Trac #17359: Hom sets for schemes and Hecke modules

In #17364, the supercategories of homsets of schemes were fixed: instead
of
{{{
sage: Schemes().Homsets().all_super_categories()
[Category of homsets of schemes, Category of objects]
}}}
we now have
{{{
sage: Schemes().Homsets().all_super_categories()
[Category of homsets of schemes,
 Category of homsets,
 Category of sets,
 Category of sets with partial maps,
 Category of objects]
}}}
There are two TODO items left from #17364, namely to determine the
correct categories for homsets of schemes and of Hecke modules.  The
purpose of this ticket is to resolve these TODOs as follows:
- Hom sets of schemes are unchanged (do not have any extra structure)
- Hom sets of Hecke modules over a ring ''A'' are now in the category of
''A''-modules.  (Note that `HeckeModules(A)` requires ''A'' to be
commutative, while Hecke algebras are not always commutative.  Hence Hom
sets are modules over ''A'' but not in general over the Hecke algebra.)

URL: http://trac.sagemath.org/17359
Reported by: pbruin
Ticket author(s): Peter Bruin
Reviewer(s): Jean-Pierre Flori

src/sage/categories/hecke_modules.py

@@ -152,12 +152,9 @@ def _Hom_(self, Y, category):
 
     class Homsets(HomsetsCategory):
         """
-        .. TODO:: shall there be any additional category structure on Homsets of hecke modules?
-
         TESTS::
 
-            sage: HeckeModules(ZZ).Homsets().super_categories()
-            [Category of homsets]
+            sage: TestSuite(HeckeModules(ZZ).Homsets()).run()
         """
 
         def base_ring(self):
@@ -169,5 +166,20 @@ def base_ring(self):
             """
             return self.base_category().base_ring()
 
+        def extra_super_categories(self):
+            """
+            TESTS:
+
+            Check that Hom sets of Hecke modules are in the correct
+            category (see :trac:`17359`)::
+
+                sage: HeckeModules(ZZ).Homsets().super_categories()
+                [Category of modules over Integer Ring, Category of homsets]
+                sage: HeckeModules(QQ).Homsets().super_categories()
+                [Category of vector spaces over Rational Field, Category of homsets]
+            """
+            from sage.categories.modules import Modules
+            return [Modules(self.base_ring())]
+
         class ParentMethods:
             pass

src/sage/categories/schemes.py

@@ -44,6 +44,11 @@ class Schemes(Category):
     TESTS::
 
         sage: TestSuite(Schemes()).run()
+
+    Check that Hom sets of schemes are in the correct category::
+
+        sage: Schemes().Homsets().super_categories()
+        [Category of homsets]
     """
 
     @staticmethod
@@ -145,16 +150,6 @@ def _call_(self, x):
             raise TypeError("No way to create an object or morphism in %s from %s"%(self, x))
 
 
-    class Homsets(HomsetsCategory):
-        """
-        TESTS::
-
-            sage: Schemes().Homsets().super_categories()
-            [Category of homsets]
-
-        .. TODO:: shall there be any additional category structure on Homsets of hecke modules?
-        """
-
 #############################################################
 # Schemes over a given base scheme.
 #############################################################