Hi Bob,
Mathematically, a classical example of a graded object is a polynomial
ring, and non-topologists often consider nonhomogeneous polynomials, for
reasons that are beyond me. But once you do that, it seems that you're
forced to live in a direct sum. Sage could have two different structures,
although they would need names: an indexed family of objects and a direct
sum of objects. I don't really want to start a fight over which deserves to
be called "graded objects," though.
Let me go back to a question I asked Sverre: what happens in your code if
you allow nonhomogeneous elements? It may not be something you would ever
want to do, but maybe it would just work without breaking anything, then
that's an easy way around.
- John
On Sunday, July 19, 2020 at 2:25:38 AM UTC-7, rrbold wrote:
>
> Hi Christian and John,
>
> Christian, your first sentence puts the finger on the correct spot: I
> take the position that a graded abelian group is not an abelian group. It
> is a sequence of abelian groups.
>
> For any category C, one can consider Gr(C), the category of graded objects
> in C, which has objects the functions from your grading monoid, frequently
> the natural numbers, to Obj(C), and morphisms the sequences of morphisms of
> C. There is no need for C to have a direct sum or categorical coproduct
> which will allow you to combine these
> into a single object in C, in order to consider such things.
>
> Mathematically, consider singular n-cochains on a space X with values in a
> module M. These are functions from the set Top(\Delta_n,X) of continuous
> maps \Delta_n --> X into the module M, i.e., elements of
> Set(Top(\Delta_n,X),M), given the natural module structure inherited from
> M. If n \neq k, then there is no sensible relation between the zero
> function Top(\Delta_n, X) --> M and the zero function Top(\Delta_k,X).
> Only inductive generalization or habit would suggest that sending all the
> elements of Top(\Delta_n,X) to zero \in M means you should also do this to
> all the elements in the entirely different set Top(Delta_k,X).
>
> The fact that this causes difficulties in the programming is a hint that
> we make an error in thinking of graded objects as their direct sum. I
> think it is better to take the mathematically sensible solution, and accept
> that there is a different 0 in each degree of a graded module.
>
> Best,
> Bob (rrb - old)
>
>
>
>
>
> On Saturday, July 18, 2020 at 5:57:21 PM UTC-4, Christian Nassau wrote:
>>
>> Hi Sverre,
>>
>> I don't think it's a good idea to have different zeroes in an algebraic
>> structure that is also categorized as an abelian group, unless you take the
>> point that a "graded abelian group" should not be an "abelian group".
>>
>> But let me also point out that something similar to what you want already
>> exists: you can take a homogeneous component of the Steenrod algebra and
>> look at its zero:
>>
>> sage: A=SteenrodAlgebra(2)
>> sage: A[18]
>> Vector space spanned by (Sq(0,1,0,1), Sq(3,0,0,1), Sq(1,1,2), Sq(4,0,2),
>> Sq(2,3,1), Sq(5,2,1), Sq(8,1,1), Sq(11,0,1), Sq(0,6), Sq(3,5), Sq(6,4),
>> Sq(9,3), Sq(12,2), Sq(15,1), Sq(18)) over Finite Field of size 2
>> sage: A[18].zero() == A.zero()
>> True
>> sage: A[18].zero() == A[17].zero()
>> False
>>
>> This suggests that "A[18].zero().degree()" could give 18, and the fact
>> that it currently gives a ValueError might be considered a bug.
>>
>> Best,
>> Christian
>>
>>
>> On 18.07.20 23:35, Sverre Lunøe-Nielsen wrote:
>>
>> Hi,
>>
>> Thank you for your comments so far. I feel I need to expand some more on
>> the issue of zero elements which is the central thing for the problem we
>> are adressing.
>>
>> It is mathematically equivalent to think of a graded k-algebra A as either
>>
>> 1) a direct sum A = \bigosum_i A_i, together with a graded k-linear map
>> from
>> the graded tensor product A\tensor_k A --> A,
>>
>> or
>>
>> 2) a sequence of k-vectorspaces {A_i}_i, together with a set of structure
>> maps
>> \{ A_i \tensor_R A_j --> A_{i+j} \}_{i,j}.
>>
>> (In both cases the structure maps should satisfy usual algebraic
>> conditions.)
>>
>> Similar for graded A-modules.
>>
>> The implementation of the SteenrodAlgebra package takes the approach of
>> 1), and never speaks about the zero element z_i \in A_i for any i. Rather,
>> they are all identified in A via the canonical injection A_i --> A. It is
>> tradition not to worry too much about this since you can "figure it out" if
>> you have to, and know how you ended up with a zero.
>>
>> However, it is arguably better, specially when writing software, to avoid
>> this simplifaction since it leads to a corner case which has to be dealt
>> with over and over again. A great share of the bugs I have corrected in
>> the package I have been editing have been caused by the wrongful assumption
>> that all elements have an integer degree. Having not to worry about this
>> would make our code cleaner, and so will all future code building on it.
>>
>> I was being rather vague about making proposals for change in the
>> SteenrodAlgebra package in my last post, so to be clear let me propose a
>> specific change and invite anyone to share their opinion on it:
>>
>> Change SteenrodAlgebra such that _all_ homogeneous elements have a well
>> defined degree. For the user, this means in particular that when
>> constructing the zero element, its degree must be given:
>>
>> sage: A = SteenrodAlgebra(p=2)
>> sage: z = A.zero(degree=2)
>> sage: Sq(1)*Sq(1) == z
>> True
>> sage: Sq(2)*Sq(1)*Sq(1) == z
>> False
>>
>> This involves adding the degree as internal data to zero elements, and
>> change the behaviour of degree() such that it raises an exception only for
>> inhomogeneous elements.
>>
>> I hope I have clearified that I am not seeking a strange new definition
>> of graded module or algebra, and that I am merely wanting to discuss the
>> possibility of changing the implementation of SteenrodAlgebra.
>>
>> E.g. are there perhaps unwanted software ramifications that our proposal
>> would bring about?
>>
>> Regards,
>>
>> Sverre
>>
>>
>>
>>
>>
>> On Saturday, July 18, 2020 at 11:31:43 PM UTC+2, John H Palmieri wrote:
>>>
>>>
>>>
>>> On Saturday, July 18, 2020 at 2:31:01 AM UTC-7, Sverre Lunøe-Nielsen
>>> wrote:
>>>>
>>>> Dear list,
>>>>
>>>> I have been involved in preparing a package by M. Catanzaro and R.
>>>> Bruner lately, which implements finitely presented modules over the mod
>>>> `p`
>>>> Steenrod algebra.
>>>>
>>>> We have encountered a conflict regarding how to present graded objects,
>>>> and I am writing to the list to get other people's opinion on how to
>>>> proceed on this matter.
>>>>
>>>> Briefly, the issue is that the Steenrod algebra allows inhomogeneous
>>>> elements and our graded modules do not. Thus, the Steenrod algebra has a
>>>> single zero element with no well defined degree, while our modules could
>>>> potentially have one zero element for each degree.
>>>>
>>>> My wish is to allow degreewise zero elements in our graded modules, so
>>>> that x.degree() would return an integer for every element x. But because
>>>> the unique zero in the Steenrod algebra has no well defined degree, I am
>>>> forced to let degree() treat all zero elements in our modules the same way
>>>> and return ``None``.
>>>>
>>>> A more precise description of the issue is found in the Sphinx note
>>>> below.
>>>>
>>>> My questions to the list are: Has similar issues been discussed and/or
>>>> resolved before? And more specificly: What acceptable changes could be
>>>> made to the Steenrod algebra package to achieve what I want?
>>>>
>>>> Regards,
>>>>
>>>> Sverre Lunøe-Nielsen
>>>>
>>>>
>>>> .. NOTE::
>>>> Our implementation treats a graded module as the disjoint union, rather
>>>> than a
>>>> direct sum, of vectorspaces of homogeneous elements. Elements are
>>>> therefore
>>>> always homogeneous, which also implies that sums between elements of
>>>> different
>>>> degrees are not allowed. This also means that acting by an
>>>> inhomogeneous
>>>> element of the Steenrod algebra makes no sense.
>>>>
>>>> In this setting there is no single zero element, but rather a zero for
>>>> every
>>>> degree. It follows that, in theory, all elements, including the zero
>>>> elements,
>>>> have a well defined degree.
>>>>
>>>> This way of representing a graded object differs from the way the
>>>> graded
>>>> Steenrod algebra is represented by :class:`sage.algebras.steenrod`
>>>> where inhomogeneous
>>>> elements are allowed and there is only a single zero element.
>>>> Consequently,
>>>> this zero element has no well defined degree.
>>>>
>>>> Thus, because of the module action, we are forced to follow the same
>>>> convention
>>>> when it comes to the degree of zero elements in a module: The method
>>>>
>>>> :meth:`sage.modules.finitely_presented_over_the_steenrod_algebra.module.fp_element.FP_Element.degree'
>>>> returns the value ``None`` for zero elements.
>>>>
>>>> An example which highlights this problem is the following::
>>>>
>>>> sage: F = FPA_Module([0], SteenrodAlgebra(p=2)) # The free module
>>>> on a single generator in degree 0.
>>>> sage: g = F.generator(0)
>>>> sage: x1 = Sq(1)*g
>>>> sage: x2 = Sq(1)*x1
>>>>
>>>> Clearly, the code implementing the module action has all the
>>>> information it needs
>>>> to conclude that the element ``x2`` is the zero element in the second
>>>> degree.
>>>> However, because of the module action, we cannot distinguish it from
>>>> the element::
>>>>
>>>> sage: x2_ = (Sq(1) * Sq(1))*g
>>>>
>>>> The latter is equal to the action of the zero element of the Steenrod
>>>> algebra on `g`, but since the zero element has no degree in the
>>>> Steenrod algebra,
>>>> the module class cannot deduce what degree the zero element `x2_`
>>>> should belong
>>>> to.
>>>>
>>>
>>> In my experience, algebraic topologists often think of graded objects as
>>> disjoint unions, and you can often get away with this, but really they're
>>> not — they're direct sums. I think you should use Sage's categories
>>> framework, graded modules with basis or whatever, to set these up. In any
>>> case where the degree matters, you should first test whether an element is
>>> zero (in which case it won't have a degree) and then perhaps whether it is
>>> homogeneous. If not, you can raise an error (to keep someone from
>>> multiplying a module element by Sq(1) + Sq(2), for example). If it is
>>> homogeneous, you can proceed the way you want.
>>>
>>> --
>>> John
>>>
>>> --
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