Is "braided monoidal" a conservative extension of "lax braided monoidal"?
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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 On 30/11/2010, at 3:34 AM, Alan Jeffrey wrote:
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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (2)
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Alan Jeffrey -
Ross Street