Andre points out: "To call a monoidal category 1-braided is kind of confusing because there is no commutation structure on a general monoidal category. A monoidal category is 0-braided." Being an outsider, with no previous neither usage or opinion on this terminology beyond just monoidal and/or tensor category, this seems to me definitive, and more than enough to settle the question. e.d. Andre Joyal wrote:
Dear John and Michael,
It all depends on where you start counting. For americans, the first floor of a buiding is the ground floor but for most europeans, it is the floor right above:
http://en.wikipedia.org/wiki/Storey#Numbering
We sometime need to recall in which part of the world we are when we take an elevator! But a ten stories building is the same for everyone.
More seriously, John wrote:
I use "k-tuply monoidal" to mean what you'd call "(k-1)-braided". This seems preferable to me, not because it sounds nicer - it doesn't - but because it starts counting at a somewhat more natural place. I believe that counting monoidal structures is more natural than counting braidings.
Michael wrote:
I am using a mixture of your terminologies: monoidal = 1-braided braided = 2-braided sylleptic = 3-braided
I understand your ideas both. Along the same line we could also use:
E1-category = Monoidal E2-category = Braided monoidal E3-category = ..... .....
John wrote:
By the way: I don't remember anyone on this mailing list ever asking if their own terminology is good. I only remember them complaining about other people's terminology. I applaud your departure from this unpleasant tradition!
My goal is to have a public discussion on terminology. It can be very difficult to agree upon because adopting one is like commiting to a rule of law, to a moral code, possibly to a social code. There is an emotional and social aspect to this commitment. There is also a psychological aspect because a terminology looks natural if you use it long enough (it is a matter of a few days). I hope that a public discussion can help peoples choosing their terminology.
I do think that my terminology for higher braided monoidal categories is quite good. Let me say a few things in its defense:
First, it extends naturally a terminology which is used by the mathematical community since many decades. Only a specialist can truly appreciate E(k)-categories or k-tuply monoidal categories. Second, a braiding is a commutation structure. To call a monoidal category 1-braided is kind of confusing because there is no commutation structure on a general monoidal category. A monoidal category is 0-braided. Third, a n-braided (topological or simplicial) group is exactly what you need to describe the homotopy type of an n-connected space (n\geq 1).
I wonder who introduced the notion of E(n)-space and the terminology?
Best regards, André
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