Re: bilax_monoidal_functors

[Jeff Egger <jeffegger@yahoo.ca>]

>> But I think "braided = doubly monoidal" is even
> > better.  After all, a
> > monoidal category has one tensor product; a braided
> > monoidal category has
> > two compatible tensor products, and a symmetric
> > monoidal category has three.
> 
> The trouble is that n-monoidal  categories already exist.
> They were
> introduced my Balteanu, Fioderowicz, Shwantzl and Vogt.
> This is why I
> also see n-tuply monoidal as confusing. I do not say that
> they sound
> identical but certainly very close to each other.

This is a strong point.  Obviously n-tuply monoidal category
should mean category with n "compatible" monoidal structures;
but there many possible meanings of "compatible".  One choice 
leads to a single monoidal structure with an (n-1)-braiding;
but a different  choice leads to the notion of BFSV.  In fact,
I think that even the BFSV  notion is too strict---it forces 
all the units to be the same, where I think one should allow
them to be different (in general).  That is, I think it would
be reasonable to use "doubly monoidal category" to mean 
(pseudo)monoid internal to LAX (rather than STRONG, or even 
NORMAL).  

Cheers,
Jeff.





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