categories: Re: Operads

[Tom Leinster <T.Leinster@dpmms.cam.ac.uk>]

Steve Vickers wrote re operads: 

> What I understand from the discussion is they capture single sorted
> algebraic theories with respect to a symmetric monoidal product ox.

I'd agree.  You could say that an operad is exactly an algebraic theory for
which it makes sense to take models in a monoidal category.  

That should be qualified/explained a bit.  By "algebraic theory" I mean to
exclude *co*algebraic and *bi*algebraic theories: e.g. I do count the theory
of monoids, but not those of comonoids or bimonoids.  And just as monoidal
categories can come equipped with symmetries or not, so operads can come
equipped with symmetric group actions or not; the choice of flavours is
yours.  (So if you're using operads with symmetries, you should also use
monoidal categories with symmetries.)  And if the objects O_n are objects of
some monoidal category other than Set, then you're talking about "enriched
algebraic theories".

> 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?

Yes: it's a PRO or a PROP (depending on whether you don't or do have
symmetries: the final P is for "permutations").  Formally, a PRO(P) is a
(symmetric) strict monoidal category whose underlying monoid of objects is
the natural numbers.  If you want your O_mn's to be objects of an arbitrary
symmetric monoidal category V (rather than just sets), then insert
"V-enriched" into the last sentence.  As far as I know, PROPs were first
thought about by Adams and Mac Lane, and subsequently developed by Boardman
and Vogt.

A model for a PRO(P) is a monoidal functor from it into some other monoidal
category.  So, for instance, there's a PRO whose models are monoids, and
another whose models are comonoids, and there's a PROP whose algebras are
bimonoids.  Thus PRO(P)s capture both the algebraic and the coalgebraic,
whereas operads only capture the algebraic.  I wouldn't interpret this as
saying that PRO(P)s exist at a more "proper" level of generality than
operads - just a different one.

(Incidentally, there's a PROP whose algebras are Hopf algebras (=bimonoids
with antipode), and an algebra for this PROP in (Set,x,1) is precisely a
group.  This contradicts the notion that it's impossible to formulate a
definition of "group" which makes sense in an arbitrary monoidal category,
although you do need your mon cat to have symmetries.)

Tom