Implement generic support for parents with (multiple) realizations

[Jason Bandlow]

This patch implement generic support for parents with (multiple) realizations See:

    Sets().WithRealizations().example()

[Mike Hansen]

Didn't we want Realization rather than Representation?

[Simon King]

Since the ticket description is rather sparse, I'm collecting here some statements that emerged from reading the patch and discussing with Nicolas, who is sitting beside me.

• It disposes of abstract categories. Hooray!
• On the one hand, one has (abstract) vectorspaces. A realization of such abstract vector space is given by specific data, for example by a basis, but it should be emphasized that not all realizations of a vector space are necessarily in terms of a basis. So, not any realization of a vectorspace is a "vectorspace with basis".
• The realizations of a vector space are related by canonical morphisms (for example, change of basis).
• Is any "vectorspace with basis" a realisation of a vector space? No! There is no gurantee that this specific vectorspace with basis is equiped with the above-mentioned canonical morphisms.

TODO

• The docstring of the class Category_realization has no examples. Its __init__ method should define its arguments and provide a test. _get_name is not documented either.
• Similarly, Realizations is undocumented.
• Realizations.super_categories() returns [Sets()]. Idea of Nicolas: Add an optional argument Realizations(parent, category=Bla), so that the default for Bla is Sets(). The given category will be returned as super category.

• I tried some example, with comments/questions inserted:

  sage: C = Algebras(QQ)
  sage: C.WithRealizations()
  Join of Category of algebras over Rational Field and Category of sets with realizations
  

  Does it really have to be a join category?

  sage: from sage.categories.category_types import Category_realization
  sage: Category_realization(QQ)
  The category of category_realization of Rational Field
  

  So, do I understand correctly that the set of all realisations of QQ forms a category, each object being a realisation of QQ, the morphisms being canonical maps between the different realisations?

  sage: Category_realization(QQ).is_subcategory(Rings().WithRealizations())
  BOOM
  sage: hasattr(Category_realization(QQ), 'super_categories')
  False
  

  So, that's a bug. Category_realization(QQ) is a category and thus MUST have a super_categories method.

Is the category of "Bla with realisations" a 2-category?

Since Nicolas is now teaching, I can not ask him directly:

• If I understand correctly, for any parent P there is a category Category_realizations(P).
• For any category C, there is a category C.WithRealizations().
• Shouldn't it be the case that for any P in C, the realisations of P (thus, the category Category_realizations(P)) is an object in C.WithRealizations()?
• But if the objects of C.WithRealizations() are categories themselves, shouldn't we first implement the notion of higher categories in Sage, before we treat the category of realizations of objects of C?

[Simon King]

Replying to SimonKing:

Is the category of "Bla with realisations" a 2-category?

No, it isn't. I had a totally wrong notion of a higher category. Sorry.

Anyway, the missing super_categories method must be provided (as for any category).

[Nicolas M. Thiéry]

100% doctest coverage; all test pass

Category_realization (now Category_realization_of_parent) does not need a super_category method: it is an abstract class.

[Florent Hivert]

Hi Nicolas,

I just posted a ​review patch on sage-combinat queue. Here are the main changes

1. This is mostly documentation, except that I changed trivially the code in

the example, by renaming some parameter names:

• in SubsetAlgebra(S) the parameter S is a set;

• in Fundamental(S), In(S) and Out(S) it is a SubsetAlgebra;

I was really confused about that reading the code so that I renamed S to SAlg in the second case and added an INPUT: field in the doc

2. The correct markup for see also is the following:

   .. seealso::
      bla bla bla
      bla bla 
   

as variants, you can use uppercase as Sphinx markup is not case sensitive. You can also put everything on one line if it fits:

.. SEEALSO:: bla bla bla

If you put a space as in ..see also:: then this become a comment and is ignored by Sphinx ! This remind me that I should finish #12078 Add an example of SEE ALSO

3. There is a missing doctest in my review patch

       sage: A.F() in A.Realizations()
   Expected:
       True
   Got:
       False
   

I left it on purpose. I realize that the correct doctest is

    sage: A.F() in A.Bases()
Expected:
    True
Got:
    False

But it took me quite a lot of thinking understanding that. I think we should explain this somewhere, probably where the broken test is.

Please review my numerous doc change, fold the patch if you are Ok with it.

Florent

[Nicolas M. Thiéry]

Replying to hivert:

I just posted a review patch on sage-combinat queue.

Thanks!

I went through it, folded it in, and added a [​http://combinat.sagemath.org/patches/file/tip/trac_7980-multiple-realizations-review-nt.patch review review patch]

1. This is mostly documentation, except that I changed trivially the code in

the example, by renaming some parameter names:

• in SubsetAlgebra(S) the parameter S is a set;

• in Fundamental(S), In(S) and Out(S) it is a SubsetAlgebra;

I was really confused about that reading the code so that I renamed S to SAlg in the second case and added an INPUT: field in the doc

Good point. I ended up renaming SAlg to A since this is what we use in all the examples.

3. There is a missing doctest in my review patch

       sage: A.F() in A.Realizations()
   Expected:
       True
   Got:
       False
   

Ouch! That's not good. I am not sure this is the best approach, but I added A.Realizations() to the super categories Bases(A). Maybe A.Realizations() should be Bases(A) instead. In any cases, that will work for now. I added A._test_realizations() which checks:

    R in A.Realizations() for R in A.realizations()

Feel free to fold in my review patch and repost after checking it out.

Cheers,

Nicolas

[Nicolas M. Thiéry]

Please update the queue first, since I took off the fix to the CategoryObject? link to put it in #9469

[Nicolas M. Thiéry]

Replying to nthiery:

Please update the queue first, since I took off the fix to the CategoryObject? link to put it in #9469

Again: I fixed a couple trivially failing doctests.

[Simon King]

Replying to jbandlow:

This patch implement generic support for parents with (multiple) realizations

What patch? This ticket does not provide any patch.

[Nicolas M. Thiéry]

Replying to SimonKing:

Replying to jbandlow:

This patch implement generic support for parents with (multiple) realizations

What patch? This ticket does not provide any patch.

I should have mentioned that it is in on the Sage-Combinat queue, and the review by Florent is occuring here. In such situations it's a waste of time to update the patch on trac all the time; and leaving an outdated patch on trac is not so good either (someone might waste time reviewing old stuff).

[Florent Hivert]

For the record: as of version ​a7993225ce8e/trac_7980-multiple-realizations-nt.patch I'm Ok with the patch upto a few documentation glitches. I pushed the following patch ​a7993225ce8e/trac_7980-multiple-realizations-review-fh.patch. If you are Ok with my review patch and the tests pass you can set positive review on my behalf. If so don't forget to fold and update.

Florent

[Nicolas M. Thiéry]

I folded Florent's patch, and we fixed together a couple remaining glitches on the Sage-Combinat queue.

Florent: the sphinx errors were clearly due to a broken sphinx state on my test machine. After nuking the doctrees, this compiled smoothly. So I set a positive review on your behalf.

[Florent Hivert]

One more ! Good !

Florent