### 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 $\infty$-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.

## 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?