Moerdijk on Infinity-Operads
Posted by Urs Schreiber
Yesterday I had reported (here) some aspects of Ieke Moerdijk’s talk at Higher structures II in Göttingen about dendroidal sets, which are to simplicial sets as operads are to categories. Hence there should be a notion of -operad (or -operads, really) which is to -categories as dendroidal sets are to simplicial sets.
In today’s talk Ieke Moerdijk looked into more details of the homotopical description of -operads within all dendroidal sets. Here are some aspects reproduced from the notes that I have taken during the talk.
A major ingredient for everything that follows is the fact – already emphasized in yesterday’s lecture, but I hadn’t mentioned it – that
Theorem. There is a symmetric closed monoidal category structure on dendroidal sets such that
a) the tensor unit is , the free operad on the tree with a single branch, which is the operad with a single color and only the identity operation on that;
b) the internal hom dendroidal set hom(A,B) is
c) for and two operads we have for the tensor product of their nerves that it is the nerve of an operad such that algebras for are precisely the -algebras in the category of -algebras.
On representables this tensor product is given by a “shuffle product of trees”. This is best described maybe in terms of diagrams and I won’t attempt to do so at the moment.
Recall that we say
Definition. An -operad (or -operad if we are being colloquial) is a dendroidal set which satisfies the weak dendroidal Kan condition: all inner dendroidal horns of have fillers. An ordinary operad is precisely an -operad for which all these lifts are unique.
In the following we will notationally identify operads with their dendroidal nerves .
Another concept from yesterday which I didn’t mention in my last installment is that of a normal dendroidal set: this is one which can be realized by iterative pushouts, so it’s the analog of a CW-complex. These have some nice properties:
Theorem. For a normal dendroidal set and an -operad, the inernal hom dendroidal set is itself an -operad.
For instance consider the symmetric monoidal category with its standard interval object with the homotopy coherent nerve of the canonical operad on (recalled last time, here the choice of interval object enters), and for the operad of associtive algebras, we have
is again an -operad. It is at the moment a conjecture that it should be true that the vertices of this dendroidal set are the double loop spaces.
More generally, the Boardman-Vogt resolution of the -fold tensor product of the associtive algebra operad with itself should be the little -cube operad :
As an example of this example: for with interval object the groupoid we have that
is the -operad whose vertices are the braided monoidal categories.
Now some more homotopyy theory description of .
Recall from the very conception of Quillen model categories that and are Quillen model equivalent. Now
Theorem (Moerdijk, Cisinski) . On there is a monoidal model category structure. in which
- normal dendroidal sets (the CW-complex-like ones) are precisely the cofibrant objects
- -categories are precisely the fibrant objects
- a morphism between normal dendroidal sets is a weak equivalence precisely if for any -operad the functor
is an equivalence of categories. Here is the left adjoint to the dendroidal nerve on ordinary operads and is pullback along the inclusion , so that applied to an operad picks out the category of unary operations inside the operad.
Observe that ordinary simplicial sets can be characterized as precisely those dendroidal sets which have a morphism to the tensor unit operad (the trivial operad on a single color), and that in fact the over-category is equivalent to . We have
Under this identification the above model structure on induces a model structure on which is the one due to Joyal whose fibrant objects are the -categories in their incarnation as quasi-categories.
With this observation in mind one can now try to generalize a couple of facts about the Joyal model structure on simplicial sets to the above model strucvture on .
Theorem The homotopy coherent nerve functor
induces a Quillen equivalence of model categories.
Re: Moerdijk on Infinity-Operads
In this talk I remembered the talk of Prof. Dr. Rainer Vogt in the Kolloquium in Hamburg two weeks ago.
He spoke about Operads and Tensorprodukt of operads. But he had the statment
W(N(Ass) ⊗n)≃D n.
as a Theorem, which Moerdijk only formulated as a conjecture (he said it should be right). More precisely Vogt had:
D_1 ⊗n ≃ D n.
Is it right, that W(N(Ass)) is the little 1-Cube (or Intervall) Operad D_1? Furthermore Vogt wrote an Operad having the Homotopy type of D_n as E_n. He pointed out, that one of the Problems is, that while
D_n ⊗ D_k ≃ D_{n+k}
the relation
E_n ⊗ E_k ≃ E_{n+k}
does NOT hold in General. This means especially that the Tensor Product is not compatible with weak equvalences. But Moerdijt told us today, that his tensor product is compatible with the model structure? Is he using another tensor product on operads (and calling it Boardman-Vogt Tensor Product)? Maybe the derrived one? Or does the Quillen Pair not resprect the monoidal structure?