Categories for finite (enumerated) sets

[nthiery]

There still remains a lot of finite (enumerated) sets that are not in the appropriate categories::

sage: CartesianProduct([1,2],[2,3]).category()
Category of enumerated sets
sage: LatticePolytope(matrix(ZZ, [[1,1],[0,1]])).category()
Category of objects

See also: #12913

The following examples were originally in the ticket description but are good on sage-6.9.beta1

sage: Permutations().category()
Category of infinite enumerated sets
sage: DirichletGroup(5).category()
Category of finitely generated finite commutative groups

Moreover, with Sage 7.4.beta0 we now have:

sage: CartesianProduct([1,2],[2,3]).category()
... DeprecationWarning: CartesianProduct is deprecated. Use cartesian_product instead
See http://trac.sagemath.org/18411 for details.
  #!/usr/bin/env python
Category of Cartesian products of finite enumerated sets
sage: cartesian_product([[1,2],[2,3]]).category()
Category of Cartesian products of finite enumerated sets
sage: Set([1,2,3]).category()
Category of finite sets

Formerly not working, ok in sage 9.3.b6:

sage: Set([1,2,3]).category()
Category of finite sets
sage: (GF(3) ^ 2) in FiniteEnumeratedSets()
True
sage: MatrixSpace(GF(3),2) in FiniteEnumeratedSets()
True

[vdelecroix]

On sage-6.9.beta1 things get better... I modified the description accordingly

[mkoeppe]

I have updated the description with the status as of version 7.4.beta0.

Regarding LatticePolytope (one of the examples in the description), I would be reluctant to consider it a finite enumerated set. There should be a distinction between the lattice polytope (an infinite set) and the set of its lattice points. The latter (which also makes sense for non-lattice polytopes) should be made a finite enumerated set.