Trac #15852: uncouple Sequence from categories

The `class Sequence` is difficult to place in the categories, as the
discussion in this ticket shows. Rather, it has its place as programming
utility, probably in `misc`. Thus, it needs not have a parent nor a
category, nor mathematical operations like +, ... This ticket will
deprecate/remove the corresponding methods.

Previous description:

The '''category `Sequence`''' is deprecated since 2009 (commit
`619aa0fa`). If I understand it correctly, the '''class `Sequence`'''
should instead have a parent from the category `Sets`.

The ticket that will remove category `Sequence` depends on this.

URL: http://trac.sagemath.org/15852
Reported by: rws
Ticket author(s): Ralf Stephan
Reviewer(s): Volker Braun

Author: Release Manager

src/doc/de/tutorial/tour_polynomial.rst

@@ -309,9 +309,6 @@ verwenden, nicht mehr funktionieren könnten)
 
 ::
 
-    sage: B.parent()
-    Category of sequences in Multivariate Polynomial Ring in x, y over Rational
-    Field
     sage: B.universe()
     Multivariate Polynomial Ring in x, y over Rational Field
     sage: B[1] = x

src/doc/en/tutorial/tour_polynomial.rst

@@ -303,9 +303,6 @@ break other routines that use the Gröbner basis).
 
 ::
 
-    sage: B.parent()
-    Category of sequences in Multivariate Polynomial Ring in x, y over Rational
-    Field
     sage: B.universe()
     Multivariate Polynomial Ring in x, y over Rational Field
     sage: B[1] = x

src/doc/fr/tutorial/tour_polynomial.rst

@@ -311,9 +311,6 @@ base de Gröbner).
 
 ::
 
-    sage: B.parent()
-    Category of sequences in Multivariate Polynomial Ring in x, y over Rational
-    Field
     sage: B.universe()
     Multivariate Polynomial Ring in x, y over Rational Field
     sage: B[1] = x

src/doc/pt/tutorial/tour_polynomial.rst

@@ -306,9 +306,6 @@ erros em outras rotinas que usam bases de Gröbner).
 
 ::
 
-    sage: B.parent()
-    Category of sequences in Multivariate Polynomial Ring in x, y over Rational 
-    Field
     sage: B.universe()
     Multivariate Polynomial Ring in x, y over Rational Field
     sage: B[1] = x

src/doc/ru/tutorial/tour_polynomial.rst

@@ -282,9 +282,6 @@ Sage использует Singular [Si]_ для вычислений НОД и 
 
 ::
 
-    sage: B.parent()
-    Category of sequences in Multivariate Polynomial Ring in x, y over Rational
-    Field
     sage: B.universe()
     Multivariate Polynomial Ring in x, y over Rational Field
     sage: B[1] = x

src/sage/categories/all.py

@@ -2,7 +2,6 @@
 
 from category_types import(
                         Elements,
-                        Sequences,
                         SimplicialComplexes,
                         ChainComplexes,
 )

src/sage/categories/category.py

@@ -2625,7 +2625,7 @@ def category_graph(categories = None):
         Graphics object consisting of 20 graphics primitives
 
         sage: sage.categories.category.category_graph().plot()
-        Graphics object consisting of 312 graphics primitives
+        Graphics object consisting of 310 graphics primitives
     """
     from sage import graphs
     if categories is None:

src/sage/categories/category_types.py

@@ -110,92 +110,6 @@ def _latex_(self):
         return "\\mathbf{Elt}_{%s}"%latex(self.__object)
 
 
-#############################################################
-# Category of sequences of elements of objects
-#############################################################
-class Sequences(Category):
-    """
-    The category of sequences of elements of a given object.
-
-    This category is deprecated.
-
-    EXAMPLES::
-
-        sage: v = Sequence([1,2,3]); v
-        [1, 2, 3]
-        sage: C = v.category(); C
-        Category of sequences in Integer Ring
-        sage: loads(C.dumps()) == C
-        True
-        sage: Sequences(ZZ) is C
-        True
-
-        True
-        sage: Sequences(ZZ).category()
-        Category of objects
-    """
-    def __init__(self, object):
-        Category.__init__(self)
-        self.__object = object
-
-    @classmethod
-    def an_instance(cls):
-        """
-        Returns an instance of this class
-
-        EXAMPLES::
-
-            sage: Elements(ZZ)
-            Category of elements of Integer Ring
-        """
-        from sage.rings.rational_field import QQ
-        return cls(QQ)
-
-    def super_categories(self):
-        """
-        EXAMPLES::
-
-            sage: Sequences(ZZ).super_categories()
-            [Category of objects]
-        """
-        return [Objects()]  # Almost FiniteEnumeratedSets() except for possible repetitions
-
-    def _call_(self, x):
-        """
-        EXAMPLES::
-
-            sage: v = Sequence([1,2,3]); v
-            [1, 2, 3]
-            sage: C = v.category(); C
-            Category of sequences in Integer Ring
-            sage: w = C([2/1, 3/1, GF(2)(5)]); w # indirect doctest
-            [2, 3, 1]
-            sage: w.category()
-            Category of sequences in Integer Ring
-        """
-        from sage.structure.sequence import Sequence
-        return Sequence(x, self.__object)
-
-    def object(self):
-        return self.__object
-
-    def __reduce__(self):
-        return Sequences, (self.__object, )
-
-    def _repr_object_names(self):
-        return "sequences in %s"%self.object()
-
-    def _latex_(self):
-        r"""
-        EXAMPLES::
-
-            sage: v = Sequence([1,2,3])
-            sage: latex(v.category()) # indirect doctest
-            \mathbf{Seq}_{\Bold{Z}}
-        """
-
-        return "\\mathbf{Seq}_{%s}"%latex(self.__object)
-
 #############################################################
 # Category of objects over some base object
 #############################################################

src/sage/coding/code_constructions.py

@@ -153,7 +153,7 @@
 from linear_code import LinearCodeFromVectorSpace, LinearCode
 from sage.modules.free_module import span
 from sage.schemes.projective.projective_space import ProjectiveSpace
-from sage.structure.sequence import Sequence
+from sage.structure.sequence import Sequence, Sequence_generic
 from sage.rings.arith import GCD,LCM,divisors,quadratic_residues
 from sage.rings.finite_rings.integer_mod_ring import IntegerModRing
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
@@ -463,7 +463,10 @@ def permutation_action(g,v):
     if isinstance(v, list):
         v_type_list = True
         v = Sequence(v)
-    V = v.parent()
+    if isinstance(v, Sequence_generic):
+        V = v.universe()
+    else:
+        V = v.parent()
     n = len(list(v))
     gv = []
     for i in range(n):

src/sage/databases/oeis.py

@@ -915,16 +915,16 @@ def natural_object(self):
             sage: fib = oeis('A000045') ; fib           # optional -- internet
             A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
 
-            sage: x = fib.natural_object() ; x.parent()         # optional -- internet
-            Category of sequences in Non negative integer semiring
+            sage: x = fib.natural_object() ; x.universe()         # optional -- internet
+            Non negative integer semiring
 
         ::
 
             sage: sfib = oeis('A039834') ; sfib         # optional -- internet
             A039834: a(n+2)=-a(n+1)+a(n) (signed Fibonacci numbers); or Fibonacci numbers (A000045) extended to negative indices.
 
-            sage: x = sfib.natural_object() ; x.parent()    # optional -- internet
-            Category of sequences in Integer Ring
+            sage: x = sfib.natural_object() ; x.universe()    # optional -- internet
+            Integer Ring
 
         TESTS::
 
@@ -933,16 +933,16 @@ def natural_object(self):
             QQ as continued fractions
 
             sage: s = oeis._imaginary_sequence('nonn')
-            sage: s.natural_object().parent()
-            Category of sequences in Non negative integer semiring
+            sage: s.natural_object().universe()
+            Non negative integer semiring
 
             sage: s = oeis._imaginary_sequence()
-            sage: s.natural_object().parent()
-            Category of sequences in Integer Ring
+            sage: s.natural_object().universe()
+            Integer Ring
         """
         if 'cofr' in self.keywords() and not 'frac' in self.keywords():
             return ContinuedFractionField()(self.first_terms())
-        elif 'cons' in self.keywords():
+        if 'cons' in self.keywords():
             offset = self.offsets()[0]
             terms = self.first_terms() + tuple([0] * abs(offset))
             return RealLazyField()('0' + ''.join(map(str, terms[:offset])) + '.' + ''.join(map(str, terms[offset:])))

src/sage/structure/parent.pyx

@@ -2618,17 +2618,6 @@ cdef class Parent(category_object.CategoryObject):
             else:
                 raise TypeError("_convert_map_from_ must return a map or callable (called on %s, got %s)" % (type(self), type(user_provided_mor)))
 
-        if not isinstance(S, type) and not isinstance(S, Parent):
-            # Sequences is not a Parent but a category!! As we would like to
-            # consider them as a parent we consider a workaround by using
-            # Set_PythonType from sage.structure.parent
-            from sage.categories.category_types import Sequences
-            from sage.structure.coerce_maps import ListMorphism
-            if isinstance(S, Sequences):
-                from sage.structure.sequence import Sequence_generic
-                SS = Set_PythonType(Sequence_generic)
-                return ListMorphism(SS, self.convert_map_from(list))
-
         mor = self._generic_convert_map(S)
         return mor