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? Steve Vickers Department of Pure Maths Faculty of Maths and Computing The Open University ----------- Tel: 01908-653144 Fax: 01908-652140 Web: http://mcs.open.ac.uk/sjv22
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
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
participants (3)
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baez＠math.ucr.edu -
S.J.Vickers＠open.ac.uk -
Tom Leinster