ScalarField.codomain: New, put scalar fields in category of continuou…

…s maps

src/sage/manifolds/differentiable/manifold.py

@@ -196,7 +196,7 @@
     Algebra of differentiable scalar fields on the 2-dimensional differentiable
      manifold S^2
     sage: f.parent().category()
-    Category of commutative algebras over Symbolic Ring
+    Join of Category of commutative algebras over Symbolic Ring and Category of homsets of topological spaces
 
 A differentiable manifold has a default vector frame, which, unless otherwise
 specified, is the coordinate frame associated with the first defined chart::
@@ -376,7 +376,7 @@
     Algebra of differentiable scalar fields on the 1-dimensional complex
      manifold C*
     sage: f.parent().category()
-    Category of commutative algebras over Symbolic Ring
+    Join of Category of commutative algebras over Symbolic Ring and Category of homsets of topological spaces
 
 A vector field on the Riemann sphere::
 

src/sage/manifolds/manifold.py

@@ -188,7 +188,7 @@
     sage: f.parent()
     Algebra of scalar fields on the 2-dimensional topological manifold S^2
     sage: f.parent().category()
-    Category of commutative algebras over Symbolic Ring
+    Join of Category of commutative algebras over Symbolic Ring and Category of homsets of topological spaces
 
 
 .. RUBRIC:: Example 2: the Riemann sphere as a topological manifold of
@@ -295,7 +295,7 @@
     Algebra of scalar fields on the Complex 1-dimensional topological
      manifold C*
     sage: f.parent().category()
-    Category of commutative algebras over Symbolic Ring
+    Join of Category of commutative algebras over Symbolic Ring and Category of homsets of topological spaces
 
 AUTHORS:
 
@@ -1833,7 +1833,7 @@ def scalar_field_algebra(self):
             sage: CU = U.scalar_field_algebra() ; CU
             Algebra of scalar fields on the Open subset U of the 3-dimensional topological manifold M
             sage: CU.category()
-            Category of commutative algebras over Symbolic Ring
+            Join of Category of commutative algebras over Symbolic Ring and Category of homsets of topological spaces
             sage: CU.zero()
             Scalar field zero on the Open subset U of the 3-dimensional topological manifold M
 

src/sage/manifolds/scalarfield.py

@@ -1580,6 +1580,21 @@ def domain(self):
         """
         return self._domain
 
+    def codomain(self):
+        r"""
+        Return the codomain of the scalar field.
+
+        EXAMPLES::
+
+            sage: M = Manifold(2, 'M', structure='topological')
+            sage: c_xy.<x,y> = M.chart()
+            sage: f = M.scalar_field(x+2*y)
+            sage: f.codomain()
+            Real Field with 53 bits of precision
+
+        """
+        return self._domain.base_field()
+
     def copy(self, name=None, latex_name=None):
         r"""
         Return an exact copy of the scalar field.

src/sage/manifolds/scalarfield_algebra.py

@@ -35,6 +35,7 @@
 from sage.structure.unique_representation import UniqueRepresentation
 from sage.misc.cachefunc import cached_method
 from sage.categories.commutative_algebras import CommutativeAlgebras
+from sage.categories.topological_spaces import TopologicalSpaces
 from sage.symbolic.ring import SR
 from sage.manifolds.scalarfield import ScalarField
 
@@ -86,11 +87,11 @@ class ScalarFieldAlgebra(UniqueRepresentation, Parent):
     :class:`~sage.symbolic.ring.SymbolicRing`)::
 
         sage: CM.category()
-        Category of commutative algebras over Symbolic Ring
+        Join of Category of commutative algebras over Symbolic Ring and Category of homsets of topological spaces
         sage: CM.base_ring()
         Symbolic Ring
         sage: CW.category()
-        Category of commutative algebras over Symbolic Ring
+        Join of Category of commutative algebras over Symbolic Ring and Category of homsets of topological spaces
         sage: CW.base_ring()
         Symbolic Ring
 
@@ -383,7 +384,7 @@ def __init__(self, domain):
         if domain.base_field_type() in ['real', 'complex']:
             base_field = SR
         Parent.__init__(self, base=base_field,
-                        category=CommutativeAlgebras(base_field))
+                        category=CommutativeAlgebras(base_field) & TopologicalSpaces().Homsets())
         self._domain = domain
         self._populate_coercion_lists_()