## February 6, 2009

#### 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 $\infty$-operad (or $(\infty,1)$-operads, really) which is to $(\infty,1)$-categories as dendroidal sets are to simplicial sets.

In today’s talk Ieke Moerdijk looked into more details of the homotopical description of $(\infty,1)$-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 $U = \Omega(|)$, 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 $hom(A,B) : T \mapsto DendroidalSets(A \otimes T , B)$

c) for $P$ and $Q$ two operads we have for the tensor product $N(P) \otimes N(Q)$ of their nerves that it is the nerve of an operad $P \otimes Q$ such that algebras for $P \otimes Q$ are precisely the $P$-algebras in the category of $Q$-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 $(\infty,1)$-operad (or $\infty$-operad if we are being colloquial) is a dendroidal set $A$ which satisfies the weak dendroidal Kan condition: all inner dendroidal horns of $A$ have fillers. An ordinary operad is precisely an $(\infty,1)$-operad for which all these lifts are unique.

In the following we will notationally identify operads $P$ with their dendroidal nerves $N(P)$.

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 $A$ a normal dendroidal set and $P$ an $(\infty,1)$-operad, the inernal hom dendroidal set $hom(A,P)$ is itself an $(\infty,1)$-operad.

For instance consider the symmetric monoidal category $E = Top$ with its standard interval object with $P = D(E)$ the homotopy coherent nerve $D(E)$ of the canonical operad on $E$ (recalled last time, here the choice of interval object enters), and for $Ass$ the operad of associtive algebras, we have

$hom(N(Ass) \otimes N(Ass) , D(E))$

is again an $(\infty,1)$-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 $n$-fold tensor product of the associtive algebra operad with itself should be the little $n$-cube operad $D_n$:

$W(N(Ass)^{\otimes n}) \simeq D_n \,.$

As an example of this example: for $E = Cat$ with interval object the groupoid $\{a \stackrel{\simeq}{\to} b\}$ we have that

$hom(N(Ass) \otimes N(Ass), D(E))$

is the $(\infty,1)$-operad whose vertices are the braided monoidal categories.

Now some more homotopyy theory description of $DendroidalSets$.

Recall from the very conception of Quillen model categories that $SimplicialSets$ and $Top$ are Quillen model equivalent. Now

Theorem (Moerdijk, Cisinski) . On $DendroidalSets$ there is a monoidal model category structure. in which

- normal dendroidal sets (the CW-complex-like ones) are precisely the cofibrant objects

- $(\infty,1)$-categories are precisely the fibrant objects

- a morphism $A \to B$ between normal dendroidal sets is a weak equivalence precisely if for any $(\infty,1)$-operad $X$ the functor

$j^* \tau hom(B,X) \to j^* \tau hom(A,X)$

is an equivalence of categories. Here $\tau : DendroidaSets \to OrdinaryOperads$ is the left adjoint to the dendroidal nerve on ordinary operads and $j^*$ is pullback along the inclusion $j : Categories \to Operads$, so that $j^*$ 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 $U = \Omega(|)$ (the trivial operad on a single color), and that in fact the over-category $DendroidalSets\downarrow U$ is equivalent to $SimplicialSets$. We have

$\array{ DendroidalSets\downarrow U &&\simeq&& SimplicialSets \\ & {}_{forgetful}\searrow && \swarrow_{Lan(i)} \\ && DendroidalSets } \,.$

Under this identification the above model structure on $DendroidalSets$ induces a model structure on $SimplicialSets$ which is the one due to Joyal whose fibrant objects are the $(\infty,1)$-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 $DendroidalSets$.

Theorem The homotopy coherent nerve functor $SimplicialOperads \simeq TopologicalOperads \stackrel{D}{\to} DendroidalSets$

induces a Quillen equivalence of model categories.

Posted at February 6, 2009 3:49 PM UTC

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In this talk I remembered the talk of Prof. Dr. Rainer Vogt in the Kolloquium in Hamburg two weeks ago.

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?

Posted by: Thomas Nikolaus on February 6, 2009 9:15 PM | Permalink | Reply to this

The tensor product of dendroidal sets is defined by the formula $\Omega[T]\otimes\Omega[S]=N(T\otimes S)$ for trees, the general case being described by the Day convolution formula. This means that the left adjoint of the nerve functor is symmetric monoidal. However, the nerve functor from operads to dendrodial sets is not monoidal, and neither is the functor $W$. So the fact that the tensor product of dendroidal sets is homotopically well behaved does not imply the Boardman-vogt tensor product of simplicial (or topological) operads is gentle (and that is one the main reasons the point of view of dendroidal sets is useful: it allows to speak of derived tensor product and of derived internal Hom without suffering too much).

Posted by: Denis-Charles Cisinski on February 7, 2009 12:12 PM | Permalink | Reply to this

Does `gentle’ have a technical meaning?

Posted by: jim stasheff on February 7, 2009 1:41 PM | Permalink | Reply to this

Well, the most gentle a tensor product on a model category might be is when it is a left Quillen bifunctor (i.e. it should preserve cofibrations and trivial cofibrations in each variable, and also the pushout-product axiom), so that it would define a tensor product and an inernal Hom in the homotopy category. The Boardman-Vogt tensor product is certainly not gentle in this sense. It can be seen already at the level of simplicial categories: this is one of the biggest technical problem the theory of simplicial categories developed by Dwyer and Kan in the 80’s had: how to get a nice (i.e. computable in some sense) internal Hom in the homotopy category of simplicial categories? In fact, even the existence of an internal Hom is not obvious at all if we remain in the setting of simplicial categories. I guess this question is one of the most important motivation for the introduction of other models like Segal categories or quasi-categories: in this case, the cartesian product is gentle in the sense defined above. And of course, the problem does not vanish by passing from categories to colored operads.

However, there is weaker notion of gentle tensor product: just the property of preserving weak equivalences in each variable. In the case of simplicial categories, the cartesian product is gentle in this weaker sense. In the case of operads, we might hope this property for cofibrant simplicial colored operads, but I confess I don’t even know if this is true or not. What seems reasonnable to hope is that, for two cofibrant dendrodial sets $X$ and $Y$, the simplicial colored operads $W(X\otimes Y)$ and $W(X)\otimes W(Y)$ are equivalent (i.e. are related by a zig-zag of equivalences of simplicial operads).

Posted by: Denis-Charles Cisinski on February 7, 2009 2:39 PM | Permalink | Reply to this

Well, the most gentle a tensor product on a model category might be is when it is a left Quillen bifunctor (i.e. it should preserve cofibrations and trivial cofibrations in each variable, and also the pushout-product axiom)

I think usually “left Quillen bifunctor” only refers to having the pushout product axiom. That implies that tensoring with a cofibrant object preserves cofibrations and trivial cofibrations in each variable, but not more generally.

Posted by: Mike Shulman on February 7, 2009 11:34 PM | Permalink | Reply to this

It was a high point of the summer of 07 to hear Ieke talk about these tensor products in Bellaterra. I’ll take this opportunity to post an update of the conversations we had with several of the other posters here about a possible extension of the ideas to the n-operads as defined by Batanin and Leinster.

Recall that n-operads generalize ordinary non-symmetric operads. Just as an ordinary operad has a set for each natural number, a 2-operad has a set for each level 2 tree (these are special trees Batanin created for the purpose).
The natural numbers are level 1 Batanin trees, they look like an n-corolla , e.g . 3 = \|/.
The main use of n-operads is to provide a way to define Batanin n-categories–as the algebras of n-operads. So bicategories are the algebras of 2-operads.
Globular pasting diagrams are pasting diagrams which correspond to (Batanin’s)trees with height. For examples of pasting diagrams together with their trees see page 8 of Batanin’s preprint .

To describe the nerve of a 2-operad, you can generalize the construction of the category Omega defined by Moerdijk and Weiss in their preprint .
Recall that the objects of Omega are trees and a morphism is an operad map from
from the operad generated by the vertices of t to that generated by the vertices of t’, for two trees t and t’.
In order to define a category Omega_2 of 2-globular pasting trees,
for each 2-globular pasting tree t we need to construct

The definitions of 2-dendroidal sets and
the nerve of a 2-operad are mostly in place. Of course pictures are the best form of communication in this case, so here is the proposed idea as a pdf.
Of course the questions that immediately present themselves, are: Can one prove that the nerve is
fully faithful? Can one construct some sort of analogue of the
Boardman-Vogt tensor product? I’d be interested in any ideas and or feedback along these lines!

Posted by: Stefan on February 9, 2009 11:01 PM | Permalink | Reply to this

Stefan wrote:

$n$-operads as defined by Batanin and Leinster.

The definition was Batanin’s, not mine. I did do a lot of work on them, and I’d like to think that I helped to elucidate them, e.g. by showing how $n$-operads in the sense that you mention are a special case of Burroni’s generalized operads. I also worked on other types of higher-dimensional operad, not necessarily globular. But the original definition of (globular) $n$-operad was Michael’s.

Posted by: Tom Leinster on February 10, 2009 1:47 PM | Permalink | Reply to this

Right! I wanted to bring out the fact that the best way to learn about these n-operads as far as I know is to read both Michael’s papers and the relevant portions of Tom’s book , as simultaneously as possible.

I would like to know how the idea of n-dendroidal sets fits into the hyperstructure discussion that Urs was just mentioning . There are several places where the definition could be tweaked. For instance, a 2-dendroidal set is a functor from Omega_2^{op} to Set. Perhaps though the image should be globular sets?

Posted by: Stefan on February 10, 2009 3:30 PM | Permalink | Reply to this

I would like to know how the idea of $n$-dendroidal sets fits into the hyperstructure discussion that Urs was just mentioning .

I also kept thinking about it, but still need to further order my thoughts, also in order to finally reply to Mike Shulman’s comments over at hyperstructures.

One aspect that makes this a bit tricky is that there are so many different ways a poset can encode cells in higher structures.

For instance every object $[n]$ of the simplicial category $\Delta$ is usefully thought of as a poset, but the posets which describe cells in a simplicial structure along the lines of the discussion at hyperstructures are the posets usually called $P \Delta^n$: the posets of sub-cell inclusions in the $n$-simplex, which is the over-category $\Delta'\downarrow [n]$ with $\Delta'$ the poset obtained from $\Delta$ by discarding all degeneracy maps.

I suppose when we are looking at trees that represent (higher) operads, then here, too, the relevant posets in the sense of hyperstructures should be suitable over-categories over these trees.

I need to think about how for instance the Kan filler condition in its usual form formulated in terms of $\Delta$ looks when we pass from the $\Delta^n$ to the $P \Delta^n$ picture and use the filler condition which I proposes at that entry on hyperstructures.

Posted by: Urs Schreiber on February 10, 2009 8:08 PM | Permalink | Reply to this

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