categories: Re: Operads

[baez@math.ucr.edu]

Steve Vickers writes:

> There's some discussion on the Universal Algebra list at present on operads.
> 
> I'm not very familiar with them. What I understand from the discussion is
> they capture single sorted algebraic theories with respect to a symmetric
> monoidal product ox. For each natural number n an object of n-ary operators
> O_n is given, and an algebra A has operations O_n ox A^(n) -> A where A^(n)
> is A ox ... ox A n times.
> 
> If you do this sort of thing with respect to categorical product, then it
> already contains the information of the Lawvere theory category (for
> single-sorted finitary algebraic theories), since hom(m,n) is hom(m,1)^n and
> you take hom(m,1) to be O_m. But with a monoidal product this doesn't work.
> It seemed to me that for proper generality the operad ought to have objects
> O_mn (m, n natural numbers) representing the object of operations from A^(m)
> to A^(n). Is there a name for that?

These are called PROPs.   People in homotopy theory have been using
operads and PROPs since the 1970's.  As you note, there's no big difference
between operads and PROPs in a Cartesian category, but there is in a
more general symmetric monoidal category.   A nice example occurs
if we use Vect with its tensor product.  We can describe coalgebras
as algebras of a PROP, but not of an operad.

Nonetheless, every PROP has an underlying operad, and I believe that 
if your symmetric monoidal category has colimits, every operad freely 
generates a PROP, giving an adjunction.  

I always forget what "PROP" is an acronym for - something like 
"projection and permutation".  We can also formulate things like
operads and PROPs in the context of a monoidal category, and they
are sometimes called "planar operads" and "PROs".

Best,
jb