10668: Explicitly document that Element.__getattr__ can also be usefu…

…l when refining categories

src/sage/rings/polynomial/polynomial_quotient_ring.py

@@ -267,9 +267,11 @@ class of the category, and store the current class of the quotient
         sage: [s for s in dir(Q.category().element_class) if not s.startswith('_')]
         ['cartesian_product', 'gcd', 'is_idempotent', 'is_one', 'is_unit', 'lcm', 'lift', 'xgcd']
 
-    As one can see, the elements are now inheriting additional methods: lcm and gcd. Even though
-    ``Q.an_element()`` belongs to the old and not to the new element class, it still inherits
-    the new methods from the category of fields::
+    As one can see, the elements are now inheriting additional
+    methods: lcm and gcd. Even though ``Q.an_element()`` belongs to
+    the old and not to the new element class, it still inherits the
+    new methods from the category of fields, thanks to
+    :meth:`Element.__getattr__`::
 
         sage: isinstance(Q.an_element(),Q.element_class)
         False
@@ -287,7 +289,6 @@ class of the category, and store the current class of the quotient
         sage: TestSuite(Q(x)).run()
         sage: isinstance(Q(x), Q.element_class)
         True
-
     """
     Element = PolynomialQuotientRingElement
 

src/sage/structure/element.pyx

@@ -277,16 +277,40 @@ cdef class Element(sage_object.SageObject):
 
     def __getattr__(self, str name):
         """
-        Let cat be the category of the parent of ``self``.  This
-        method emulates ``self`` being an instance of both ``Element``
-        and ``cat.element_class``, in that order, for attribute
-        lookup.
+        Lookup a method or attribute from the category abstract classes.
 
-        NOTE:
+        Let ``P`` be a parent in a category ``C``. Usually the methods
+        of ``C.element_class`` are made directly available to elements
+        of ``P`` via standard class inheritance. This is not the case
+        any more if the elements of ``P`` are instances of an
+        extension type. See :class:`Category`. for details.
+
+        The purpose of this method is to emulate this inheritance: for
+        ``e`` and element of ``P``, if an attribute or method
+        ``e.foo`` is not found in the super classes of ``e``, it's
+        looked up manually in ``C.element_class`` and bound to ``e``.
+
+        .. NOTES::
+
+            - Attributes beginning with two underscores but not ending
+              with an unnderscore are considered private and are thus
+              exempted from the lookup in ``cat.element_class``.
 
-        Attributes beginning with two underscores but not ending with
-        an unnderscore are considered private and are thus exempted
-        from the lookup in ``cat.element_class``.
+            - The attribute or method is actually looked up in
+              ``P._abstract_element_class``. In most cases this is
+              just an alias for ``C.element_class``, but some parents,
+              notably homsets, customizes this to let elements also
+              inherit from other abstract classes. See
+              :meth:`Parent._abstract_element_class` and
+              :meth:`Homset._abstract_element_class` for details.
+
+            - This mechanism may also enter into action when the
+              category of `P` is refined on the fly, leaving
+              previously constructed elements in an outdated element
+              class.
+
+              See :class:`~sage.rings.polynomial.polynomial_quotient_ring.PolynomialQuotientRing_generic`
+              for an example.
 
         EXAMPLES:
 
@@ -330,7 +354,6 @@ cdef class Element(sage_object.SageObject):
             Traceback (most recent call last):
             ...
             AttributeError: 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint' object has no attribute '__foo'
-
         """
         if (name.startswith('__') and not name.endswith('_')):
             dummy_error_message.cls = type(self)