symmetric monoidal (∞,1)-category of spectra
Dendroidal sets are a geometric model for higher operads (precisely: multi-coloured symmetric operads / symmetric multicategories). They are to operads and to (∞,1)-operads as simplicial sets are to categories and (∞,1)-categories.
A dendroidal set is something that consists of trees or “dendrices” in the way that a simplicial set consists of simplices: the trees represent the free Set-operads over them, and so a dendroidal set is a structure defined as having consistent probes by free $Set$-operads.
More precisely, the category $dSet$ of dendroidal sets serves to complete the following commuting diagram of functors
where the left vertical functor is the nerve $N :$ Cat $\to$ sSet, and where the top horizontal functor includes categories into Set-enriched operads as those operads with only unary operations. Moreover, $dSet$ completes this diagram even as a diagram of most of the evident pairs of adjoint functors. For details see The full diagram of relations below.
Finally, there is a model structure on dendroidal sets which is the operadic analog of the model structure for quasi-categories.
Write Operad for the category of operads, specifically for
operads, also known as symmetric multicategories.
The symmetric tree category is the full subcategory
of Operad on those symmetric operads that are free on finite rooted non-planar trees.
See (below) for details on the trees appearing here.
A dendroidal set is a presheaf on the tree category $\Omega$.
The category of dendroidal sets is the functor category
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 tree, the set $dSet(\Omega[T],X) \simeq X(T)$ (using the Yoneda lemma) is the set of $T$-shaped dendrices in $X$. One of its elements is called a $T$-shaped dendrex, analogous to a simplex in a simplicial set.
We expand on the notion of symmetric/non-planar trees used in definition 3 of dendroidal sets (see for instance (Weiss, section 2.1)).
A finite symmetric rooted tree – or just tree for short, in the following – is a finite poset $(T, \leq)$ such that
it has a bottom element;
for each element $e \in T$ the set $\{y \in T | T \leq e\}$ is a linear order under $\leq$;
and equipped with a subset $L \subset max(T)$ of the maximal elements of $T$.
One has the following terminology
Let $((T, \leq), L)$ be a tree.
An element $e \in T$ is called an edge of the tree.
The bottom element is called the root of the tree.
An edge in $L \subset T$ is a leaf of the tree.
An edge which is either a leaf or the root is called an outer edge.
An edge which is neither a leaf nor the root is called an inner edge.
Given a non-leaf edge $e$, another edge $e'$ is called an incoming edge of $e$ if it is a direct succesor of $e$, hence if
$e \leq e'$;
for any $e \in T$ with $e \leq x \leq e'$ either $x = e$ or $x = e'$.
Write $in(e)$ for the set of incoming edges of $e$.
For $e$ a non-leaf edge, the subset $v_e := \{e\} \cup in(e)$ is called a vertex of $(T, \leq)$. $e$ is called the outgoing edge of $v$, and the ingoing edges of $e$ are also called ingoing edges of $v_e$, $in(v_e) := in(e)$.
The valence of a vertex $v$ is the cardinality ${\vert in(v)\vert}$.
A typical tree as above is
where
the arrows depict the edges;
the bullets depict vertices;
the arrows not starting at a bullet depict leaf edges.
the arrow not ending at a bullet depicts the root edge.
The set of incoming edges of $v$ is $\{e,f\}$, the outgoing edge of $v$ is $b$. The set of incoming edges of $u$ is $\{b,c,d\}$, its outgoing edge is the root edge $r$. The set of ingoing edges of $e$ is the empty set, its outgoing edge is $d$.
The valence of $v$ is 2, that of $u$ is 3 and that of $w$ is 0.
Notice that there is no order on the incoming edges of any vertex implied, the fact that there is one in the above picture is an artefact of being a planar diagram for visualization purposes. As a tree, the above is equal to, for instance
For $((T, \leq ), L)$ a tree, the corresponding symmetric operad over Set is the one
whose set of colours is is $T$ (the set of edges);
whose operations are freely generated from the set of vertices of $T$, where the generating operation corresponding to a vertex $v$ goes from $in(v)$ to the outgoing edge $c$ of $v$.
More precisely, for every vertex $v$ and every choice of order $(c_1, \cdots, c_n)$ on $in(v)$ there is a generating operation
and the action of an element of the symmetric group $\sigma \in S_n$ takes this to the generator
The operad corresponding to the tree from example 1
has six colours;
has precisely six non-identity operations:
the generators $v$, $u$, $w$;
the composites $u \circ (v, id_c, id_d)$ and $u \circ (id_b, id_c, w)$ and $u \circ (v, id_c, w)$.
The inclusion $\Omega \hookrightarrow$ Operad is the full subcategory on those operads that arise from trees in this way.
Some non-typical trees of importance are the following.
For every $n \in \mathbb{N}$ there is the linear tree
with $(n+1)$ colors and only unary operations.
The map that sends the simplicial simplex $\Delta^n$ to $L_n$ extends to a functor
which exhibits the simplex category as a full subcategory of the tree category.
For each $n \in \mathbb{N}$ there is the corolla tree
with $n$ leaves and precisely one vertex.
The trees $L_0$ and $C_0$ differ. $L_0$ is the tree
with no vertex, while $C_0$ is the tree
with a single vertex, which has valence 0.
A dendroidal set encodes composition of operations in analogy to how a simplicial set encodes composition of edges: by way of horn extensions. In order to formalize this one uses the dendroidal analog of faces of a simplex, generalized from the simplex category to the tree category.
To motivate the definition of dendroidal face maps, first consider the reformulation of simplicial faces under the map $i_! : sSet \to dSet$.
Under this embedding the $n$-simplex $\Delta[n]$ becomes the operad corresponding to the linear tree $L_n$, example 3
with unary operations $\bullet_{i (i+1)}$.
The face inclusion $\delta_i : \Delta[n-1] \to \Delta[n]$ which simplicially omits the $i$th vertex, operadically contracts away the $i$th color, i.e it is the morphism of operads that sends the unary operation $\bullet_{(i-1)i}$ to the unary operation
and sends all other generating unary operation to generating unary operations.
From this it is clear that for any tree the map that exhibits the contraction of one edge should be a dendroidal face map. However, there are also some more cases to be taken care of. One has
inner face maps – obtained by contracting an inner edge
outer face maps – obtained by
removing an outer input vertex
removing a vertex whose output is the root and which has precisely one inner incoming edge (which becomes the new root)
a corolla face – any one of the inclusions of the tree with no vertex into a tree with precisely one vertex
For $T \in \Omega$ a tree and $e$ an inner edge , write $T/e$ for the tree obtained by contracting/discarding this inner edge. There is a canonical inclusion
in $\Omega$.
This is called an inner face map of $T$.
Let
then
and the inclusion $T/e \to T$ is the operad morphism that is the identity on the operation $\array{ \searrow \\ & \bullet & \to \\ \nearrow }$ and which sends the trinary operation $\array{ \searrow && \swarrow \\ \to & \bullet & \to }$ to the composite trinary operation $\array{ \searrow && \swarrow \\ & \bullet \\ & & \searrow^e \\ & &\to& \bullet &\to& \\ }$
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
This is called an outer face map.
With $T$ as from example 5, we have
Let $T$ be a corolla tree, example 4. There are $n+1$ injections of the tree $L_0$ with no vertex $|$ into the corolla with $n$ inputs.
These are called corolla face maps and counted as outer face maps.
In terms of these face maps there is now a notion of boundary and horn of a tree in direct analogy to the notion of boundary of a simplex and horn in a simplex.
Let $T$ be a tree and let $Faces(T) \subset \Omega_{/T}$ be the set of all its face maps $\partial_\alpha \Omega[T] \to \Omega[T]$ , as defined above. The boundary of $T$ is the dendroidal set
given by the union of all these faces in $\Omega[T]$.
For given $\alpha \in Faces(T)$, the $\alpha$-horn $\Lambda^\alpha[T] \in dSet$ of $T$ is the union of all faces except this one:
A horn is called an outer horn or an inner horn depending on whether the omitted face is outer or inner, respectively.
The inner horn corresponding to the inner face map given by contraction an edge $e$ is canonically denoted $\Lambda^a \Omega[T]$.
These definitions are due to (Weiss (thesis)) and (MoerdijkWeiss).
This is a genuine generalization of the notion of horns and boundaries of simplices. The outer/inner horns of the $n$-simplex $\Delta[n]$ are taken by $i_! : sSet \to dSet$ precisely to the outer/inner hors of the linear tree $L_n = i_!(\Delta^n)$.
For $\Lambda^\alpha \Omega[T] \to \Omega[T]$ a horn inclusion and for $X \in dSet$ a dendroidal set, an $\alpha$-horn in $X$ is a morphism of simplicial set
A filler of this horn in $X$ is an extension $\sigma$ in
Choices of dendroidal inner horn fillers correspond to choices of composites of operations.
A dendrex $\Omega[T] \to X$ encodes a collection of operations and choices of theor composite in $X$.
An inner horn $\Lambda^e \Omega[T] \to X$ encodes a choice of operations in $X$ and their composites except a choice for the composition of operations along $e$. Picking a filler for this inner horn is picking such a choice.
The outer horn fillers have different interpretation. They correspond to choices of a) inverses of linear operations 2) invertible elements of $n$-ary operation.
For more on this see model structure on dendroidal sets.
For $X \in dSet$ a dendroidal set and $n \in \mathbb{N}$, write $Sk_n(X) \in dSet$ for the dendroidal set which is generated from the non-degenerate $T$-dendrices for ${\vert T\vert} \leq n$.
The sequence of inclusions
is called the skeletal filtration of $X$.
A key fact in the theory of simplicial sets is that the monomorphisms there are generated under pushout, transfinite composition and retracts from the boundary inclusions (indeed the model structure on simplicial sets is a cofibrantly generated model category with generating cofibrations the boundary inclusions).
For dendroidal sets the boundary inclusions turn out not to generate all monomorphisms, but just a subclass called the normal monomorphisms. We discuss now the definition and some basic propoerties of normal monomorphisms. Most of these are a specialization of the general notion of normal morphisms over a generalized Reedy category to the generalized Reedy category $\Omega$.
The inner and outer face morphisms $\partial_e$ and $\partial_v$ are precisely the monomorphisms in the tree category $\Omega$.
Lemma 3.1 in MoeWei07.
The boundary of a tree is the union of all its face dendroidal sets
Compare to boundary of a simplex.
Analogously then to the notion of horn of a simplex, for $d$ an edge of $T$ the union
of dendroidal sets is the inner horn of $T$ at $e$, and for $w$ an outer face the union
is the outer horn at $w$.
We have the canonical boundary inclusions
and horn inclusions
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 subgroup $Aut(T)_y \subset Aut(T)$ of the automorphism group of $T$.
A dendroidal set $X$ is called normal if $\emptyset \hookrightarrow X$ is a normal monomorphism.
This is unrelated to the notion of normal monomorphism in a context with zero morphisms.
Here are some equivalent characterizations of normality.
A dendroidal set $X$ is normal precisely if for every $n \in \mathbb{N}$ the canonical commuting diagram
is a pushout diagram, where $\{Sk_n(X)\}$ is the skeletal filtration, def. 12 and where the coproduct is over isomorphism classes of non-degenerate dendrices.
A monomorphism $f : X \to Y$ in $dSet$ is normal precisely if for every $T \in \Omega$ the action of the automorphism group $Aut(T)$ on $Y(T)-X(T)$ is a free action.
Accordingly, a dendroidal set $Y$ is normal precisely if for every $T \in \Omega$ the action of $Aut(T)$ on $Y(T)$ is free.
This is (CisMoer09, prop 1.5).
For $f : X \to Y$ any morphism between dendroidal sets, then
if $Y$ is normal, then $X$ is normal;
if $Y$ is normal and $f$ is monic, then $f$ is normal.
For every tree $T$, the dendroidal set $\Omega[T]$ is normal.
For every simplicial set, the dendroidal set $i_!(X)$ is normal.
The class of normal morphisms in $dSet$ is generated from the boundary inclusions under
In particular it is closed under these operations.
This is (CisMoer09, prop 1.4).
As any category of presheaves, $dSet$ is a cartesian monoidal category. However, the cartesian tensor product is not the natural one with respect to the inclusion of operads into dendroidal sets. The natural monoidal structure on Operad is rather a generalization of the Boardman-Vogt tensor product.
Write
for the functor that forms the Boardman-Vogt tensor product of the free operads given by two trees, and then regards the result as a dendroidal set by the dendroidal nerve, def. 15.
The tensor product of dendroidal sets is the Yoneda extension
of this functor, hence the unique such functor which preserves colimits in both variables and coincides with the BV-tensor product of operads on $\Omega$.
With respect to this tensor product is $dSet$ a closed monoidal category. This is discussed below.
For $A \to B$ and $X \to Y$ in $dSet$ two normal monomorphisms, def. 13, the canonical morphism out of their pushout product
is also normal.
This is (CisMoer09, prop 1.9).
Write $0$ for the tree consisting of a single edge. Let $\Omega/0$ be the over category. By inspection one sees that
There is an equivalence of categories
of the slice with the simplex category.
Moreover, the over-category of all dendroidal sets over $\eta := \Omega[0]$ is the category sSet of simplicial sets
By general properties of Kan extension we have the following
The corresponding canonical forgetful functor
is
The functor $i : \Delta \to \Omega$ induces an adjoint triple
where
$i^*$ is given by precomposition with $i$;
$i_!$ is a full and faithful functor defined equivalently as
the left adjoint to $i^*$;
the left Kan extension of $i$;
the canonical functor $sSet \stackrel{\simeq}{\to} dSet/\Omega[0] \to dSet$;
the functor that sends $X \in sSet$ to
Hence $i_!$ is a full and faithful functor, and hence (see the discussion at adjoint triple) so is $i_*$.
Accordingly, $i : sSet \hookrightarrow dSet$ is an open geometric embedding of presheaf toposes.
The dendroidal set $i_!([n])$ that corresponds to the $n$-simplex is the linear tree; example 3, with $(n+1)$-edges:
We think of each bullet as a unary operation – an ordinary morphism – $\stackrel{n}{\to}\bullet \stackrel{n+1}{\to}$ from the object $n$ to the object $(n+1)$.
Notice that this is hence Poincaré-dual to how one tends to visualize $[n]$ itself
and how one tends to visualize morphisms $n \to (n+1)$.
The embedding of simplicial sets into dendroidal sets compatibly relates the Cartesian monoidal structure $(sSet, \times)$ with the Boradman-Vogt monoidal structure $(dSet, \otimes_{BV})$
The functor
is a strong monoidal functor, in that for all $X, Y \in sSet$ there is a natural isomorphism
in $dSet$.
Moreover, $(i_! \dashv i^*)$ respects the internal hom of dendroidal sets, prop. 12, in that for all $X,Y \in sSet$ and $D \in dSet$
and hence in particular
This appears as (Moerdijk-Weiss, prop. 5.3).
By the general notion of nerve and realization, the inclusion $\Omega \hookrightarrow Operad$ from def. 2 induces a nerve operation on operads with values in simplicial sets.
For $O \in Operad$ an operad, its dendroidal nerve is the dendroidal set given by
This extends to a functor
By general properties of nerve and realization we have:
The dendroidal nerve has the following properties.
It extends the simplicial nerve of operads in that
It has a left adjoint
given by left Kan extension.
$N_d$ is a full and faithful functor, equivalently there is a natural isomorphism
See for instance (Moerdijk-Weiss, section 4).
For $X$ a dendroidal set, $\tau_d(X)$ is also called the operad generated by $X$.
In conclusion we find reflective subcategory
This is compatible with the Boardman-Vogt tensor product as defined above:
The functor $\tau_d : (dSet, \otimes) \to (Operad, \otimes_{BV})$ is a strong monoidal functor, in that for all $X, Y \in dSet$ there is a natural isomorphism
This appears as (Moerdijk-Weiss, prop. 5.2).
Since $\tau_d N_d \simeq id$ in particular we have for all $P, Q \in Operad$ an isomorphism
The dendroidal nerve of Set-operads discussed above is important for setting up the model of dendroidal sets. But the usefulness of the model comes from its relation to sSet/Top-enriched operads (topological operads) via an operadic generalization of the homotopy coherent nerve that sends sSet-enriched categories to simplicial sets.
Write $Operad_sSet$ for the category of sSet-enriched operads (simplicial operads). Write
for the Boardman-Vogt resolution functor.
The restriction of this along the inclusion of free Set-operads
which will also be denoted just $W_H$ in the following, canonically induces nerve and realization functors:
The dendroidal homotopy coherent nerve of a simplicial operad $P$ is the dendroidal set $hcN_d(P)$ given by
This extends to a functor
We write
(or $W_!$ as here) for the corresponding left adjoint realization.
Via this adjunction
we may understand generally the theory of dendroidal sets as being about BV-resolutions of simplicial operads.
Let $P \in Operad_{sSet}$.
There is an isomorphism of simplicial operads
between the Boardman-Vogt resolution $W_H(P)$ of $P$ and the homotopy-realization of its homotopy-coherent dendroidal nerve.
All of this is an operadic generalization of the relation between quasi-categories and simplicial categories.
The above functors between dendroidal sets and simplicial sets (here) and operads (here) arrange into the following diagram
where functors on the left and on top are left adjoints to those on the right and on the bottom, respectively.
This commutes in three of four possible ways, up to natural isomorphism in that
$j_! \tau \simeq \tau_d i_!$;
$N j^* \simeq i^* N_d$;
$i_! N \simeq N_d j_!$.
There is also a natural transformation
but not all of its components are isomorphisms.
Moreover, $N, N_d$,$i_!, j_!$ are full and faithful functors and hence (see the properties of adjoint functors)
$\tau N \simeq id$;
$\tau_d N_d \simeq id$;
$i^* i_! \simeq id$;
$j^* j_! \simeq id$.
There exists an essentially unique symmetric closed monoidal category structure $(dSet, \otimes)$ on $dSet$ such that for all $S, T \in \Omega \hhokrightarrow Operad$ there is a natural isomorphism
with $\otimes_{BV}$ the Boardman-Vogt tensor product on operads, and with $N_d$ the dendroidal nerve, def. 15.
This is given as discussed above. The corresponding internal hom $[-,-]_{BV} : dSet^{op} \times dSet \to dSet$ is given by the formula
This appears as (Moerdijk-Weiss, prop. 5.1).
Using the fact that $dSet$ is a closed monoidal category with internal hom dendroidal sets $[C,D]$ for dendroidal sets $C$ and $D$, and using the functor $i^* : dSet \to SSet$ we obtain canonically the structure of an simplicially enriched category / sSet-enriched category on $dSet$ with the hom-simplicial set between $C$ and $D$ being $i^*[C,D]$.
The category $dSet$ carries the Cisinski-Moerdijk model structure on dendroidal sets. With this model structure it forms a monoidal model category.
Together with the fact that $i^*: dSet \to sSet$ is a right Quillen functor (with respect to the model structure for quasi-categories) this imples that $dSet$ is an $sSet_{Joyal}$-enriched model category (but not, without further work, an $sSet_{Quillen}$-enriched model category!).
Surveys of the theory as developed currently include:
Ieke Moerdijk, Lectures on dendroidal sets , lectures given at the Barcelona workshop on Simplicial methods in higher categories (2008) (preliminary writeup)
Ittay Weiss, From operads to dendroidal sets , in Mathematical Foundations of Quantum Field and Perturbative String Theory, AMS (2011)
Dendroidal sets were introduced in
A discussion of dendroidal inner Kan complexes (see also at model structure on dendroidal sets) appeared in
The thesis
contains essentially the material of these two articles, together with a discussion of broad posets.
The model structure for dendroidal complete Segal spaces and the model structure for Segal operads was constructed in
Normal morphisms of dendroidal sets are discussed for instance around prop. 1.4 of