Re: Re: bilax_monoidal_functors

Re: Re: bilax_monoidal_functors

[Michael Shulman]

On Fri, May 14, 2010 at 8:05 PM, Joyal, Andre <joyal.andre@uqam.ca> wrote:
> I guess that in the category of R-modules over a commutative ring R,
> a module M has a (good) dual iff it is finitely generated projective
> iff the endo-functor functor Hom(M,-) preserves all colimits
> (M is *compact* in a strong sense).

Indeed, but in this case it is the objects of the category which are
"compact," not the category itself.  So if this is the argument, then
a more natural term would be "locally compact" (clashing with "locally
small," of course, but agreeing with "locally presentable" categories
in which all objects are presentable).

(I am *not* proposing to *actually* use "locally compact" -- I don't
want to introduce yet another name for something that already has at
least four names, even if none of the existing four are optimal.)

Mike


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RE : bilax monoidal functors

[John Baez]

André Joyal wrote:

I am using the following terminology for
> higher braided monoidal (higher) categories:
>
> Monoidal< braided < 2-braided <.......<symmetric
>
> A (n+1)-braided n-category is symmetric
> according to your stabilisation hypothesis.
>
> Is this a good terminology?
>

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.

For example, a doubly monoidal n-category, one with two compatible monoidal
structures, is a braided monoidal n-category.    I believe this is a theorem
proved by you and Ross when n = 1.  This way of thinking clarifies the
relation between braided monoidal categories and double loop spaces.

Various numbers become more complicated when one counts braidings rather
than monoidal structures:

An n-tuply monoidal k-category is (conjecturally) a special sort of
(n+k)-category... while an n-braided category is a special sort of
(n+k+1)-category.

Similarly: n-dimensional surfaces in (n+k)-dimensional space are n-morphisms
in a k-tuply monoidal n-category... but they are n-morphisms in an
(k-1)-braided n-category.

And so on.

On the other hand, if it's braidings that you really want to count, rather
than monoidal structures, your terminology is perfect.

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!

Best,
jb


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