#10963: More functorial constructions
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Reporter: nthiery | Owner: stumpc5
Type: enhancement | Status: needs_info
Priority: major | Milestone: sage-6.2
Component: categories | Resolution:
Keywords: days54 | Merged in:
Authors: Nicolas M. Thiéry | Reviewers: Simon King, Frédéric
Report Upstream: N/A | Chapoton
Branch: | Work issues: merge with #15801
public/ticket/10963-doc- | once things stabilize
distributive | Commit:
Dependencies: #11224, #8327, | 16d530dfc1838a6b497afac03e1e2b13be24795d
#10193, #12895, #14516, #14722, | Stopgaps:
#13589, #14471, #15069, #15094, |
#11688, #13394, #15150, #15506, |
#15757, #15759 |
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Comment (by nthiery):
Replying to [comment:583 darij]:
> In other news, I'm still trying to understand why `Sets().Algebras(QQ)`
needs to make sense. This here is seriously weird:
> {{{
> sage: Sets().example()
> Set of prime numbers (basic implementation)
> sage: Sets().example().algebra(QQ)
> Free module generated by Set of prime numbers (basic implementation)
over Rational Field
> sage: Sets().example().algebra(QQ) in Algebras(QQ)
> False
> sage: Sets().example().algebra(QQ) in Sets().Algebras(QQ)
> True
> }}}
>
> Is it possible that "set algebra" is just a misnomer for "set module"
(free module of a set)? I'm *not* asking you to change this; but it really
needs to be documented both in the primer and at module level.
The above results are correct but, as you point out, the name
"algebra" is certainly a misnomer.
In general, if S is a parent and F a field, S.algebra(F) constructs
the F-free module FS with basis indexed by S, endowed with whatever
structure can be induced from that of S. Typically, if S is a
magma/monoid/group, you get the magma/monoid/group algebra. For a
group, it actually gives a Hopf algebra. Same thing for additive
magmas/monoids/groups. With #14102, if S is a root lattice, the action
of the Weyl group and the like get lifted to FS too.
In other words, in most practical use cases, you indeed get an
algebra; actually "the algebra"; hence the name. But I agree that in
the other cases the name is misleading. Still we need a uniform name,
and so far nobody came up with something better ...
In the mean time, I am all for improving the documentation of the
algebra method of parents and Algebras methods of the
categories. Should I just throw in the above paragraphs there?
This probably does not need to be discussed in the primer though,
given that it barely mentions the algebra construction.
Cheers,
Nicolas
--
Ticket URL: <http://trac.sagemath.org/ticket/10963#comment:585>
Sage <http://www.sagemath.org>
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