Jason (jcreed) wrote,

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...
Tags: categories, math

  • (no subject)

    Something that's bugged me for a long time is this: How many paths, starting at the origin, taking N steps either up, down, left or right, end up at…

  • (no subject)

    Still sad that SAC seems to end up being as complicated as it is. Surely there's some deeper duality between…

  • (no subject)

    I had already been meaning to dig into JaneSt's "Incremental" library, which bills itself as a practical implementation (in ocaml) of the ideas in…

  • Post a new comment


    Anonymous comments are disabled in this journal

    default userpic

    Your reply will be screened

    Your IP address will be recorded