* Is "braided monoidal" a conservative extension of "lax braided monoidal"?
@ 2010-11-29 16:34 Alan Jeffrey
[not found] ` <E1PNQxX-0002w5-Gy@mlist.mta.ca>
0 siblings, 1 reply; 2+ messages in thread
From: Alan Jeffrey @ 2010-11-29 16:34 UTC (permalink / raw)
To: categories
Hi everyone,
Is there a known result (or counterexample) saying that the equational
theory of a braided monoidal category is a conservative extension of a
lax braided monoidal category?
As a reminder, a lax braid is a natural b : A * B -> B * A satisfying
coherence conditions with the units and associators, and a braid is a
lax braid where b is an isomorphism. The motivation for asking this
question is that string diagrams are a proof technique for braids, and
I'd like to be able to use them for lax braids too, but this requires
braiding to be a conservative extension of lax braiding.
Cheers,
Alan Jeffrey.
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* Re: Is "braided monoidal" a conservative extension of "lax braided monoidal"?
[not found] ` <E1PNQxX-0002w5-Gy@mlist.mta.ca>
@ 2010-11-30 23:43 ` Alan Jeffrey
0 siblings, 0 replies; 2+ messages in thread
From: Alan Jeffrey @ 2010-11-30 23:43 UTC (permalink / raw)
To: Ross Street; +Cc: categories
Indeed, a categorification of Garside's embedding of the positive braid
monoid into the braid group is exactly what I'm after. Unfortunately,
the nearest published result I've been able to find is:
Left-Garside categories, self-distributivity, and braids
Patrick Dehornoy
http://ambp.cedram.org/item?id=AMBP_2009__16_2_189_0
which shows this result for "the positive braid category", which has
natural numbers as objects, and braids as morphisms (and hence there are
only morphisms m --> n when m = n). The case of a freely generated lax
braided monoidal category isn't covered, although it may follow similarly.
There's a couple of related works:
Garside categories, periodic loops and cyclic sets
David Bessis
http://arxiv.org/abs/math/0610778
Garside and locally Garside categories
François Digne and Jean Michel
http://arxiv.org/abs/math/0612652
but they don't appear to have exactly the categorification of Garside
either.
A.
On 11/29/2010 04:24 PM, Ross Street wrote:
> Dear Alan
> The positive braid monoid is a Garside monoid and it embeds
> in the braid group.
> (It is not generally true that a monoid embeds in its group of
> fractions.
> There is a literature on Garside monoids.)
> I think this may be what you need.
> Ross
>
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2010-11-29 16:34 Is "braided monoidal" a conservative extension of "lax braided monoidal"? Alan Jeffrey
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2010-11-30 23:43 ` Alan Jeffrey
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