#10963: More functorial constructions
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Reporter: nthiery |
Owner: stumpc5
Type: enhancement |
Status: needs_review
Priority: major |
Milestone:
Component: categories |
Resolution:
Keywords: |
Work issues: Rebase wrt. #13589
Report Upstream: N/A |
Reviewers: Simon King
Authors: Nicolas M. Thiéry |
Merged in:
Dependencies: #11224, #8327, #10193, #12895, #14516, #14722, #13589 |
Stopgaps:
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Comment (by SimonKing):
Replying to [comment:26 nthiery]:
> However in many textbooks and other pieces of literature
> "algebra" implicitly includes "associative" and "unital"
Certainly there also exist textbooks that will for simplicity say
"algebra" when they in fact mean "commutative algebra". But I would expect
that all these textbooks state at some point the definition of (plain)
algebras and later say that "for simplicity" or "unless stated otherwise"
they assume whatever additional axioms.
And even "better": There were times when a certain algebraic community
would only talk about ''finite'' groups. I recently heard colleagues talk
about these times. It was like "they provided generators and relations and
then needed to prove that it is a group", which in today's language is
"they provided a group presentation and needed to prove that the group is
finite".
You see: There ''are'' certain conventions peculiar to certain fields of
research.
But I think a general computer algebra system should not be biased towards
any of these peculiar conventions. Hence, it should use the "greatest
common divisor" of the notions, which is: An R-algebra is a an R-module
and a multiplicative magma, such that multiplication is R-bilinear.
> (for the same
> reason that it will be heavy for us to write almost everywhere
> Algebras().Associative().Unital()).
We can certainly have a short-cut for defining this thing.
> More importantly: changing the semantic current "Algebras" in Sage
> would be seriously backward incompatible.
Backward compatibility is indeed important. It would be difficult to
switch from Algebras in the current Sage-use to Algebras in the (I think)
normal mathematical use.
However, I ''do'' think that to the very very least we should let
`Algebras()` print as "Category of unital associative algebras".
> And we would have to think
> about what we want to do about categories like "HopfAlgebras" to keep
> things consistent.
Wikipedia does not assume associativity for algebras, but it does assume
co-associativity for co-algebras. Weird.
> So I definitely see your point but at this point I am not keen on
> opening yet another can of worms (both technical and social) to this
> already too big patch.
Concerning social: I vividly remember many talks in the séminaire
quantique in Strasbourg, entitled along the lines of "quasi-commutative
quasi-cocommutative quasi-Hopf algebras". I think ''these'' guys would be
unhappy about tacitly assuming too many axioms for algebras. And I just
checked: There also is the notion of quasi-associative algebras in
literature...
> What about, at least as a temporary measure, going for:
>
> {{{
> magmatic algebras
> / \
> associative magmatic algebras unital magmatic algebras
> \ /
> algebras
> }}}
>
> (or any other not-yet-used name you like instead of "magmatic
> algebra")
I have never heard about "magmatic algebras" before. But I have no better
idea ("plain algebras"?).
Sage-devel poll? Sage-algebra poll (although this list seems dead)? Sage-
combinat-devel poll?
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10963#comment:27>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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