wiki:CategoriesRoadMap

Version 40 (modified by nthiery, 5 years ago) (diff)

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#5891: Categories for the working mathematics programmer (status and roadmap)

This page describes the status and roadmap for the category patches.

This (series of) patch(es) extends the Sage category framework as a design pattern for organizing generic code.

(with special thanks to Robert Bradshaw, Mike Hansen, Florent Hivert, David Roe, William Stein, ... for design discussions).

The latest version of all the patches are available from: http://combinat.sagemath.org/patches/file/ -> browse Please review them from there!

Small technical patches the category patch depends on:

  • #5449: cached_in_parent_method-5449-submitted.patch
  • #5783: lazy_attributes-fixes-5783-final.patch
  • #5120: unique_representation-5120-submitted.patch
  • #5967: element_wrapper-5967-submitted.patch
  • #6000: transitive_ideal-6000-submitted.patch
  • #5483: explain-pickle-v1.patch
  • #6097: abstract_method-6097-nt.patch
  • #6343: sage_object-test-6343-nt.patch (with patch, needs review)
  • #5979: parent-element_constructor-fix-5979-submitted.patch
  • #6809: abstract_methods_of_class
  • #5985: customized class pickling
  • #5986: pickling of nested classes
  • #5991: dynamic_class
  • #5598: 5598-coerce-declare.patch (Robert should polish by 17/10; marked as depending on #5597?)
  • #7251: Allow for Integer(3, parent = MyParent?)

Main patches #5891:

Second batch of patches that will need to go in at once (in separate tickets). Up to final polishing, they are ready for inclusion.

Over patches basically with positive review, and depending on the previous ones:

  • #3663 (affine crystals) to be reviewed by Dan
  • #5794 branching rules (positive review: Brant)

Later patches using categories:

Current status:

  • All test pass
  • Categories:
    • Infrastructure:
      • sage.categories.category (500 lines of code)
      • sage.structure.parent (100 lines of code)
      • class manipulation: (25 lines of code)
    • All the mathematical categories of Axiom and MuPAD (Courtesy of Teresa Gomez Diaz) (except *WithSeveralBases? which are not needed anymore; see .abstract_category()) See the category graph: attachment:sage-category-graph.pdf
    • (Infinite/Finite?)EnumeratedSets? (with example) (Courtesy of Florent Hivert)
    • Semigroups (with example, basic methods, subquotients)
    • FiniteSemigroups? (with example, cayley graphs, basic representation theory, ...)
    • ModulesWithBasis? (with example, morphisms)
    • HopfAlgebras? & friends (with a couple examples)
    • Cleanup:
      • Have unique representation by default (no need to inherit from Category_uniq)
      • Have construction / reduce by default
      • Can systematically use the idiom P in Rings()
  • Functorial constructions:
    • cartesian product, tensor_product
    • subquotient, subset, quotient (in progress)
    • dual (very preliminary)
    • isomorphism type / abstract (very preliminary)
  • Homomorphisms:
    • Integrates with current sage morphisms
    • Adds morphisms for some categories
    • Some general infrastructure
    • Hom:
      • Simplification of the logic. Hopefully completely compatible.
      • X._Hom_(Y, category) may now return NotImplemented?
      • failed attempt at using cached_function (to be pursued)
    • Systematic use of category = ... instead of cat = This may be backward incompatible.
  • Generic test framework: Functional. See Todo's in Sets.Parent.check? for possible improvements
  • Lazy attributes: Non essential, and essentially invisible from the user interface Powerful tool for:
    • Conditionally defined methods
    • Enriched methods
  • Categories and Cython: Proof-of-concept for Cython classes to inherit code from categories See: sage.categories.examples.semigroups_cython
  • Reorganization of the Sage library to start using the category framework:
    • Combinatorial free modules
      • Have unique representation, use coercion, are more robust
      • Improved _repr_
      • Handle combinatorial classes that can't be sorted as indices
    • Groups:
    • NumberFields?: bug fix (categories-numberfield_homset-nt.patch / needs review): Hom(SomeNumberField?,SomeVectorSpace?) returned a numberfield homset
    • Fixed some import loops
    • Added temporary list() methods to:
    • Added sage.sets.finite_enumerated_set
    • Added aliases one = one_element, zero = zero_element to start the transition (to be discussed)
    • Naming conventions and cleanup:
      • parent.product(x,y) parent.product parent.product_on_basis (was: multiply, _multiply, multiply_basis, _multiply_basis)
      • parent.summation(x,y) parent.summation # risk of confusion with infinite summation / ...
  • parent.sum ([x,y,z,y])
  • parent.prod([x,y,z,y])
  • parent.coproduct, parent.coproduct_on_basis, parent.coproduct_on_generators
  • parent.antipode, parent.antipode_on_basis, parent.antipode_on_generators
  • cat.example() cat.counter_example()
  • A.one() A.zero() a.is_one() a.is_zero() A(1) A(0) when it makes sense A.one_element() A.zero_element() deprecated in the doc; fully deprecated later
  • Use class.an_instance() whenever meaningful
  • parent.an_element() parent.some_elements(); possibly parent.example() parent.counterexample()
  • all_weakly_super_categories -> all_super_categories(proper=False)
  • Todo for after:
    • Add more examples of parents (HELP!): Ultimately, this should be for all categories, Those are needed urgently:
      • Modules (for doctests for mul / rmul / ...)
      • Algebras (for doctests for mul / rmul / ...)
      • CommutativeAdditiveSemigroups? (for doctesting add / sub / ...)
    • Allow for super classes to also provide .Element
    • Make sure that ? and ?? on categories / examples / ... point to the right place e.g.: Semigroups??, Semigroups().example()?? Semigroups().parent_class?, Semigroups.element_class?
    • abstract / isomorphism type / with_several_representations
    • Choose, implement, and use systematically a mantra for testing if P is an algebra (over whatever ring):
      • P in Algebras
      • P in Algebras()
  • Long run discussions:
    • init_add
    • Writing and linking to generic documentation
    • Standardize the functorial constructions, and streamline with variants (finite / finite dimensional / graded / abstract / ...)
    • Hom is *not* a functorial construction. This works for now, but the design and user interface needs to be discussed for the long run implementation.
    • What's the right inheritance structure for homsets? Also, how should one specify that a category's homsets are enriched in another category? Via a method like "enriched_in()" on the category? By extra_super_categories() in a custom HomCategory??

Current status: if B is a subcategory of A, we want HomCategory?(B).element_class to inherit from HomCategory?(A), and a coercion Hom(P1, P2, B) -> Hom(P1, P2, A). Such coercions cannot be handled well by the standard coercion mechanism, as Hom(P1, P2, Sets) will have *lot* of coercion sources. But Hom sets could have special rules (like tensor products will have).

  • Support for multivariate morphisms, i.e. morphisms (A x B) -> C where the specializations (A x b) -> C and and (a x B) -> C are morphisms for possibly different categories
  • Defining new inline operators, at least within the sage interpreter
  • Convention for the order in super_categories (see primer)
  • Cythonification, and merging of the code with sage.structure.element.*Element?
  • Inheritance from existing parents
  • How to specify the category of a parent (overloading of method / argument to init)
  • Rings() could be a subcategory of Algebras(ZZ) (or coincide with it?), Groups() of Modules(ZZ). In a similar vein, a field F could be in the category Modules(F). However, this can cause some trouble by introducing cross dependencies: ZZ involves constructing the category Rings() which uses Modules(ZZ) which itself uses ZZ. To avoid this, the current setup makes the basic categories (Groups(), Rings(), ...) independent of any parent. This should probably be fixable later on by introducing more lazyness.

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