#10667: Morphisms and Objects of Categories
---------------------------+------------------------------------------------
Reporter: SimonKing | Owner: nthiery
Type: enhancement | Status: needs_review
Priority: major | Milestone:
Component: categories | Keywords: objects morphisms containment
Work_issues: | Upstream: N/A
Reviewer: | Author: Simon King
Merged: | Dependencies: #9138, #11115
---------------------------+------------------------------------------------
Changes (by SimonKing):
* status: needs_work => needs_review
* dependencies: => #9138, #11115
Comment:
I created a new version of my patch. The aim is to make the performance of
containment tests even better. I did the following, compared with the old
patch:
* I make the patch dependent on #9138 and #11115 (both help to come to
speed).
* I've put power series rings into the category and coercion framework.
* I introduced base() for join categories: If at least one of the
underlying categories has a base and if there is no conflict with
different bases, then the join shall have that base as well. That is
needed because some algebras have a join category by #9138.
* SimplicialComplex has not been derived from CategoryObject, even though
its instances are objects of a category! I corrected it.
* GroupAlgebras should not only be Hopf algebras with basis but also
group algebras. Hence, I made it a join of the two.
* I implemented is_supercategory (there has only been is_subcategory),
and use it to make containment tests even faster. It depends on #11115,
because I use the Cython class of cached methods.
I guess the best news is that the containment test via category framework
can now compete with a pure class check, if that class check is done in
Python. I take, for example, commutative rings:
{{{
sage: from sage.rings.commutative_ring import is_CommutativeRing
sage: is_CommutativeRing??
Type: function
Base Class: <type 'function'>
String Form: <function is_CommutativeRing at 0x118c488>
Namespace: Interactive
File:
/mnt/local/king/SAGE/sandbox/sage-4.7.2.alpha2/local/lib/python2.6/site-
packages/sage/rings/commutative_ring.py
Definition: is_CommutativeRing(R)
Source:
def is_CommutativeRing(R):
return isinstance(R, CommutativeRing)
sage: is_CommutativeRing(QQ)
True
sage: s = SymmetricGroup(4)
sage: is_CommutativeRing(s)
False
sage: %timeit is_CommutativeRing(QQ)
625 loops, best of 3: 1.09 µs per loop
sage: %timeit is_CommutativeRing(s)
625 loops, best of 3: 3.51 µs per loop
}}}
Since `is_CommutativeRing` just tests the class, it is supposed to be very
fast. But let us compare with the generic containment test in the category
of commutative rings and in the class of objects of that category:
{{{
sage: C = CommutativeRings()
sage: O = C.objects()
sage: QQ in C
True
sage: QQ in O
True
sage: s in C
False
sage: s in O
False
sage: %timeit QQ in C
625 loops, best of 3: 4.62 µs per loop
# Timing in sage-4.6.2: 12.9 µs per loop
sage: %timeit QQ in O
625 loops, best of 3: 1.5 µs per loop
sage: %timeit s in C
625 loops, best of 3: 4.69 µs per loop
# Timing in sage-4.6.2: 10.2 µs per loop
sage: %timeit s in O
625 loops, best of 3: 1.46 µs per loop
}}}
Hence, `is_CommutativeRing(s)` is slower than `s in O`, where `O =
CommutativeRings().objects()`.
The reason for that speedup is Cython. While `is_CommutativeRing` is a
Python function, the objects of a category are implemented in Cython.
Moreover, category containment is tested by the cached method
`is_supercategory`, which also is in Cython by #9138.
Caveat: I did not run the full tests, yet, and I would like to try and
remove some custom containment tests in the category framework, that tend
to be slower than the generic test and might not be needed with #9138.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10667#comment:12>
Sage <http://www.sagemath.org>
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