Re: bilax_monoidal_functors?=

[Andre Joyal <joyal.andre@uqam.ca>]

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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