
the only component.
Examples
========
>>> from sympy.categories import Object, IdentityMorphism
>>> A = Object("A")
>>> id_A = IdentityMorphism(A, 'id_A')
>>> id_A.components
(IdentityMorphism(Object("A"), "id_A"),)
"""
return Tuple(self)
def compose(self, g, new_name=""):
"""
If ``g`` is a morphism with codomain equal to the domain of
this identity morphisms, returns ``g``.
The argument ``new_name`` is not used.
Examples
========
>>> from sympy.categories import Object, Morphism, IdentityMorphism
>>> A = Object("A")
>>> B = Object("B")
>>> f = Morphism(A, B, "f")
>>> id_A = IdentityMorphism(A)
>>> f.compose(id_A) == f
True
"""
if g.codomain != self.domain:
raise ValueError("Uncomponsable morphisms.")
return g
def __hash__(self):
return hash(self.domain)
class Category(Basic):
r"""
An (abstract) category.
A category [JoyOfCats] is a quadruple `\mbox{K} = (O, \hom, id,
\circ)` consisting of
* a (set-theoretical) class `O`, whose members are called
`K`-objects,
* for each pair `(A, B)` of `K`-objects, a set `\hom(A, B)` whose
members are called `K`-morphisms from `A` to `B`,
* for a each `K`-object `A`, a morphism `id:A\rightarrow A`,
called the `K`-identity of `A`,
* a composition law `\circ` associating with every `K`-morphisms
`f:A\rightarrow B` and `g:B\rightarrow C` a `K`-morphism `g\circ
f:A\rightarrow C`, called the composite of `f` and `g`.
Composition is associative, `K`-identities are identities with
respect to composition, and the sets `\hom(A, B)` are pairwise
disjoint.
This class knows nothing about its objects and morphisms.
Concrete cases of (abstract) categories should be implemented as
classes derived from this one.
Certain instances of :class:`Diagram` can be asserted to be
commutative in a :class:`Category` by supplying the argument
``commutative`` in the constructor.
Examples
========
>>> from sympy.categories import Object, Morphism, Diagram, Category
>>> from sympy import FiniteSet
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = Morphism(A, B, "f")
>>> g = Morphism(B, C, "g")
>>> d = Diagram([f, g])
>>> K = Category("K", commutative=[d])
>>> K.commutative == FiniteSet(d)
True
See Also
========
Diagram
"""
def __new__(cls, name, objects=EmptySet(), commutative=EmptySet()):
new_category = Basic.__new__(cls, Symbol(name), objects,
FiniteSet(commutative))
return new_category
@property
def name(self):
"""
Returns the name of this category.
Examples
========
>>> from sympy.categories import Category
>>> K = Category("K")
>>> K.name
'K'
"""
return self.args[0].name
@property
def objects(self):
"""
Returns the class of objects of this category.
Examples
========
>>> from sympy.categories import Object, Category
>>> from sympy import FiniteSet
>>> A = Object("A")
>>> B = Object("B")
>>> K = Category("K", FiniteSet(A, B))
>>> K.objects
{Object("B"), Object("A")}
"""
return self.args[1]
@property
def commutative(self):
"""
Returns the :class:`FiniteSet` of diagrams which are known to
be commutative in this category.
>>> from sympy.categories import Object, Morphism, Diagram, Category
>>> from sympy import FiniteSet
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = Morphism(A, B, "f")
>>> g = Morphism(B, C, "g")
>>> d = Diagram([f, g])
>>> K = Category("K", commutative=[d])
>>> K.commutative == FiniteSet(d)
True
"""
return self.args[2]
def hom(self, A, B):
raise NotImplementedError(
"hom-sets are not implemented in Category.")
def all_morphisms(self):
raise NotImplementedError(
"Obtaining the class of morphisms is not implemented in Category.")
class Diagram(Basic):
r"""
Represents a diagram in a certain category.
Informally, a diagram is a collection of objects of a category and
certain morphisms between them. A diagram is still a monoid with
respect to morphism composition; i.e., identity morphisms, as well
as all composites of morphisms included in the diagram belong to
the diagram. For a more formal approach to this notion see
[Pare1970].
A commutative diagram is often accompanied by a statement of the
following kind: "if such morphisms with such properties exist,
then such morphisms which such properties exist and the diagram is
commutative". To represent this, an instance of :class:`Diagram`
includes a collection of morphisms which are the premises and
another collection of conclusions. ``premises`` and
``conclusions`` associate morphisms belonging to the corresponding
categories with the :class:`FiniteSet`'s of their properties.
The set of properties of a composite morphism is the intersection
of the sets of properties of its components. The domain and
codomain of a conclusion morphism should be among the domains and
codomains of the morphisms listed as the premises of a diagram.
No checks are carried out of whether the supplied object and
morphisms do belong to one and the same category.
Examples
========
>>> from sympy.categories import Object, Morphism, Diagram
>>> from sympy import FiniteSet
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = Morphism(A, B, "f")
>>> g = Morphism(B, C, "g")
>>> d = Diagram([f, g])
>>> Morphism(A, A, identity=True) in d.premises.keys()
True
>>> g * f in d.premises.keys()
True
>>> d = Diagram([f, g], {g * f:"unique"})
>>> d.conclusions[g * f] == FiniteSet("unique")
True
References
==========
[Pare1970] B. Pareigis: Categories and functors. Academic Press,
1970.
"""
@staticmethod
def _set_dict_union(dictionary, key, value):
"""
If ``key`` is in ``dictionary``, set the new value of ``key``
to be the union between the old value and ``value``.
Otherwise, set the value of ``key`` to ``value.
Returns ``True`` if the key already was in the dictionary and
``False`` otherwise.
"""
if key in dictionary:
dictionary[key] = dictionary[key] | value
return True
else:
dictionary[key] = value
return False
@staticmethod
def _add_morphism(morphisms, morphism, props, add_identities=True):
"""
Adds a morphism and its attributes to the supplied dictionary
``morphisms``. If ``add_identities`` is True, also adds the
identity morphisms for the domain and the codomain of
``morphism``.
"""
if Diagram._set_dict_union(morphisms, morphism, props) == False:
# We have just added a new morphism.
if morphism.identity:
return
if add_identities:
empty = EmptySet()
id_dom = Morphism(morphism.domain, morphism.domain, identity=True)
id_cod = Morphism(morphism.codomain, morphism.codomain, identity=True)
Diagram._set_dict_union(morphisms, id_dom, empty)
Diagram._set_dict_union(morphisms, id_cod, empty)
for existing_morphism, existing_props in morphisms.items():
new_props = existing_props & props
if morphism.domain == existing_morphism.codomain:
left = morphism * existing_morphism
Diagram._set_dict_union(morphisms, left, new_props)
if morphism.codomain == existing_morphism.domain:
right = existing_morphism * morphism
Diagram._set_dict_union(morphisms, right, new_props)
def __new__(cls, *args):
premises = {}
conclusions = {}
# Here we will keep track of the objects which appear in the
# premises.
objects = EmptySet()
if len(args) >= 1:
# We've got some premises in the arguments.
premises_arg = args[0]
if isinstance(premises_arg, list):
# The user has supplied a list of morphisms, none of
# which have any attributes.
empty = EmptySet()
for morphism in premises_arg:
objects |= FiniteSet(morphism.domain, morphism.codomain)
Diagram._add_morphism(premises, morphism, empty)
elif isinstance(premises_arg, dict) or isinstance(premises_arg, Dict):
# The user has supplied a dictionary of morphisms and
# their properties.
for morphism, props in premises_arg.items():
objects |= FiniteSet(morphism.domain, morphism.codomain)
Diagram._add_morphism(premises, morphism, FiniteSet(props))
if len(args) >= 2:
# We also have some conclusions.
conclusions_arg = args[1]
if isinstance(conclusions_arg, list):
# The user has supplied a list of morphisms, none of
# which have any attributes.
empty = EmptySet()
for morphism in conclusions_arg:
# Check that no new objects appear in conclusions.
if (morphism.domain in objects) and \
(morphism.codomain in objects):
# No need to add identities this time.
Diagram._add_morphism(conclusions, morphism, empty, False)
elif isinstance(conclusions_arg, dict) or \
isinstance(conclusions_arg, Dict):
# The user has supplied a dictionary of morphisms and
# their properties.
for morphism, props in conclusions_arg.items():
# Check that no new objects appear in conclusions.
if (morphism.domain in objects) and \
(morphism.codomain in objects):
# No need to add identities this time.
Diagram._add_morphism(conclusions, morphism,
FiniteSet(props), False)
return Basic.__new__(cls, Dict(premises), Dict(conclusions), objects)
@property
def premises(self):
"""
Returns the premises of this diagram.
Examples
========
>>> from sympy.categories import Object, Morphism, Diagram
>>> from sympy import EmptySet, Dict
>>> A = Object("A")
>>> B = Object("B")
>>> f = Morphism(A, B, "f")
>>> id_A = Morphism(A, A, identity=True)
>>> id_B = Morphism(B, B, identity=True)
>>> d = Diagram([f])
>>> d.premises == Dict({f:EmptySet(), id_A:EmptySet(), id_B:EmptySet()})
True
"""
return self.args[0]
@property
def conclusions(self):
"""
Returns the conclusions of this diagram.
Examples
========
>>> from sympy.categories import Object, Morphism, Diagram
>>> from sympy import FiniteSet
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = Morphism(A, B, "f")
>>> g = Morphism(B, C, "g")
>>> d = Diagram([f, g])
>>> Morphism(A, A, identity=True) in d.premises.keys()
True
>>> g * f in d.premises.keys()
True
>>> d = Diagram([f, g], {g * f:"unique"})
>>> d.conclusions[g * f] == FiniteSet("unique")
True
"""
return self.args[1]
@property
def objects(self):
"""
Returns the :class:`FiniteSet` of objects that appear in this
diagram.
Examples
========
>>> from sympy.categories import Object, Morphism, Diagram
>>> from sympy import FiniteSet, Dict
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = Morphism(A, B, "f")
>>> g = Morphism(B, C, "g")
>>> d = Diagram([f, g])
>>> d.objects == FiniteSet(A, B, C)
True
"""
return self.args[2]
def hom(self, A, B):
"""
Returns a 2-tuple of sets of morphisms between objects A and
B: one set of morphisms listed as premises, and the other set
of morphisms listed as conclusions.
Examples
========
>>> from sympy.categories import Object, Morphism, Diagram
>>> from sympy import FiniteSet
>>> A = Object("A")
>>> B = Object("B")
>>> C = Object("C")
>>> f = Morphism(A, B, "f")
>>> g = Morphism(B, C, "g")
>>> d = Diagram([f, g], {g * f: "unique"})
>>> d.hom(A, C) == (FiniteSet(g * f), FiniteSet(g * f))
True
See Also
========
Object, Morphism
"""
premises = EmptySet()
conclusions = EmptySet()
for morphism in self.premises.keys():
if (morphism.domain == A) and (morphism.codomain == B):
premises |= FiniteSet(morphism)
for morphism in self.conclusions.keys():
if (morphism.domain == A) and (morphism.codomain == B):
conclusions |= FiniteSet(morphism)
return (premises, conclusions)
def __eq__(self, other):
if not isinstance(other, Diagram):
return False
return (self.premises == other.premises) and \
(self.conclusions == other.conclusions)
def __ne__(self, other):
return not (self == other)