nLab
dendroidal set

Idea

Dendroidal sets are to operads and (∞,1)-operads as simplicial sets are to categories and (∞,1)-categories.

A dendroidal set is something that consists of trees in the way a simplicial set consists of simplices.

Definition

A dendroidal set is a presheaf on the tree category $\Omega$.

The category of dendroidal sets is the functor category

$$\mathrm{dSet}=[{\Omega }^{\mathrm{op}},\mathrm{Set}].$$dSet = [\Omega^{op}, Set] \,.

Some terminology and notation:

• For $T\in \Omega$ a tree, write $\Omega [T]:=\Omega (-,T)$ for the dendroidal set that it represents.

• For $X$ a dendroidal set and $T$ a tress, the set $\mathrm{dSet}(T,X)\simeq X(T)$ (using the Yoneda lemma) is the set o $T$-shaped dendrices in $X$. One of its elements is called a $T$-shaped dendrex, analogous to a simplex in a simplicial set.

Relation to simplicial sets

Write $0$ for the tree consisting of a single edge. Let $\Omega /0$ be the over category. This is canonically isomorphic to the simplex category

$$\Omega /0\simeq \Delta .$$\Omega/0 \simeq \Delta \,.

Moreover, the over-category of all dendroidal sets over $\eta :=\Omega [0]$ is the category SSet of simplicial sets

$$\mathrm{SSet}\simeq \mathrm{dSet}/\Omega [0].$$SSet \simeq dSet/\Omega[0] \,.

The corresponding canonical functor

$$i:\Delta =\Omega /0\to \Omega$$i : \Delta = \Omega/0 \to \Omega

is a full and faithful functor and induces an adjunction

$$i_{*}:\mathrm{SSet}:\overleftarrow{\to }:\mathrm{dSet}:i^{*},$$i_* : SSet : \stackrel{\leftarrow}{\to} : dSet : i^* \,,

where $i_{*}$ is a full and faithful functor defined equivalently as

• the left adjoint to $i^{*}$;

• the left Kan extension of $i$;

• the canonical functor $\mathrm{SSet}\stackrel{=}{\to }\mathrm{dSet}/\Omega [0]\to \mathrm{dSet}$.

A morphism $X\to i_{*}Y$ of dendroidal sets exists only for $X$ itself a simplicial set, i.e. $X$ itself in the image of $i_{*}$.

The dendroidal set $i([n])$ that corresponds to the $n$-simplex is to be visualized as the linear tree with $(n+1)$-edges:

$$i([n])=(\stackrel{0}{\to }•\stackrel{1}{\to }•\stackrel{2}{\to }\cdots •\stackrel{n}{\to }).$$i([n]) = \left( \stackrel{0}{\to} \bullet \stackrel{1}{\to} \bullet \stackrel{2}{\to} \cdots \bullet \stackrel{n}{\to} \right) \,.

We think of each bullet as a unary operation – an ordinary morphism – $\stackrel{n}{\to }•\stackrel{n+1}{\to }$ from the object $n$ to the object $(n+1)$.

Notice that this is hence Poincaré-dual to how one tends to visualize $[n]$ itself

$$[n]=(0\to 1\to 2\to \cdots \to n)$$[n] = (0 \to 1 \to 2 \to \cdots \to n)

and how one tends to visualize morphisms $n\to (n+1)$.

Faces of trees and dendroidal sets

The face maps on trees, regarded as dendroidal sets, are morphisms that generalize the face maps in the simplex category $\Delta$. Defining them in $\Omega$ requires a few case distinctions:

there are

1. inner face maps – obtained by contracting an inner edge

2. outer face maps – obtained by

   1. removing an outer input vertex

   2. removing a vertex whose output is the root and which has precisely one inner incoming edge (which becomes the new root)

   3. a corolla face – any one of the inclusions of the tree with no vertex into a tree with precisely one vertex

Inner faces

For $T\in \Omega$ and $e$ an inner edge (in the obvious sense), write $T/e$ for the tree obtained by contracting/discarding this inner edge. There is then a canonical inclusion

$${\partial }_e:T/e\to T.$$\partial_e : T/e \to T \,.

This is called an inner face map.

For example let

$$T=\left(\begin{array}{ccc}\searrow & & \swarrow \\ & v\\ \searrow & & {\searrow }^e\\ & •& \to & •& \to & \\ \nearrow \end{array}\right)$$T = \left( \array{ \searrow && \swarrow \\ & v \\ \searrow & & \searrow^e \\ & \bullet &\to& \bullet &\to& \\ \nearrow } \right)

then

$$T/e=\left(\begin{array}{ccccc}\searrow & & \searrow & & \swarrow \\ & •& \to & •& \to & \\ \nearrow \end{array}\right)$$T/e = \left( \array{ \searrow && \searrow& & \swarrow \\ & \bullet &\to& \bullet &\to& \\ \nearrow } \right)

and the inclusion $T/e\to T$ the the operad morphism that is the identity on the operation $\begin{array}{c}\searrow \\ & •& \to \\ \nearrow \end{array}$ and which sends the trinary operation $\begin{array}{ccc}\searrow & & \swarrow \\ \to & •& \to \end{array}$ to the composite trinary operation $\begin{array}{ccc}\searrow & & \swarrow \\ & •\\ & & {\searrow }^e\\ & & \to & •& \to & \\ \end{array}$

Outer face maps

In contrast to that, outer edges are always removed together:

if $v$ is a vertex of $T$ such that all but one edge incident on it are outer, then denote by $T/v$ the tree obtained by discarding $v$ and all the outer edges indicent on it. There is then again a canonical inclusion

$${\partial }_v:T/v\to T.$$\partial_v : T/v \to T \,.

This is called an outer face map.

In the above example for $T$ we have

$$T/v=\left(\begin{array}{ccc}\searrow & & {\searrow }^e\\ & •& \to & •& \to & \\ \nearrow \end{array}\right)$$T/v = \left( \array{ \searrow & & \searrow^e \\ & \bullet &\to& \bullet &\to& \\ \nearrow } \right)

Corolla faces

If the tree has precisely one vertex it is called a corolla. There are $n+1$ injections of the tree with no vertex $\mid$ into the corolla with $n$ inputs. All these are outer face maps.

$$\mid \hookrightarrow \begin{array}{ccc}\searrow & & \swarrow \\ & *\\ & \downarrow \end{array}.$$| \hookrightarrow \array{ \searrow && \swarrow \\ & {*} \\ & \downarrow } \,.

Properties

Lemma (face maps are the monomorphisms)

The inner and outer face morphisms ${\partial }_e$ and ${\partial }_v$ are precisely the monomorphisms in the tree category $\Omega$.

Proof

Lemma 3.1 in MoeWei07.

The boundary of a tree is the union of all its face dendroidal sets

$$\partial \Omega [T]={\cup }_{e\in \mathrm{Edges}(T)}\Omega [{\partial }_{e}T]{\cup }_{v\in \mathrm{Vertices}(T)}\Omega [{\partial }_{v}T].$$\partial \Omega[T] = \cup_{e \in Edges(T)} \Omega[\partial_e T] \cup_{v \in Vertices(T)} \Omega[\partial_v T] \,.

Compare to boundary of a simplex.

Analogously then to the notion of horn of a simplex, for $d$ an edge of $T$ the union

$${\Lambda }^e\Omega [T]={\cup }_{e\ne d\in \mathrm{Edges}(T)}\Omega [{\partial }_{e}T]{\cup }_{v\in \mathrm{Vertices}(T)}\Omega [{\partial }_{v}T]$$\Lambda^e \Omega[T] = \cup_{e \neq d \in Edges(T)} \Omega[\partial_e T] \cup_{v \in Vertices(T)} \Omega[\partial_v T]

of dendroidal sets is the inner horn of $T$ at $e$, and for $w$ an outer face the union

$${\Lambda }^w\Omega [T]={\cup }_{e\ne d\in \mathrm{Edges}(T)}\Omega [{\partial }_{e}T]{\cup }_{v\ne w\in \mathrm{Vertices}(T)}\Omega [{\partial }_{v}T]$$\Lambda^w \Omega[T] = \cup_{e \neq d \in Edges(T)} \Omega[\partial_e T] \cup_{v \neq w \in Vertices(T)} \Omega[\partial_v T]

is the outer horn at $w$.

We have the canonical boundary inclusions

$$\partial \Omega [T]\to \Omega [T]$$\partial \Omega[T] \to \Omega[T]

and horn inclusions

$${\Lambda }^{\alpha }\Omega [T]\to \Omega [T].$$\Lambda^\alpha \Omega[T] \to \Omega[T] \,.

Definition

A monomorphism $X\to Y$ of dendroidal sets is called normal if for any tree $T$, any non-degenerate dendrex $y\in Y(T)$ which does not belong to the image of $X(T)$ has a trivial stabilizer $\mathrm{Aut}(T{)}_y\subset \mathrm{Aut}(T)$.

A dendroidal set $X$ is normal if $\varnothing \hookrightarrow X$ is a normal monomorphism.

For instance for any tree $T$, the dendroidal set $\Omega [T]$ is normal.

Remark

This has nothing to do with the notion of normal monomorphism in a context with zero morphisms.

Proposition

The class of morphisms in $\mathrm{dSet}$ generated from the boundary inclusions under pushout and transfinite composition is precisely the class of normal monomorphisms.

Proof

This is prop 1.4 in CisMoer09.

Structure on $\mathrm{dSet}$

Closed symmetric monoidal structure

The category $\mathrm{dSet}$ of dendroidal sets carries the structure of a symmetric monoidal category, defined as follows:

in analogy to the nerve adjunction

$$\tau :\mathrm{SSet}\overleftarrow{\to }\mathrm{Cat}:N$$\tau : SSet \stackrel{\leftarrow}{\to} Cat : N

between ordinary categories and simplicial sets, there is an adjunction

$${\tau }_d:\mathrm{dSet}\overleftarrow{\to }\mathrm{Operad}:N_d$$\tau_d : dSet \stackrel{\leftarrow}{\to} Operad : N_d

between operads and dendroidal sets.

The category $\mathrm{Operad}$ carries the Boardman-Vogt tensor product? ${\otimes }_{\mathrm{BV}}$ and the symmetric monoidal structure on $\mathrm{dSet}$ is taken to be the unique one

• that makes ${\tau }_d$ a symmetric monoidal functor;

• such that for any two trees $T,S$ we have

  $$\Omega [T]\otimes \Omega [S]=N_d(T{\otimes }_{\mathrm{BV}}S).$$\Omega[T] \otimes \Omega[S] = N_d(T \otimes_{BV} S) \,.

With respect to this monoidal struucture $\mathrm{dSet}$ is a closed monoidal category.

$\mathrm{SSet}$-enriched structure

Using the fact that $\mathrm{dSet}$ is a closed monoidal category with internal hom dendroidal sets $[C,D]$ for dendroidal sets $C$ and $D$, and using the functor $i^{*}:\mathrm{dSet}\to \mathrm{SSet}$ we obtain canonically the structure of an SSet-enriched category on $\mathrm{dSet}$ with the hom-simplicial set between $C$ and $D$ being $i^{*}[C,D]$.

Model category structure

The category $\mathrm{dSet}$ also carries the Cisinski-Moerdijk model structure on dendroidal sets. With this model structure it forms a monoidal model category. Together with the fact that $i^{*}:\mathrm{dSet}\to \mathrm{SSet}$ is a right Quillen functor (with respect to the Joyal model structure on simplicial sets) this imples that $\mathrm{dSet}$ is a Joyal-SSet enriched model category.

References

A good survey of the theory as developed currently is

• Ieke Moerdijk, Lectures on dendroidal sets , lectures given at the Barcelona workshop on Simplicial methods in higher categories (2008) (preliminary writeup)

An expanded and polished version has meanwhile been written up by Javier Guitiérrez and should appear in print soon. An electronic copy is probably available on request.

The PhD thesis that gave the original definition of dendroidal sets is:

• Ittay Weiss, Dendroidal sets PhD thesis (web)

The publication derived from that:

• Ieke Moerdijk Ittay Weiss, Dendroidal sets (web)

A discussion of inner Kan complexes (see also model structure on dendroidal sets):

• Ieke Moerdijk Ittay Weiss, On inner Kan complexes in the category of dendroidal sets (web)

Here two blog entries with some summaries and pointers to the literature

• Dendroidal sets and infinity-operads

• Moerdijk on infinity-operads