Adding p-adics to metric spaces and some cleanup.

src/doc/en/reference/categories/index.rst

@@ -56,6 +56,7 @@ Individual Categories
    sage/categories/coxeter_group_algebras
    sage/categories/coxeter_groups
    sage/categories/crystals
+   sage/categories/cw_complexes
    sage/categories/discrete_valuation
    sage/categories/distributive_magmas_and_additive_magmas
    sage/categories/division_rings
@@ -98,6 +99,7 @@ Individual Categories
    sage/categories/graded_hopf_algebras_with_basis
    sage/categories/graded_modules
    sage/categories/graded_modules_with_basis
+   sage/categories/graphs
    sage/categories/group_algebras
    sage/categories/groupoid
    sage/categories/groups
@@ -109,10 +111,13 @@ Individual Categories
    sage/categories/integral_domains
    sage/categories/lattice_posets
    sage/categories/left_modules
+   sage/categories/lie_groups
    sage/categories/magmas
    sage/categories/magmas_and_additive_magmas
    sage/categories/magmatic_algebras
+   sage/categories/manifolds
    sage/categories/matrix_algebras
+   sage/categories/metric_spaces
    sage/categories/modular_abelian_varieties
    sage/categories/modules
    sage/categories/modules_with_basis
@@ -139,6 +144,8 @@ Individual Categories
    sage/categories/sets_cat
    sage/categories/sets_with_grading
    sage/categories/sets_with_partial_maps
+   sage/categories/simplicial_complexes
+   sage/categories/topological_spaces
    sage/categories/unique_factorization_domains
    sage/categories/unital_algebras
    sage/categories/vector_spaces
@@ -187,6 +194,7 @@ Examples of parents using categories
    sage/categories/examples/commutative_additive_semigroups
    sage/categories/examples/coxeter_groups
    sage/categories/examples/crystals
+   sage/categories/examples/cw_complexes
    sage/categories/examples/facade_sets
    sage/categories/examples/finite_coxeter_groups
    sage/categories/examples/finite_dimensional_algebras_with_basis
@@ -196,8 +204,10 @@ Examples of parents using categories
    sage/categories/examples/finite_weyl_groups
    sage/categories/examples/graded_connected_hopf_algebras_with_basis
    sage/categories/examples/graded_modules_with_basis
+   sage/categories/examples/graphs
    sage/categories/examples/hopf_algebras_with_basis
    sage/categories/examples/infinite_enumerated_sets
+   sage/categories/examples/manifolds
    sage/categories/examples/monoids
    sage/categories/examples/posets
    sage/categories/examples/semigroups_cython

src/sage/categories/metric_spaces.py

@@ -81,6 +81,28 @@ class MetricSpaces(MetricSpacesCategory):
     - `d(a, b) = 0` if and only if `a = b`.
 
     A metric space is a set `S` with a distinguished metric.
+
+    .. RUBRIC:: Implementation
+
+    Objects in this category must implement either a ``dist`` on the parent
+    or the elements or ``metric`` on the parent; otherwise this will cause
+    an infinite recursion.
+
+    .. TODO::
+
+        - Implement a general geodesics class.
+        - Implement a category for metric unital additive magmas and
+          move the generic distance `d(a, b) = |a - b|` there.
+        - Incorperate the length of a geodesic as part of the default
+          distance cycle.
+
+    EXAMPLES::
+
+        sage: from sage.categories.metric_spaces import MetricSpaces
+        sage: C = MetricSpaces()
+        sage: C
+        Category of metric spaces
+        sage: TestSuite(C).run()
     """
     def _repr_object_names(self):
         """
@@ -114,7 +136,7 @@ def _test_metric(self, **options):
             for a in S:
                 for b in S:
                     d = dist(a, b)
-                    if a is not b:
+                    if a != b:
                         tester.assertGreater(d, 0)
                     else:
                         tester.assertEqual(d, 0)
@@ -132,9 +154,8 @@ def metric(self):
                 sage: m(p1, p2)
                 2.23230104635820
             """
-            return lambda a,b: self.dist(a, b)
+            return lambda a,b: a.dist(b)
 
-        @abstract_method
         def dist(self, a, b):
             """
             Return the distance between ``a`` and ``b`` in ``self``.
@@ -150,9 +171,35 @@ def dist(self, a, b):
                 sage: PD = HyperbolicPlane().PD()
                 sage: PD.dist(PD.get_point(0), PD.get_point(I/2))
                 arccosh(5/3)
+
+            TESTS::
+
+                sage: RR.dist(-1, pi)
+                4.14159265358979
+                sage: RDF.dist(1, -1/2)
+                1.5
+                sage: CC.dist(3, 2)
+                1.00000000000000
+                sage: CC.dist(-1, I)
+                1.41421356237310
+                sage: CDF.dist(-1, I)
+                1.4142135623730951
             """
+            return (self(a) - self(b)).abs()
 
     class ElementMethods:
+        def abs(self):
+            """
+            Return the absolute value of ``self``.
+
+            EXAMPLES::
+
+                sage: CC(I).abs()
+                1.00000000000000
+            """
+            P = self.parent()
+            return P.metric()(self, P.zero())
+
         def dist(self, b):
             """
             Return the distance between ``self`` and ``other``.

src/sage/rings/complex_double.pyx

@@ -199,19 +199,6 @@ cdef class ComplexDoubleField_class(sage.rings.ring.Field):
         return 561162115
         #return hash(self.str())
 
-    def dist(self, a, b):
-        """
-        Return the distance between ``a`` and ``b``.
-
-        EXAMPLES::
-
-            sage: CDF.dist(3, 2)
-            1.0
-            sage: CDF.dist(-1, I)
-            1.4142135623730951
-        """
-        return (self(a) - self(b)).abs()
-
     def characteristic(self):
         """
         Return the characteristic of the complex double field, which is 0.

src/sage/rings/complex_field.py

@@ -745,16 +745,3 @@ def _factor_univariate_polynomial(self, f):
         from sage.structure.factorization import Factorization
         return Factorization([(R(g).monic(),e) for g,e in zip(*F)], f.leading_coefficient())
 
-    def dist(self, a, b):
-        """
-        Return the distance between ``a`` and ``b``.
-
-        EXAMPLES::
-
-            sage: CC.dist(3, 2)
-            1.00000000000000
-            sage: CC.dist(-1, I)
-            1.41421356237310
-        """
-        return self(a - b).abs()
-

src/sage/rings/integer_ring.pyx

@@ -1250,19 +1250,6 @@ cdef class IntegerRing_class(PrincipalIdealDomain):
         else:
             raise ValueError("no nth root of unity in integer ring")
 
-    def dist(self, a, b):
-        """
-        Return the distance between ``a`` and ``b``.
-
-        EXAMPLES::
-
-            sage: ZZ.dist(3, 2)
-            1
-            sage: ZZ.dist(-1, 1)
-            2
-        """
-        return self(a - b).abs()
-
     def parameter(self):
         r"""
         Return an integer of degree 1 for the Euclidean property of `\ZZ`,

src/sage/rings/padics/local_generic.py

@@ -55,6 +55,7 @@ def __init__(self, base, prec, names, element_class, category=None):
             category = CompleteDiscreteValuationFields()
         else:
             category = CompleteDiscreteValuationRings()
+        category = category.Metric().Complete()
         if default_category is not None:
             category = check_default_category(default_category, category)
         Parent.__init__(self, base, names=(names,), normalize=False, category=category, element_constructor=element_class)

src/sage/rings/padics/padic_base_leaves.py

@@ -231,7 +231,7 @@ def __init__(self, p, prec, print_mode, names):
             sage: TestSuite(R).run(elements = [R.random_element() for i in range(3^3)])
 
             sage: R = ZpCR(3, 2)
-            sage: TestSuite(R).run(elements = [R.random_element() for i in range(3^6)])
+            sage: TestSuite(R).run(elements = [R.random_element() for i in range(3^6)]) # long time
 
             sage: R = ZpCR(next_prime(10^60))
             sage: TestSuite(R).run()
@@ -320,7 +320,7 @@ def __init__(self, p, prec, print_mode, names):
             sage: TestSuite(R).run(elements = [R.random_element() for i in range(3^3)])
 
             sage: R = ZpCA(3, 2)
-            sage: TestSuite(R).run(elements = [R.random_element() for i in range(3^6)])
+            sage: TestSuite(R).run(elements = [R.random_element() for i in range(3^6)]) # long time
 
             sage: R = ZpCA(next_prime(10^60))
             sage: TestSuite(R).run()
@@ -411,7 +411,7 @@ def __init__(self, p, prec, print_mode, names):
             sage: TestSuite(R).run(elements = [R.random_element() for i in range(3^3)])
 
             sage: R = ZpFM(3, 2)
-            sage: TestSuite(R).run(elements = [R.random_element() for i in range(3^6)])
+            sage: TestSuite(R).run(elements = [R.random_element() for i in range(3^6)]) # long time
 
             sage: R = ZpFM(next_prime(10^60))
             sage: TestSuite(R).run()
@@ -525,10 +525,12 @@ def __init__(self, p, prec, print_mode, names):
             sage: TestSuite(R).run(elements = [R.random_element() for i in range(2^10)], max_runs = 2^12) # long time
 
             sage: R = Qp(3, 1)
-            sage: TestSuite(R).run(elements = [R.random_element() for i in range(3^6)])
+            sage: TestSuite(R).run(elements = [R.random_element() for i in range(3^6)]) # long time
 
             sage: R = Qp(3, 2)
-            sage: TestSuite(R).run(elements = [R.random_element() for i in range(3^9)])
+            sage: TestSuite(R).run(elements=[R.random_element() for i in range(3^9)],
+            ....:                  skip="_test_metric") # Skip because too long
+            sage: R._test_metric(elements=[R.random_element() for i in range(3^3)])
 
             sage: R = Qp(next_prime(10^60))
             sage: TestSuite(R).run()

src/sage/rings/padics/padic_generic.py

@@ -56,6 +56,7 @@ def __init__(self, base, p, prec, print_mode, names, element_class, category=Non
                 category = Fields()
             else:
                 category = PrincipalIdealDomains()
+        category = category.Metric().Complete()
         LocalGeneric.__init__(self, base, prec, names, element_class, category)
         self._printer = pAdicPrinter(self, print_mode)
 

src/sage/rings/rational_field.py

@@ -934,19 +934,6 @@ def algebraic_closure(self):
         from sage.rings.all import QQbar
         return QQbar
 
-    def dist(self, a, b):
-        """
-        Return the distance between ``a`` and ``b``.
-
-        EXAMPLES::
-
-            sage: QQ.dist(3, 2)
-            1
-            sage: QQ.dist(-1, 1/2)
-            3/2
-        """
-        return self(a - b).abs()
-
     def order(self):
         r"""
         Return the order of `\QQ` which is `\infty`.

src/sage/rings/real_double.pyx

@@ -236,19 +236,6 @@ cdef class RealDoubleField_class(Field):
             return 0
         return cmp(type(self), type(x))
 
-    def dist(self, a, b):
-        """
-        Return the distance between ``a`` and ``b`` in ``self``.
-
-        EXAMPLES::
-
-            sage: RDF.dist(5, 1/2)
-            4.5
-            sage: RDF.dist(-1, 1/2)
-            1.5
-        """
-        return (self(a) - self(b)).abs()
-
     def construction(self):
         r"""
         Returns the functorial construction of ``self``, namely, completion of

src/sage/rings/real_mpfr.pyx

@@ -801,19 +801,6 @@ cdef class RealField_class(sage.rings.ring.Field):
         """
         return self.complex_field()
 
-    def dist(self, a, b):
-        """
-        Return the distance between ``a`` and ``b``.
-
-        EXAMPLES::
-
-            sage: RR.dist(3, 2)
-            1.00000000000000
-            sage: RR.dist(-1, pi)
-            4.14159265358979
-        """
-        return self(a - b).abs()
-
     def ngens(self):
         """
         Return the number of generators.