#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 nbruin):
Replying to [comment:585 nthiery]:
> 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.
Doesn't that mean you construct a free F-module that happens to have some
extra structure thanks to S? The free F-algebra indexed by S would seem to
me to be the free F-module spanned by the free monoid on S. Hence, this
"associative algebra" functor would belong on monoids, not on sets.
> Same thing for additive magmas/monoids/groups.
Is that the same thing? If S is an additive group and the addition induces
the additive relations on the F-module generators, it seems to me you'd
not get the free F-module FS. If S is a free abelian group on X, you'd get
the free F-module FX.
> With #14102, if S is a root lattice, the action
> of the Weyl group and the like get lifted to FS too.
Nice! So in this case we have a free abelian group S with a W-action (a
W-group, I guess) and the functor now gives us W-F-algebras. That makes
perfect sense, but now we are working in yet another category (I don't
think that in this case the term "algebra" will lead to confusion, because
you're getting something with *more* structure, not less than the name
suggests)
> In other words, in most practical use cases, you indeed get an
> algebra; actually "the algebra"; hence the name.
No, only when you do this with a monoid. (or a magma if you like your
algebras non-associative)
> Still we need a uniform name,
I don't think you do. The construction is nice and functorial over
multiplicative monoids (magmas if you want to treat associativity as an
axiom).
Outside monoids there is a construction that produces the same setwise
result (the free F-module), but it's a functor into a different category.
F-algebras do not form a full subcategory of F-modules, so treating the
two functors as separate makes a lot of sense.
(it doesn't have much to do with this ticket, though).
--
Ticket URL: <http://trac.sagemath.org/ticket/10963#comment:587>
Sage <http://www.sagemath.org>
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