Fix element and parent classes of Hom categories to be abstract, and simplify the Hom logic.

[nthiery]

This patch fixes the parent and element classes for Hom categories to
be purely abstract, and simplifies the Hom logic:

• Unified the logic for selecting the class when building a Homset
  (e.g. Homset, RingHomset, HeckeModuleHomspace, ...). This is now
  systematically done through the _Hom_ hook. The logic still has a
  fundamental flaw, but that's for the later Refactor category support for morphisms (Hom is not a functorial construction!) #10668.
• The cache for Hom is handled at a single point in Hom
  In particular, homsets created via the _Hom_ hook are now unique.
• If category is None, Hom simply calls itself with the meet of the
  categories of the parent, which removes a cache handling duplication
  in the code
• Parent.Hom calls Hom directly (removes duplicate _Hom_ logic).
• ParentWithBase.Hom was redundant and is gone.
• Reduce the footprint of the current trick to delegate
  Hom(F,F)(on_basis=...) to module_morphism, allow for the diagonal
  option too, an make sure the homset category is set properly.
• Update a doctest in sage.modules.vector_space_homspace to take into
  account that homsets created via _Hom_ are now unique.
• Scheme is (apparently) an abstract base class; so it should not be
  instantiated. I changed some doctests in
  sage.schemes.generic.SchemeMorphism to use instead the concrete
  Spec(ZZ). Those doctests were breaking because Scheme does not
  implement equality, which is required for Hom caching.

As a byproduct, the HeckeModules category does not import any more
HeckeModulesHomspace, which was a recurrent source of import loops.

#11935 depends on this ticket

Apply:

• attachment: trac_12876_category_abstract_classes_for_hom.patch

Depends on #715
Depends on #11521
Depends on #12215
Depends on #12313
Depends on #13412
Depends on #13145
Depends on #14159
Depends on #13184
Depends on #14287
Depends on #14217

CC: @sagetrac-sage-combinat @simon-king-jena

Component: categories

Keywords: Hom

Author: Nicolas M. Thiéry

Reviewer: Simon King

Merged: sage-5.11.beta0

Issue created by migration from https://trac.sagemath.org/ticket/12876

[nthiery]

comment:1

Note: this might depend on #12875

[nthiery]

Description changed:

--- 
+++ 
@@ -24,3 +24,4 @@
 As a byproduct, the HeckeModules category does not import any more
 HeckeModulesHomspace, which was a recurrent source of import loops.
 
+#11935 depends on this ticket

[nthiery]

Reviewer: Simon King

[nthiery]

comment:3

Note: there are two doctests failures that I don't know how to handle, related to the refcounting of Singular rings::

sage -t  devel/sage-combinat/sage/rings/polynomial/multi_polynomial_libsingular.pyx
**********************************************************************
File "/opt/sage-5.0.beta11/devel/sage-combinat/sage/rings/polynomial/multi_polynomial_libsingular.pyx", line 418:
    sage: len(ring_refcount_dict) == n
Expected:
    True
Got:
    False
sage -t  devel/sage-combinat/sage/libs/singular/ring.pyx
**********************************************************************
File "/opt/sage-5.0.beta11/devel/sage-combinat/sage/libs/singular/ring.pyx", line 469:
    sage: ring_ptr in ring_refcount_dict
Expected:
    False
Got:
    True

Help fixing those welcome!

[nthiery]

comment:4

The updated patch includes two small related improvements I had in later patches. I moved them here to resolve the conflict.

[nthiery]

Description changed:

--- 
+++ 
@@ -12,7 +12,8 @@
 - Parent.Hom calls  Hom directly (removes duplicate `_Hom_` logic).
 - ParentWithBase.Hom was redundant and is gone.
 - Reduce the footprint of the current trick to delegate
-  Hom(F,F)(on_basis=...) to module_morphism.
+  Hom(F,F)(on_basis=...) to module_morphism, allow for the diagonal
+  option too, an make sure the homset category is set properly.
 - Update a doctest in sage.modules.vector_space_homspace to take into
   account that homsets created via `_Hom_` are now unique.
 - Scheme is (apparently) an abstract base class; so it should not be

[nthiery]

comment:6

Oops. The updated patch includes two further hunks I had forgotten.

[nthiery]

comment:7

Arr, yet another missing hunk ... Time to go to bed!