### Dendroidal Sets and Infinity-Operads

#### Posted by Urs Schreiber

This morning I was in Hamburg chatting with David Ben-Zvi about his work (see his recent guest post for a pedagogical introduction) then I jumped on the train and arrived just in time in Göttingen at the workshop *Higher Structures II* (see this post for the announcement) to hear Ieke Moerdijk’s talk on **$\infty$-Operads and dendroidal sets**.

Now after conference dinner it’s already late, but I thought I’d try to produce at least parts of my notes of Ieke Moerdijk’s talk. I am doing this with an eye towards our discussion about hyperstructures over at the $n$Lab (which started at this blog entry) which is clearly somehow related (inconclusive as it is at this stage) to the idea underlying dendroidal sets.

From one point of view the simple underlying idea is that

- simplicial sets and hence in particular for instance Kan complexes and $(\infty,1)$-categories are presheaves on the simplex category $\Delta$, which can be regarded as the full subcategory of $Cat$ on those posets which are “linear” (totally ordered) $\cdots \leftarrow \leftarrow \leftarrow \cdots$

- a dendroidal set is a presheaf on a subcategory of $Cat$ (not a full one, though) on slightly more general posets, namely those which are “tree shaped” $\array{ &&& \swarrow & \leftarrow & \cdots \\ \cdots &\leftarrow &\leftarrow &\leftarrow& \cdots \\ & \nwarrow \\ && \leftarrow & \cdots }$

- while there are some situations which are naturally modeled by presheaves on more general posets, for instance those of the form $\left( \array{ && \top \\ & \nearrow && \nwarrow \\ a &&&& b } \right)^{\times n}$ as considered in Marco Grandis’s work on Cospans in Algebraic Topology for describing extended cobordisms

- or even entirely general ones, possibly, as considered tentatively at $n$Lab. hyperstructure.

One way to think about this is that every (finite) poset can be regarded as defining one of the geometric shapes for higher structures. For instance the posets $G'\downarrow [n]$ arising as over-categories of the globe category, which locally look like the poset
that Tom Leinster recently mentioned here, are the poset incarnation of the $n$-globe: $c_i$ and $d_i$ represent its two sub-$i$-globes and the morphisms indicate which subglobe sits inside which higher sub-globe (the poset depicted above is actually that of the boundary of the $(n+1)$-globe, with the top $(n+1)$-cell missing). These *globe-posets* play a crucial role in Michael Batanin’s work (see his comment here).

As Ronnie Brown kindly points out at hyperstructure, the idea of defining higher structures *without* commiting oneself to a single or to one of the standard shapes (globes, simplices, cubes) is an old one (I am being told that this goes back to Grothendieck’s *dérivateurs*, but am lacking currently further information on that) which has for instance been studied by D. Jones in *A general theory of polyhedral sets and their corresponding T-complexes*.

All this may or may not be directly related to the main point of *dendroidal sets and $\infty$-operads*, but I felt like mentioning it in any case. More concretely, dendroidal sets are supposed to be precisely the notion that completes the analogy

$\array{ category & (\infty,1)-category & simplicial set & weak Kan complex \\ operad & \infty-operad & dendroidal set & Kan dendroidal set } \,.$

The following is a reproduction of some of the notes that I took in Ieke Moerdijk’s talk today. A closely related survey talk I had recently reproduced here.

[*some aspects of notes I had taken*]

(The following is a description of parts of joint work of Ieke Moerdijk with Ittay Weiss.)

Noticing that every category is a colored operad with only a unary operation, we have a canonical inclusion of the collection of categories into that of operads

$Categories \stackrel{j}{\hookrightarrow} ColoredOperads (= Multicategories)$

A the same time categories are related via the simplicial nerve to simplicial sets. Dendroidal sets are supposed to be the notion that sensibly completes the square

$\array{ Categories &\stackrel{Nerve}{\to}& SimplicialSets \\ \downarrow^j && \downarrow \\ ColoredOperads &\stackrel{}{\to}& DendroidalSets } \,.$

This is realized in terms of the **dendroid category** $\Omega$ whose objects are planar rooted trees. Every planar rooted tree $T$ defines freely a colored operad $\Omega(T)$ whose colors are the edges of the tree and whose operations are the vertices of the tree, subject to the obvious relations. A morphism $T \to S$ in $\Omega$ is precisely a morphism $\Omega(T) \to \Omega(S)$ of operads.

A **dendroidal set** is a presheaf on $\Omega$:

$DendroidalSets := [\Omega^{op}, Sets] \,.$

The full subcategory of the dendroid category $\Omega$ on linear trees is precisely the simplex category $\Delta$

$\Delta \stackrel{i}{\hookrightarrow} \Omega$

and the moprhism $SimplicialSets \to DendroidalSets$ is defined to be the left Kan extension along this inclusion $i^{op}$:

$Lan(i^{op}) : SimplicialSets \to DendroidalSets \,.$

Given any (colored) operad $P$ (= multicategory) its **dendroidal nerve** is, as usual, the dendroidal set $N(P)$ whose value on the dendroid $T$ is the Hom-set
$N(P) := Operads(\Omega(-), P) : T \mapsto Operads(\Omega(T), P)
\,,$

i.e. the collection of all ways to label $T$ by colors and operations of $T$.

Recalling that every symmetric monoidal category $E$ induces an operad colored by the objects of $E$ with $n$-ary operations colored by $(c_1, \cdots, c_n)$ incoming colors and $c_0$ outgoing being the Hom-set $E(c_1 \otimes \cdots \otimes c_n, c_0)$ from the tensor product of all these objects, we observe that for $P$ any operad the Hom-set $DendroidalSets(N(P), N(E))$ is the set of $P$-algebras of $E$ (you can take this as the definition, if you like).

If we are in a homotopy theoretic context such as that of Top we have derived versions of all of this. For instance for $P$ any topological operad let $W(\Omega(T))$ be the Boardman-Vogt resolution of the operad $\Omega(T)$, then we have the **homotopy coherent nerve** of $P$ defined by

$D(P) ::= TopologicalColoredOperads(W(\Omega(-)), P) \,.$

Everything goes through analogously as before, for instance $P$-algebras up to homotopy in a symmetric monoidal category $E$ are the elements of

$TopologicalColoredOperads( D(P), D(E) ) \,.$

**Theorem.** There exists a model category structure on $DendroidalSets$ such that the homotopy coherent nerve $D : TopologicalColoredOperads \to DendroidalSets$ is a (right) Quillen equivalence.

(This is joint work by Ieke Moerdijk and Denis-Charles Cisinski, involving a contribution by Clemens Berger.)

We want to characterize the image of this functor: those dendroidal sets in which one can “compose up to homotopy”. The idea for that is a more or less straightforward generalization of the Kan condition:

- a **face** of a dendroid $T$ is a subdendroid $f \hookrightarrow T$ with exactly one vertex less that $T$ obtained from $T$ by contracting precisely one edge. The **inner dendroid horn** $\Lambda^e(T)$ is the union of dendroidal sets given by all the faces of the dendroid $T$ except the one coming from contracting the inner edge $e$.

**Definition.** A dendroidal set $X$ is an **weak dendroidal Kan complex** if for any dendroid $T$ and any one of its inner edges $e$ and any morphism $\Lambda^e(T) \to X$ there exists the lift $\phi$ in

$\array{ \Lambda^e(T) &\to& X \\ \downarrow & \nearrow_{\exists \phi} \\ \Omega(T) } \,.$

It is clear from this that if $X$ is just a simplicial set in that it is in the image of the inclusion $Lan i : SimplicialSets \to DendroidalSets$ then $X$ is a weak dendroidal Kan complex precisely if it is a weak Kan complex, i.e. an $(\infty,1)$-category.

So much for today. tomorrow we’ll here in the next part about The homotopy theory of infinity-categories and infinity-operads.

## Re: Dendroidal Sets and Infinity-Operads

Ieke Moerdijk gave a very similar talk in January at the fourth “Categories, Logic and Foundations of Physics” workshop at Imperial. A video can be found here.