Categories for the working mathematics programmer (status and roadmap)
This page describes the status and roadmap for the category framework in Sage
With special thanks to Robert Bradshaw, Volker Braun, Nils Bruin, Peter Bruin, Teresa GomezDiaz, Darij Grinberg, Mike Hansen, Florent Hivert, Simon King, David Roe, Travis Scrimshaw, William Stein, and so many others ... for design discussions. Special kudos go to Simon King for all his work on the category infrastructure, in particular toward optimizations.
References
 Quickref card: sage.categories?
 Primer: sage.categories.primer?
 Technical background: Category?
 Tutorial: How to implement new algebraic structures in Sage
Design discussions:
 on sagedevel, November 2009
 Long series of comments on #10963
 ... TODO ...
Technical dependencies
 abstract methods: #6097, #6809
 lazy attributes: #4371, #5793
 cached methods: ..., #11115, #15759
 support for nested classes and dynamic classes: #5483, #5985, #5986, #5991, #9107
 support for dynamic classes: #5783
 support for lazy import: #11224, #14722, #15648
 support for large hierarchy of classes and C3: #11943, #13589
 unique representation: #5120
 testsuites: #6343, #7478, #8001, #16244
 misc: #5449, #5598, #5967, #5979, #6000, #7251, #7259, #7420, #7421
 cache and memory management: #15069, #13394, #13901, #15506, ...
Original ticket: #5891 (2009)
Typical use cases
 Combinatorial Hopf algebras and generalizations of symmetric functions: #8899, #11929, #15150, #6889
 #6812 Enumerate integer list up to the action of a Permutation Group
 Semigroups (upcoming), #11111
Categorification tickets
 polynomial rings: #9944
 (combinatorial) modules and algebras: #6136, #6137, #6138
 enumerated sets: #7403, #7395, #7396, #7397, #10193
 families: #7208
 root systems: #4326, #3663, #5794
 rings: #9138
 misc: #7921, #8001, #8044, #8562, #8576, #8911, #9138, #9944, #12877
TODO:
 make a census of the parents that are not yet categorified
 Document that the old hierarchy of abstract classes (Ring, Field, ...) is deprecated (except for some of the element classes whose presence is critical for the performance of arithmetic)
 Actually start deprecating it. This requires all/most parents to be
categorified.
 #804 Matrix should not inherit from AlgebraElement?
 #15947: Weaken types for _rmul_ and _lmul_ (Element instead of RingElement?)
 ...
 Move a lot of methods like base_ring, list, ... from Parent, Element, SageObject? to the categories #????
Category hierarchy
The current hierarchy includes all the mathematical categories of Axiom and MuPAD (courtesy of Teresa Gomez Diaz)
TODO: add link to there + to the category graph
 #15696 Implement an_instance for more categories, and extend category_sample. This could benefit from #15801.
 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 / ...)
Functorial constructions
 TensorProducts?, Algebras, Quotients, Subquotients, Subobjects, IsomorphicImages?: #8881, #10963
 WithRealizations?: #7980
 Dual, graded dual: #8881, #10963, #15647
 Graded: #8881, #10963, #11688
 Cartesian products: #8881, #16269, #10963, #15425
 Upcoming: CharacterRings?
Axioms
sage: Groups() & Sets().Finite() Category of finite groups sage: (CommutativeAdditiveGroups() & Monoids()).Distributive().Division() Category of division rings sage: _ & Sets().Finite() Category of finite fields
 Original infrastructure: #10963
 Alternative infrastructure proposal: #15701
 Finite, Infinite, Associative, Unital, Commutative, Inverse (and additive variants), Division, Distributive, ...: #10963
 FiniteDimensional?: #10963, #11111
 #15475 Reenable broken doctests in #15473 and #15476 when #10963 is merged
Morphisms, homsets
 favor *_generators over gens(): #15381
use cached_function for handling the cache of Hom?
 Hom is *not* a covariant functorial construction. This kind of works for now, but the design and user interface needs to be discussed for the long run implementation. See #10668.
 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 (see also #8900)
 Defining new inline operators, at least within the sage interpreter
Handling of the category hierarchy
 Support for subcategory methods: #12895
 Refinement of the category of a parent: ..., #14471
 Support for full subcategories: #16340
 Improve the terminology for the hierarchy relation between categories: #16183
 join and meet are misnomers: since we speak of subcategories, then the implicit order is A <= B if A is a subcategory of B. So join and meet should be exchanged. Maybe it's too late for backward compatibility. Then maybe we could use "intersect" instead of join. For meet, there does not seem to be an obvious name, but it's also very seldom used at the user interface, so the backward compatibility issue is not that relevant.
As a first step, #10963 implements the shortcuts & and  for join and meet.
Containment
 Clarify in the doc and implement systematically the distinction
between
P in MyCategory()
andP.is_...()
Support for modules and algebras
 Modules(field) returns VectorSpaces?(field): #8881
 Mantra
P in Algebras
for testing if P is an algebra (over whatever ring): done; TODO: find the ticket that implemented that!  Categories over base ring categories: #15801
 Make modules, algebras, polynomial algebra (free commutative algebra?), free algebra, exterior algebras, ... into covariant functorial constructions.
 Clarification of the specifications of modules: #16247
 Modules(R) should be restricted to R commutative, with r * m = m * r ???
 Algebras(R) should be a subcategory of Bimodules, not Modules???
 What about ModulesWithBasis??
 #15043: Rename MagmaticAlgebras?? to Algebras and Algebras to AssociativeUnitalAlgebras??
 The handling of multiplication (mul, ...) should be lifted to left / right modules
 Rings() could be a subcategory of Algebras(ZZ) (or coincide with it). CommutativeAdditiveGroups?() could coincide with 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(), Modules(Rings())...) independent of any parent. This should probably be fixable later on by introducing more lazyness.
 #11068 Basic implementation of one and twosided ideals of noncommutative rings, and quotients by twosided ideals. Since there are rings that do not inherit from sage.rings.rings.Ring, it involves moving stuff to
sage.categories.rings.Rings.ParentMethods
(depends on #9138 and #11115)
TODO: reference the related discussions on sagedevel
Optimization of the infrastructre
 #11900, #11935:
 #15801: categories over base ring categories
 #16296: Speed improvements for categories with axioms
Support for extension types
Works, up to some caveats (isinstance checks, ...). See also sage.categories.examples.semigroups_cython for further experiments.
 #7921: Original approach via getattr
 #20686: optimizing
 Better wouldbe approach: treat extension types like the others
This would require to add support for multiple inheritance from abstract classes in Cython (Robert Bradshaw: there is no theoretical obstructions; just someone needs to take care of it).
 #15718: init_extra for Cython classes
Documentation
 Main index page: #7443 TODO: refactor it (maybe as in #16256); in particular emphasize the quickref
 Primer: #5891, #10963
 Make sure that ? and ?? on categories / examples / ... point to the right place e.g.: Semigroups??, Semigroups().example()?? Semigroups().parent_class?, Semigroups.element_class?
 Improve the documentation of the implementation of axioms: #15643
 Explain category: Autogenerated overview documentation of a category: #16363
 Writing and linking to generic documentation???
Misc
 Add ObjectMethods? for category objects that are not parents?
 Allow for super classes to also provide .Element?
 Infrastructure for sharing code between the following features:
 Calculating whether an object provably belongs to a category (is_* methods)
 Testing whether an object is in the category it claims to belong to (_test_* methods)
 Inheritance from existing parents
 How to specify the category of a parent (overloading of method / argument to init) See #7921
 #15303 Coercion discovery fails to be transitive
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Sage's category graph
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