Is "braided monoidal" a conservative extension of "lax braided monoidal"?

Is "braided monoidal" a conservative extension of "lax braided monoidal"?

[Alan Jeffrey]

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"?

[Alan Jeffrey]

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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