Source code for sympy.categories.baseclasses

from __future__ import print_function, division

from sympy.core import S, Basic, Dict, Symbol, Tuple, sympify
from sympy.core.compatibility import iterable
from sympy.sets import Set, FiniteSet, EmptySet


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


[docs]class Object(Symbol): """ The base class for any kind of object in an abstract category. While technically any instance of :class:`Basic` will do, this class is the recommended way to create abstract objects in abstract categories. """
[docs]class Morphism(Basic): """ The base class for any 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 morphisms between the same pair of objects are considered to be the same morphisms. To distinguish between morphisms between the same objects use :class:`NamedMorphism`. It is prohibited to instantiate this class. Use one of the derived classes instead. See Also ======== IdentityMorphism, NamedMorphism, CompositeMorphism """ def __new__(cls, domain, codomain): raise(NotImplementedError( "Cannot instantiate Morphism. Use derived classes instead.")) @property def domain(self): """ Returns the domain of the morphism. Examples ======== >>> from sympy.categories import Object, NamedMorphism >>> A = Object("A") >>> B = Object("B") >>> f = NamedMorphism(A, B, "f") >>> f.domain Object("A") """ return self.args[0] @property def codomain(self): """ Returns the codomain of the morphism. Examples ======== >>> from sympy.categories import Object, NamedMorphism >>> A = Object("A") >>> B = Object("B") >>> f = NamedMorphism(A, B, "f") >>> f.codomain Object("B") """ return self.args[1]
[docs] def compose(self, other): r""" Composes self with the supplied morphism. The order of elements in the composition is the usual order, i.e., to construct `g\circ f` use ``g.compose(f)``. Examples ======== >>> from sympy.categories import Object, NamedMorphism >>> A = Object("A") >>> B = Object("B") >>> C = Object("C") >>> f = NamedMorphism(A, B, "f") >>> g = NamedMorphism(B, C, "g") >>> g * f CompositeMorphism((NamedMorphism(Object("A"), Object("B"), "f"), NamedMorphism(Object("B"), Object("C"), "g"))) >>> (g * f).domain Object("A") >>> (g * f).codomain Object("C") """ return CompositeMorphism(other, self)
def __mul__(self, other): r""" Composes self with the supplied morphism. The semantics of this operation is given by the following equation: ``g * f == g.compose(f)`` for composable morphisms ``g`` and ``f``. See Also ======== compose """ return self.compose(other)
[docs]class IdentityMorphism(Morphism): """ Represents 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, NamedMorphism, IdentityMorphism >>> A = Object("A") >>> B = Object("B") >>> f = NamedMorphism(A, B, "f") >>> id_A = IdentityMorphism(A) >>> id_B = IdentityMorphism(B) >>> f * id_A == f True >>> id_B * f == f True See Also ======== Morphism """ def __new__(cls, domain): return Basic.__new__(cls, domain, domain)
[docs]class NamedMorphism(Morphism): """ Represents a morphism which has a name. Names are used to distinguish between morphisms which have the same domain and codomain: two named morphisms are equal if they have the same domains, codomains, and names. Examples ======== >>> from sympy.categories import Object, NamedMorphism >>> A = Object("A") >>> B = Object("B") >>> f = NamedMorphism(A, B, "f") >>> f NamedMorphism(Object("A"), Object("B"), "f") >>> f.name 'f' See Also ======== Morphism """ def __new__(cls, domain, codomain, name): if not name: raise ValueError("Empty morphism names not allowed.") return Basic.__new__(cls, domain, codomain, Symbol(name)) @property def name(self): """ Returns the name of the morphism. Examples ======== >>> from sympy.categories import Object, NamedMorphism >>> A = Object("A") >>> B = Object("B") >>> f = NamedMorphism(A, B, "f") >>> f.name 'f' """ return self.args[2].name
[docs]class CompositeMorphism(Morphism): r""" Represents a morphism which is a composition of other morphisms. Two composite morphisms are equal if the morphisms they were obtained from (components) are the same and were listed in the same order. The arguments to the constructor for this class should be listed in diagram order: to obtain the composition `g\circ f` from the instances of :class:`Morphism` ``g`` and ``f`` use ``CompositeMorphism(f, g)``. Examples ======== >>> from sympy.categories import Object, NamedMorphism, CompositeMorphism >>> A = Object("A") >>> B = Object("B") >>> C = Object("C") >>> f = NamedMorphism(A, B, "f") >>> g = NamedMorphism(B, C, "g") >>> g * f CompositeMorphism((NamedMorphism(Object("A"), Object("B"), "f"), NamedMorphism(Object("B"), Object("C"), "g"))) >>> CompositeMorphism(f, g) == g * f True """ @staticmethod def _add_morphism(t, morphism): """ Intelligently adds ``morphism`` to tuple ``t``. If ``morphism`` is a composite morphism, its components are added to the tuple. If ``morphism`` is an identity, nothing is added to the tuple. No composability checks are performed. """ if isinstance(morphism, CompositeMorphism): # ``morphism`` is a composite morphism; we have to # denest its components. return t + morphism.components elif isinstance(morphism, IdentityMorphism): # ``morphism`` is an identity. Nothing happens. return t else: return t + Tuple(morphism) def __new__(cls, *components): if components and not isinstance(components[0], Morphism): # Maybe the user has explicitly supplied a list of # morphisms. return CompositeMorphism.__new__(cls, *components[0]) normalised_components = Tuple() for current, following in zip(components, components[1:]): if not isinstance(current, Morphism) or \ not isinstance(following, Morphism): raise TypeError("All components must be morphisms.") if current.codomain != following.domain: raise ValueError("Uncomposable morphisms.") normalised_components = CompositeMorphism._add_morphism( normalised_components, current) # We haven't added the last morphism to the list of normalised # components. Add it now. normalised_components = CompositeMorphism._add_morphism( normalised_components, components[-1]) if not normalised_components: # If ``normalised_components`` is empty, only identities # were supplied. Since they all were composable, they are # all the same identities. return components[0] elif len(normalised_components) == 1: # No sense to construct a whole CompositeMorphism. return normalised_components[0] return Basic.__new__(cls, normalised_components) @property def components(self): """ Returns the components of this composite morphism. Examples ======== >>> from sympy.categories import Object, NamedMorphism >>> A = Object("A") >>> B = Object("B") >>> C = Object("C") >>> f = NamedMorphism(A, B, "f") >>> g = NamedMorphism(B, C, "g") >>> (g * f).components (NamedMorphism(Object("A"), Object("B"), "f"), NamedMorphism(Object("B"), Object("C"), "g")) """ return self.args[0] @property def domain(self): """ Returns the domain of this composite morphism. The domain of the composite morphism is the domain of its first component. Examples ======== >>> from sympy.categories import Object, NamedMorphism >>> A = Object("A") >>> B = Object("B") >>> C = Object("C") >>> f = NamedMorphism(A, B, "f") >>> g = NamedMorphism(B, C, "g") >>> (g * f).domain Object("A") """ return self.components[0].domain @property def codomain(self): """ Returns the codomain of this composite morphism. The codomain of the composite morphism is the codomain of its last component. Examples ======== >>> from sympy.categories import Object, NamedMorphism >>> A = Object("A") >>> B = Object("B") >>> C = Object("C") >>> f = NamedMorphism(A, B, "f") >>> g = NamedMorphism(B, C, "g") >>> (g * f).codomain Object("C") """ return self.components[-1].codomain
[docs] def flatten(self, new_name): """ Forgets the composite structure of this morphism. If ``new_name`` is not empty, returns a :class:`NamedMorphism` with the supplied name, otherwise returns a :class:`Morphism`. In both cases the domain of the new morphism is the domain of this composite morphism and the codomain of the new morphism is the codomain of this composite morphism. Examples ======== >>> from sympy.categories import Object, NamedMorphism >>> A = Object("A") >>> B = Object("B") >>> C = Object("C") >>> f = NamedMorphism(A, B, "f") >>> g = NamedMorphism(B, C, "g") >>> (g * f).flatten("h") NamedMorphism(Object("A"), Object("C"), "h") """ return NamedMorphism(self.domain, self.codomain, new_name)
[docs]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_diagrams`` in the constructor. Examples ======== >>> from sympy.categories import Object, NamedMorphism, Diagram, Category >>> from sympy import FiniteSet >>> A = Object("A") >>> B = Object("B") >>> C = Object("C") >>> f = NamedMorphism(A, B, "f") >>> g = NamedMorphism(B, C, "g") >>> d = Diagram([f, g]) >>> K = Category("K", commutative_diagrams=[d]) >>> K.commutative_diagrams == FiniteSet(d) True See Also ======== Diagram """ def __new__(cls, name, objects=EmptySet(), commutative_diagrams=EmptySet()): if not name: raise ValueError("A Category cannot have an empty name.") new_category = Basic.__new__(cls, Symbol(name), Class(objects), FiniteSet(*commutative_diagrams)) 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 Class({Object("A"), Object("B")}) """ return self.args[1] @property def commutative_diagrams(self): """ Returns the :class:`FiniteSet` of diagrams which are known to be commutative in this category. >>> from sympy.categories import Object, NamedMorphism, Diagram, Category >>> from sympy import FiniteSet >>> A = Object("A") >>> B = Object("B") >>> C = Object("C") >>> f = NamedMorphism(A, B, "f") >>> g = NamedMorphism(B, C, "g") >>> d = Diagram([f, g]) >>> K = Category("K", commutative_diagrams=[d]) >>> K.commutative_diagrams == 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.")
[docs]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]. The components of composite morphisms are also added to the diagram. No properties are assigned to such morphisms by default. 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, NamedMorphism, Diagram >>> from sympy import FiniteSet, pprint, default_sort_key >>> A = Object("A") >>> B = Object("B") >>> C = Object("C") >>> f = NamedMorphism(A, B, "f") >>> g = NamedMorphism(B, C, "g") >>> d = Diagram([f, g]) >>> premises_keys = sorted(d.premises.keys(), key=default_sort_key) >>> pprint(premises_keys, use_unicode=False) [g*f:A-->C, id:A-->A, id:B-->B, id:C-->C, f:A-->B, g:B-->C] >>> pprint(d.premises, use_unicode=False) {g*f:A-->C: EmptySet(), id:A-->A: EmptySet(), id:B-->B: EmptySet(), id:C-->C: EmptySet(), f:A-->B: EmptySet(), g:B-->C: EmptySet()} >>> d = Diagram([f, g], {g * f: "unique"}) >>> pprint(d.conclusions) {g*f:A-->C: {unique}} 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_closure(morphisms, morphism, props, add_identities=True, recurse_composites=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 not Diagram._set_dict_union(morphisms, morphism, props): # We have just added a new morphism. if isinstance(morphism, IdentityMorphism): if props: # Properties for identity morphisms don't really # make sense, because very much is known about # identity morphisms already, so much that they # are trivial. Having properties for identity # morphisms would only be confusing. raise ValueError( "Instances of IdentityMorphism cannot have properties.") return if add_identities: empty = EmptySet() id_dom = IdentityMorphism(morphism.domain) id_cod = IdentityMorphism(morphism.codomain) Diagram._set_dict_union(morphisms, id_dom, empty) Diagram._set_dict_union(morphisms, id_cod, empty) for existing_morphism, existing_props in list(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) if isinstance(morphism, CompositeMorphism) and recurse_composites: # This is a composite morphism, add its components as # well. empty = EmptySet() for component in morphism.components: Diagram._add_morphism_closure(morphisms, component, empty, add_identities) def __new__(cls, *args): """ Construct a new instance of Diagram. If no arguments are supplied, an empty diagram is created. If at least an argument is supplied, ``args[0]`` is interpreted as the premises of the diagram. If ``args[0]`` is a list, it is interpreted as a list of :class:`Morphism`'s, in which each :class:`Morphism` has an empty set of properties. If ``args[0]`` is a Python dictionary or a :class:`Dict`, it is interpreted as a dictionary associating to some :class:`Morphism`'s some properties. If at least two arguments are supplied ``args[1]`` is interpreted as the conclusions of the diagram. The type of ``args[1]`` is interpreted in exactly the same way as the type of ``args[0]``. If only one argument is supplied, the diagram has no conclusions. Examples ======== >>> from sympy.categories import Object, NamedMorphism >>> from sympy.categories import IdentityMorphism, Diagram >>> A = Object("A") >>> B = Object("B") >>> C = Object("C") >>> f = NamedMorphism(A, B, "f") >>> g = NamedMorphism(B, C, "g") >>> d = Diagram([f, g]) >>> IdentityMorphism(A) in d.premises.keys() True >>> g * f in d.premises.keys() True >>> d = Diagram([f, g], {g * f: "unique"}) >>> d.conclusions[g * f] {unique} """ 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_closure(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_closure( premises, morphism, FiniteSet(*props) if iterable(props) else 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 ((sympify(objects.contains(morphism.domain)) is S.true) and (sympify(objects.contains(morphism.codomain)) is S.true)): # No need to add identities and recurse # composites this time. Diagram._add_morphism_closure( conclusions, morphism, empty, add_identities=False, recurse_composites=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 and recurse # composites this time. Diagram._add_morphism_closure( conclusions, morphism, FiniteSet(*props) if iterable(props) else FiniteSet(props), add_identities=False, recurse_composites=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, NamedMorphism >>> from sympy.categories import IdentityMorphism, Diagram >>> from sympy import pretty >>> A = Object("A") >>> B = Object("B") >>> f = NamedMorphism(A, B, "f") >>> id_A = IdentityMorphism(A) >>> id_B = IdentityMorphism(B) >>> d = Diagram([f]) >>> print(pretty(d.premises, use_unicode=False)) {id:A-->A: EmptySet(), id:B-->B: EmptySet(), f:A-->B: EmptySet()} """ return self.args[0] @property def conclusions(self): """ Returns the conclusions of this diagram. Examples ======== >>> from sympy.categories import Object, NamedMorphism >>> from sympy.categories import IdentityMorphism, Diagram >>> from sympy import FiniteSet >>> A = Object("A") >>> B = Object("B") >>> C = Object("C") >>> f = NamedMorphism(A, B, "f") >>> g = NamedMorphism(B, C, "g") >>> d = Diagram([f, g]) >>> IdentityMorphism(A) 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, NamedMorphism, Diagram >>> A = Object("A") >>> B = Object("B") >>> C = Object("C") >>> f = NamedMorphism(A, B, "f") >>> g = NamedMorphism(B, C, "g") >>> d = Diagram([f, g]) >>> d.objects {Object("A"), Object("B"), Object("C")} """ return self.args[2]
[docs] 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, NamedMorphism, Diagram >>> from sympy import pretty >>> A = Object("A") >>> B = Object("B") >>> C = Object("C") >>> f = NamedMorphism(A, B, "f") >>> g = NamedMorphism(B, C, "g") >>> d = Diagram([f, g], {g * f: "unique"}) >>> print(pretty(d.hom(A, C), use_unicode=False)) ({g*f:A-->C}, {g*f:A-->C}) 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)
[docs] def is_subdiagram(self, diagram): """ Checks whether ``diagram`` is a subdiagram of ``self``. Diagram `D'` is a subdiagram of `D` if all premises (conclusions) of `D'` are contained in the premises (conclusions) of `D`. The morphisms contained both in `D'` and `D` should have the same properties for `D'` to be a subdiagram of `D`. Examples ======== >>> from sympy.categories import Object, NamedMorphism, Diagram >>> A = Object("A") >>> B = Object("B") >>> C = Object("C") >>> f = NamedMorphism(A, B, "f") >>> g = NamedMorphism(B, C, "g") >>> d = Diagram([f, g], {g * f: "unique"}) >>> d1 = Diagram([f]) >>> d.is_subdiagram(d1) True >>> d1.is_subdiagram(d) False """ premises = all([(m in self.premises) and (diagram.premises[m] == self.premises[m]) for m in diagram.premises]) if not premises: return False conclusions = all([(m in self.conclusions) and (diagram.conclusions[m] == self.conclusions[m]) for m in diagram.conclusions]) # Premises is surely ``True`` here. return conclusions
[docs] def subdiagram_from_objects(self, objects): """ If ``objects`` is a subset of the objects of ``self``, returns a diagram which has as premises all those premises of ``self`` which have a domains and codomains in ``objects``, likewise for conclusions. Properties are preserved. Examples ======== >>> from sympy.categories import Object, NamedMorphism, Diagram >>> from sympy import FiniteSet >>> A = Object("A") >>> B = Object("B") >>> C = Object("C") >>> f = NamedMorphism(A, B, "f") >>> g = NamedMorphism(B, C, "g") >>> d = Diagram([f, g], {f: "unique", g*f: "veryunique"}) >>> d1 = d.subdiagram_from_objects(FiniteSet(A, B)) >>> d1 == Diagram([f], {f: "unique"}) True """ if not objects.is_subset(self.objects): raise ValueError( "Supplied objects should all belong to the diagram.") new_premises = {} for morphism, props in self.premises.items(): if ((sympify(objects.contains(morphism.domain)) is S.true) and (sympify(objects.contains(morphism.codomain)) is S.true)): new_premises[morphism] = props new_conclusions = {} for morphism, props in self.conclusions.items(): if ((sympify(objects.contains(morphism.domain)) is S.true) and (sympify(objects.contains(morphism.codomain)) is S.true)): new_conclusions[morphism] = props return Diagram(new_premises, new_conclusions)