April 19th, 2008

beartato phd

(no subject)

There is a move mentioned in week49 of TWF that I hadn't thought much about before.

It's the observation that if you have a k-tuply monoidal n-category, you can get a (k+1)-tuply monoidal n-category from it, one step down in the periodic table. For brevity say a k-tuply monoidal n-category is a (k,n)-category.

So start with a (k,n)-category C. Notice that it is a special kind of (0,k+n)-category. But the totality of (0,k+n)-categories is the prototypical example of a (0,k+n+1)-category. So take the full (0,k+n+1)-category hovering around C and id_C and id_(id_C) ... and so on, k+1 times. There's your (k+1,n)-category!

If we start with a set, we get the monoid of all its endomorphisms.
If we start with a monoid, we get the commutative monoid that is its center.
If we start with a commutative monoid, we stay in place.

If we start with a category, we get the monoidal category of its endofunctors and natural transformations between them (the monoidal category in which a monoid object is a monad, as wjl may remember)
If we start with a monoidal category, we must get a braided monoidal category somehow, but I can't quite unravel the definitions far enough to see clearly what it's like...