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 -operads.
where the left vertical functor is the nerve Cat sSet, and where the top horizontal functor includes categories into Set-enriched operads as those operads with only unary operations. Moreover, 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.
operads, also known as symmetric multicategories.
See (below) for details on the trees appearing here.
Some terminology and notation:
For a tree, write for the dendroidal set that it represents.
and equipped with a subset of the maximal elements of .
One has the following terminology
Let be a tree.
An element is called an edge of the tree.
The bottom element is called the root of the tree.
An edge in 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 lead nor the root is called an inner edge.
Given a non-leaf edge , another edge is called an incoming edge of if it is a direct succesor of , hence if
for any with either or .
Write for the set of incoming edges of .
For a non-leaf edge, the subset is called a vertex of . is called the outgoing edge of , and the ingoing edges of are also called ingoing edges of , .
The valence of a vertex is the cardinality .
A typical tree as above is
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 is , the outgoing edge of is . The set of incoming edges of is , its outgoing edge is the root edge . The set of ingoing edges of is the empty set, its outgoing edge is .
The valence of is 2, that of is 3 and that of 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
whose set of colours is is (the set of edges);
whose operations are freely generated from the set of vertices of , where the generating operation corresponding to a vertex goes from to the outgoing edge of .
More precisely, for every vertex and every choice of order on there is a generating operation
and the action of an element of the symmetric group 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 , , ;
the composites and and .
Some non-typical trees of importance are the following.
For every there is the linear tree
with colors and only unary operations.
For each there is the corolla tree
with leaves and precisely one vertex.
The trees and differ. is the tree
with no vertex, while 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 .
Under this embedding the -simplex becomes the operad corresponding to the linear tree , example 3
with unary operations .
The face inclusion which simplicially omits the th vertex, operadically contracts away the th color, i.e it is the morphism of operads that sends the unary operation 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 a tree and an inner edge , write for the tree obtained by contracting/discarding this inner edge. There is a canonical inclusion
This is called an inner face map of .
and the inclusion is the operad morphism that is the identity on the operation and which sends the trinary operation to the composite trinary operation
In contrast to that, outer edges are always removed together:
If is a vertex of such that all but one edge incident on it are outer, then denote by the tree obtained by discarding and all the outer edges indicent on it. There is then again a canonical inclusion
This is called an outer face map.
With as from example 5, we have
Let be a corolla tree, example 4. There are injections of the tree with no vertex into the corolla with inputs.
These are called corolla face maps and counted as outer face maps.
Let be a tree and let be the set of all its face maps , as defined above. The boundary of is the dendroidal set
given by the union of all these faces in .
For given , the -horn of 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 is canonically denoted .
This is a genuine generalization of the notion of horns and boundaries of simplices. The outer/inner horns of the -simplex are taken by precisely to the outer/inner hors of the linear tree .
For a horn inclusion and for a dendroidal set, an -horn in is a morphism of simplicial set
A filler of this horn in is an extension in
Choices of dendroidal inner horn fillers correspond to choices of composites of operations.
A dendrex encodes a collection of operations and choices of theor composite in .
An inner horn encodes a choice of operations in and their composites except a choice for the composition of operations along . 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 -ary operation.
For more on this see model structure on dendroidal sets.
For a dendroidal set and , write for the dendroidal set which is generated from the non-degenerate -dendrices for .
The sequence of inclusions
is called the skeletal filtration of .
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 .
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.
of dendroidal sets is the inner horn of at , and for an outer face the union
is the outer horn at .
We have the canonical boundary inclusions
and horn inclusions
A dendroidal set is called normal if is a normal monomorphism.
Here are some equivalent characterizations of normality.
A dendroidal set is normal precisely if for every the canonical commuting diagram
Accordingly, a dendroidal set is normal precisely if for every the action of on is free.
This is (CisMoer09, prop 1.5).
For any morphism between dendroidal sets, then
if is normal, then is normal;
if is normal and is monic, then is normal.
For every tree , the dendroidal set is normal.
For every simplicial set, the dendroidal set is normal.
The class of normal morphisms in 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, 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.
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 .
is also normal.
This is (CisMoer09, prop 1.9).
Write for the tree consisting of a single edge. Let be the over category. By inspection one sees that
There is an equivalence of categories
of the slice with the simplex category.
By general properties of Kan extension we have the following
The functor induces an adjoint triple
is given by precomposition with ;
is a full and faithful functor defined equivalently as
Notice that this is hence Poincaré-dual to how one tends to visualize itself
and how one tends to visualize morphisms .
and hence in particular
This appears as (Moerdijk-Weiss, prop. 5.3).
This extends to a functor
By general properties of nerve and realization we have:
The dendroidal nerve has the following properties.
See for instance (Moerdijk-Weiss, section 4).
For a dendroidal set, is also called the operad generated by .
In conclusion we find reflective subcategory
This appears as (Moerdijk-Weiss, prop. 5.2).
Since in particular we have for all 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.
for the Boardman-Vogt resolution functor.
The restriction of this along the inclusion of free Set-operads
which will also be denoted just in the following, canonically induces nerve and realization functors:
This extends to a functor
Via this adjunction
we may understand generally the theory of dendroidal sets as being about BV-resolutions of simplicial operads.
There is an isomorphism of simplicial operads
between the Boardman-Vogt resolution of 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.
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
There is also a natural transformation
but not all of its components are isomorphisms.
This appears as (Moerdijk-Weiss, prop. 5.1).
Using the fact that is a closed monoidal category with internal hom dendroidal sets for dendroidal sets and , and using the functor we obtain canonically the structure of an simplicially enriched category / sSet-enriched category on with the hom-simplicial set between and being .
Together with the fact that is a right Quillen functor (with respect to the model structure for quasi-categories) this imples that is an -enriched model category (but not, without further work, an -enriched model category!).
Surveys of the theory as developed currently include:
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
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