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model structure for dendroidal complete Segal spaces

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Contents

Idea

The model structure for dendroidal complete Segal spaces is an operadic generalization of the model structure for complete Segal spaces. It serves to present the (∞,1)-category of (∞,1)-operads.

A complete dendroidal Segal space XX is much like a dendroidal set, only that it has for each tree TT not just a set of dendrices, but a simplicial set X TsSetX_T \in sSet, subject to some conditions. The model structure discussed here is defined on the category of all simplicial presheaves over the tree category, such that the fibrant objects are precisely the dendroidal complete Segal spaces.

Definition

Write Ω\Omega for the tree category, the site for dendroidal sets

dSet:=[Ω op,Set]. dSet := [\Omega^{op}, Set] \,.

Write \otimes for the Boardman-Vogt tensor product on dendroidal sets (see there for details).

Definition

Let dsSet gReedy:=[Ω op,sSet]dsSet_{gReedy} := [\Omega^{op}, sSet] be the category of dendroidal simplicial sets, equipped with the generalized Reedy model structure induced from the generalized Reedy category Ω\Omega.

Write

dsSet SegaldsSet gReedy dsSet_{Segal} \stackrel{\leftarrow}{\to} dsSet_{gReedy}

for the left Bousfield localization at the set of dendroidal spine (“Segal core”) inclusions {Sp[T]Ω[T]} TΩ\{Sp[T] \to \Omega[T]\}_{T \in \Omega}, to be called the model structure for dendroidal Segal spaces.

A fibrant object in this category is called a dendroidal Segal space.

Write

dsSet cSegaldsSet Segal dsSet_{cSegal} \stackrel{\leftarrow}{\to} dsSet_{Segal}

for the further left Bousfield localization at the set of morphisms {Ω[T](J dη)} TΩ,\{\Omega[T]\otimes (J_d \to \eta) \}_{T \in \Omega, }, where J dJ_d is the dendroidal groupoidal interval

J d:=i !(N({01})). J_d := i_!(N(\{0 \stackrel{\simeq}{\to} 1\})) \,.

Call this the model structure for complete dendroidal Segal spaces.

A fibrant object in here is called a complete dendroidal Segal space.

This is (Cisinski-Moerdijk, def. 5.4, def. 6.2).

Proposition

The localization at the dendroidal spine inclusions is equivalently the left Bousfield localization at the set of dendroidal inner horn inclusions.

This is Cisinski-Moerdijk, prop. 5.5, def. 6.2.

Proof

By the nature of left Bousfield localization, it is sufficient to show that one localizing set of morphisms is contained in the weak equivalences of the other.

In one direction, it is clear that every inner anodyne morphism of dendroidal sets is a weak equivalence in the localization at the horn inclusions. By the discussion at spine, the spine inclusions are indeed inner anodyne.

Conversely, one checks that the weak equivalences generated by the spine inclusions contain all inner anodyne morphisms (Cisinski-Moerdijk, prop. 2.8)

Definition

Write ηΩ\eta \in \Omega for the tree with a single edge and no vertex. For nn \in \mathbb{N} write C nΩC_n \in \Omega for the nn-corolla, the tree with a single vertex and nn leaves (and the root).

Definition

For XX a dendroidal Segal space, and for (x 1,,x n;x)(X(η) 0) n+1(x_1, \cdots, x_n; x) \in (X(\eta)_0)^{n+1}, write X(x 1,,x n;x)sSetX(x_1, \cdots, x_n; x) \in sSet for the pullback

X(x 1,,x n;x) X(C n) * (x 1,,x n;x) (X(η)) n+1. \array{ X(x_1, \cdots, x_n; x) &\to& X(C_n) \\ \downarrow && \downarrow \\ * &\stackrel{(x_1, \cdots, x_n; x)}{\to}& (X(\eta))^{n+1} } \,.
Remark

These simplicial sets X(x 1,,x n;x)X(x_1, \cdots, x_n; x) are Kan complexes and in fact are the homotopy fibers of the right vertical morphism.

Proof

The inclusion η n+1Ω(C n)\eta^{n+1} \to \Omega(C_n) is a cofibration in [Ω op,sSet] Segal[\Omega^{op}, sSet]_{Segal}. So in this simplicial model category the right vertical morphism in def. 3 are Kan fibrations. These are stable under ordinary pullback, and their ordinary pullback is a homotopy pullback (as discussed there).

Definition

A morphism f:XYf : X \to Y between dendroidal Segal spaces is fully faithful if for all (x 1,,x n;x)X(η) n+1(x_1, \cdots, x_n; x) \in X(\eta)^{n+1}, for all nn \in \mathbb{N} the corresponding morphism

X(x 1,,x n;x)Y(f(x 1),,f(x n);f(x)) X(x_1, \cdots, x_n; x) \to Y(f(x_1), \cdots, f(x_n); f(x))

is a homotopy equivalence.

Properties

Basic technical properties

As for any category of simplicial presheaves we have

Remark

The category [Ω op,sSet][\Omega^{op}, sSet] is canonically tensored, cotensored and enriched over sSet.

The tensoring is given by the degreewise cartesian product in sSet:

:sSet×[Ω op,sSet][Ω op,sSet] \cdot : sSet \times [\Omega^{op}, sSet] \to [\Omega^{op}, sSet]
(S,X)(SX:TS×X(T)). (S, X) \mapsto (S \cdot X : T \mapsto S \times X(T)) \,.

For XdSetX \in dSet a dendroidal set, the hom object functor restricted along dSet[Ω op,sSet]dSet \hookrightarrow [\Omega^{op}, sSet]

X ():dSet opsSet X^{(-)} : dSet^{op} \to sSet

is the essentially unique limit-preserving functor such that for all TΩT \in \Omega

X Ω[T]=X(T). X^{\Omega[T]} = X(T) \,.

We will often write “×\times” also for the tensoring “\cdot”.

Proof

The essential uniqueness in the last clause follows, because by the co-Yoneda lemma every dendroidal set SS may be written as a colimit over its cells

S= isolim (Ω[T]S)Ω[T]. S =_{iso} {\lim_{\underset{(\Omega[T] \to S)}{\to}}} \Omega[T] \,.

Therefore

X S=lim (Ω[T]S)X(T). X^S = {\lim_{\underset{(\Omega[T] \to S)}{\leftarrow}}} X(T) \,.
Proposition

For X[Ω op,sSet]X \in [\Omega^{op}, sSet], and TΩT \in \Omega, the matching object of XX at TT (in the sense of generalized Reedy model structure) is

Match TX=X Ω[T]. Match_T X = X^{\partial \Omega[T]} \,.

For f:XYf : X \to Y a morphism, the relative matching morphism

X(T)Match TX× Match TYY(T) X(T) \to Match_T X \times_{Match_T Y} Y(T)

is the universal morphism induced from the commutativity of the diagram

X Ω[T] f Ω[T] Y Ω[T] X Ω[T] f Ω[T] Y Ω[T]. \array{ X^{\Omega[T]} &\stackrel{f^{\Omega[T]}}{\to}& Y^{\Omega[T]} \\ \downarrow && \downarrow \\ X^{\partial \Omega[T]} &\stackrel{f^{\partial \Omega[T]}}{\to}& Y^{\partial \Omega[T]} } \,.
Proof

By definition

Match TX=lim (TT)X(T), Match_T X = {\lim_{\leftarrow}}_{(T' \hookrightarrow T)} X(T') \,,

where the limit is over faces of TT. By remark 2 this is

X (lim (TT)Ω[T]). \cdots \simeq X^{({\lim_\to}_{(T' \hookrightarrow T)} \Omega[T'])} \,.

By the discussion at dendroidal set, the exponent is the boundary of the tree TT.

Similarly one finds that the morphism X(T)Match TXX(T) \to Match_T X is

X Ω[T]Ω[T]:X Ω[T]X Ω[T]. X^{\partial \Omega[T] \hookrightarrow \Omega[T]} : X^{\Omega[T]} \to X^{\partial \Omega[T]} \,.

Equivalent localization

We discuss a different set of morphisms, such that the model structure [Ω op,sSet] gReedy[\Omega^{op}, sSet]_{gReedy} localized at it still coincides with the localization [Ω op,sSet] cSegal[\Omega^{op}, sSet]_{cSegal} from def. 1. This different localization makes more immediate the Quillen equivalence to the model structure on dendroidal sets that we discuss below in in Relation to dendroidal sets.

Notice that, by the discussion there, the model structure on dendroidal sets, dSet CMdSet_{CM}, is a cofibrantly generated model category. Accordingly, there is a set of generating acyclic cofibrations, which we will write

S={AB}Mor(dSet). S = \{A \to B\} \subset Mor(dSet) \,.

While its existence is known, no explicit description is presently available, but we do know that we may assume that

  1. domain and codomain of all elements of SS are normal dendroidal sets, hence cofibrant;

  2. it contains a morphism ηJ\eta \to J, where JJ is the codiscrete groupoid on two objects, regarded as a unary operad, regarded as a dendroidal set.

Proposition

The model structure [Ω op,sSet] cSegal[\Omega^{op}, sSet]_{cSegal} coincides with the left Bousfield localization of [Ω op,sSet] gReedy[\Omega^{op}, sSet]_{gReedy} at the set of pushout-product morphisms

{(AB)¯(Δ[n]Δ[n])} (AB)S,n. \left\{ (A \to B) \bar \cdot ( \partial \Delta[n] \to \Delta[n] ) \right\}_{(A \to B) \in S, n \in \mathbb{N}} \,.

(Cis-Moer, cor. 6.5).

Lemma

For every normal dendroidal set AA, the morphism A BV(Jη)A \otimes_{BV}(J \to \eta) is a weak equivalence in [Ω op,sSet] cSegal[\Omega^{op}, sSet]_{cSegal}.

Moreover, every “JJ-anodyne extension” is a weak cSegal-equivalence, meaning every morphism generated by pushout, transfinite composition and retracts from the pushout-products of {e}J\{e\} \to J with tree boundary inclusions.

(Cis-Moer, prop. 6.3)

Proof

For the first statement, it is sufficient to show that the morphism is a weak equivalence regarded in the slice model structure

[Ω op,sSet] cSegal/A[(Δ×Ω/A) op,Set]. [\Omega^{op}, sSet]_{cSegal} / A \simeq [(\Delta \times \Omega / A)^{op}, Set] \,.

The category of elements Δ×Ω/A\Delta \times \Omega/A is a “regular skeletal category” in the sense of Cisinski model structure theory. By a lemma there, natural transformations between functors preserving colimits and monomorphisms are componentwise weak equivalences is they are so on representables.

Now J()J \otimes (-) does preserve colimits and monomorphisms, and on representables the transformation J()η()J \otimes (-) \to \eta \otimes (-) is a cSegal-equivalence by definition.

The second statement now follows using that [Ω op,sSet] cSegal[\Omega^{op}, sSet]_{cSegal} is a left proper model category, being the left Bousfield localization of a left proper model category. Using this we have that with (ηJ d)Ω[T](\eta \to J_d) \otimes \partial \Omega[T] also its pushout Ω[T]Ω[T]J dΩ[T]\Omega[T] \to \Omega[T] \cup J_d \otimes \partial \Omega[T] is a weak equivalence, and so by two-out-of-three with the composite

(ηJ d)Ω[T]=Ω[T]Ω[T]J dΩ[T](ηJ d)¯(Ω[T]Ω[T])J dΩ[T] (\eta \to J_d)\otimes \Omega[T] = \Omega[T] \to \Omega[T] \cup J_d \otimes \partial \Omega[T] \stackrel{(\eta \to J_d)\bar \otimes (\partial \Omega[T] \to \Omega[T])}{\to} J_d \otimes \Omega[T]

also the pushout-product itself.

Proof

of prop. 3

By this proposition the acyclic cofibrations between normal dendroidal sets are generated from the JJ-anodyne extensions and closure under left cancellation property. Therefore by lemma 1 and two-out-of-three, they are all complete weak equivalences.

Therefore by the pushout-product axiom in the simplicial model category [Ω op,sSet] cSegal[\Omega^{op}, sSet]_{cSegal}, their powering into a fibration is an acyclic Kan fibration. By Joyal-Tierney calculus this means that all the pushout-products (AB)¯(Δ[n]Δ[n])(A \to B) \bar \cdot (\partial \Delta[n] \to \Delta[n]) have the left lifting property against fibrations, hence that they are weak equivalences in [Ω op,sSet] cSegal[\Omega^{op}, sSet]_{cSegal}.

Since therefore all the morphisms (AB)¯(Δ[n]Δ[n])(A \to B) \bar \cdot ( \partial \Delta[n] \to \Delta[n] ) are weak equivalences in [Ω op,sSet] cSegal[\Omega^{op}, sSet]_{cSegal}, it is now sufficient to show, conversely, that the morphisms that define the complete Segal localization are weak equivalence in the localization at these morphisms. For the tree horn inclusions this is clear, since they are among the localizing maps for n=0n = 0. For the morphisms (J dη)Ω[T](J_d \to \eta) \otimes \Omega[T] observe that

(ηJ d)¯(Ω[T])=(ηJ d)Ω[T] (\eta \to J_d) \bar \otimes (\emptyset \to \Omega[T]) = (\eta \to J_d) \otimes \Omega[T]

is JJ-anodyne (see Cisinski model structure), hence by 2-out-of-3 its retraction (J dη)Ω[T] (J_d \to \eta ) \otimes \Omega[T] is a weak equivalence.

Fibrations and Cofibrations

Proposition

A morphism f:XYf : X \to Y in [Δ op,sSet] gReedy[\Delta^{op}, sSet]_{gReedy} is a fibration or acyclic fibration, precisely if for all trees TΩT \in \Omega, the morphism of hom objects

X Ω[T]X Ω[T]× Y ΩTY Ω[T] X^{\Omega[T]} \to X^{\partial \Omega[T]} \times_{Y^{\partial \Omega{T}}} Y^{\Omega[T]}

is a Kan fibration or acyclic Kan fibration, respectively.

Proof

By definition of generalized Reedy model structure and using prop. 2.

Proposition

The generalized Reedy model structure [Ω op,sSet] gReeedy[\Omega^{op}, sSet]_{gReeedy} is a cofibrantly generated model category with set of generating cofibrations

I:={Δ[n]Ω[T]Δ[n]Ω[T]Δ[n]Ω[T]} n,TΩ I := \{\partial \Delta [n] \cdot \Omega[T] \cup \Delta[n] \cdot \partial \Omega[T] \to \Delta[n] \cdot \Omega[T]\}_{n \in \mathbb{N}, T \in \Omega}

and with set of acyclic generating cofibrations

J:={Λ k[n]Ω[T]Δ[n]Ω[T]Δ[n]Ω[T]} n,TΩ. J := \{\Lambda^k[n] \cdot \Omega[T] \cup \Delta[n] \cdot \partial \Omega[T] \to \Delta[n] \cdot \Omega[T]\}_{n \in \mathbb{N}, T \in \Omega} \,.

The statement is (Cisinski-Moerdijk, prop. 5.2). The following proof proceeds in view of remark 5.3 there.

Proof

By prop. 4 we have that a morphism f:XYf : X \to Y in [Ω op,sSet] gReedy[\Omega^{op}, sSet]_{gReedy} is a fibration or acyclic fibration precisely if for all trees TT the canonical morphism

X Ω[T]X Ω[T]× Y Ω[T]Y Ω[T] X^{\Omega[T]} \to X^{\partial \Omega[T]} \times_{Y^{\partial \Omega[T]}} Y^{\Omega[T]}

is a Kan fibration or acyclic Kan fibration, respectively.

This means equivalently that every diagram

Λ kΔ[n] X Ω[T] Δ[n] X Ω[T]× Y Ω[T]Y Ω[T] \array{ \Lambda^k \Delta[n] &\to& X^{\Omega[T]} \\ \downarrow && \downarrow \\ \Delta[n] &\to& X^{\partial \Omega[T]} \times_{Y^{\partial \Omega[T]}} Y^{\Omega[T]} }

or, respectively,

Δ[n] X Ω[T] Δ[n] X Ω[T]× Y Ω[T]Y Ω[T] \array{ \partial \Delta[n] &\to& X^{\Omega[T]} \\ \downarrow && \downarrow \\ \Delta[n] &\to& X^{\partial \Omega[T]} \times_{Y^{\partial \Omega[T]}} Y^{\Omega[T]} }

has a lift. A little reflection shows (see Joyal-Tierney calculus) that this, in turn, is equivalent to that every diagram

Λ k[n]×Ω[T]Δ[n]×Ω[T] X Δ[n]×Ω[T] Y \array{ \Lambda^k[n] \times \Omega[T] \cup \Delta[n]\times \partial \Omega[T] &\to& X \\ \downarrow && \downarrow \\ \Delta[n] \times \Omega[T] &\to& Y }

or, respectively,

Δ[n]×Ω[T]Δ[n]×Ω[T] X Δ[n]×Ω[T] Y \array{ \partial \Delta[n] \times \Omega[T] \cup \Delta[n]\times \partial \Omega[T] &\to& X \\ \downarrow && \downarrow \\ \Delta[n] \times \Omega[T] &\to& Y }

has a lift.

The statement follows by using the small object argument.

Remark

Being a category of presheaves, [Ω op,sSet][\Omega^{op}, sSet] is a locally presentable category. Together with the cofibrant generation of the model structure from prop. 5 this means that [Ω op,sSet] gReedy[\Omega^{op}, sSet]_{gReedy} is a combinatorial model category. This implies that it has a good theory of left Bousfield localization at sets of morphisms.

Proposition

The cofibrations in [Ω op,sSet] gReedy[\Omega^{op}, sSet]_{gReedy} are precisely the simplicial-degree-wise normal monomorphisms of dendroidal sets (see here).

This is (Cisinski-Moerdijk, cor. 4.3).

Proof

The generating inclusions in prop. 5 are the boundary inclusions of representables in the product site Δ×Ω\Delta \times \Omega, regarded as a Cisinski-generalized Reedy category. By the discussion there, these generate the normal monomorphisms on Δ×Ω\Delta \times \Omega. But since Δ\Delta contains no non-trivial automorphisms, this are just the degreewise dendroidal normal monomorphisms.

Proposition

The generalized Reedy model structure [Ω op,sSet] gReedy[\Omega^{op}, sSet]_{gReedy} equipped with the sSet-enrichment from remark 2 is an enriched model category over the standard model structure on simplicial sets – a simplicial model category.

Proof

It is sufficient to check the pushout-product axiom for the tensoring operation. So for a:STa : S \to T a monomorphism of simplicial sets and f:XY f : X \to Y a degreewise normal monomorphisms in [Ω op,sSet][\Omega^{op}, sSet], we need to check, by prop 6, that the canonical morphism

(SY) (SX)(TX)TY (S \cdot Y) \coprod_{(S \cdot X)} (T \cdot X) \to T \cdot Y

is a simplicial-degreewise normal monomorphism, which is a weak equivalence if either of aa or ff is. Since this coproduct is computed objectwise, this morphism is over [n]Δ[n] \in \Delta the pushout of simplicial sets

(S nY n) (S nX n)(T nX n)T nY n, (S_n \cdot Y_n) \coprod_{(S_n \cdot X_n)} (T_n \cdot X_n) \to T_n \cdot Y_n \,,

where now the tensoring is that of dendroidal sets over sets, which is given by coproduct of dendroidal sets, S nY n= sS nY nS_n \cdot Y_n = \coprod_{s \in S_n} Y_n. It is clear that this is a monomorphism.

Moreover, the image of this morphism contains the image of T nf nT_n \cdot f_n, which for each summand tT nt \in T_n is the image of ff. Therefore the dendrices not in this image are also summand-wise not in the image of ff, hence have trivial stabilizer groups, by the assumption that ff is a normal monomorphism.

Finally, to see that the above morphism out of the pushout is a weak equivalence if either of aa or ff is, use that in [Ω op,sSet] fReedy[\Omega^{op}, sSet]_{fReedy} the weak equivalences are tree-wise those of simplicial sets. The statement then follows by sSet QuillensSet_{Quillen} being a monoidal model category with respect to its cartesian monoidal category structure.

Some of these properties are inherited by the actual model structure for dendroidal complete Segal spaces

Corollary

The model structures [Ω op,sSet] Segal[\Omega^{op}, sSet]_{Segal} and [Ω op,sSet] cSegal[\Omega^{op}, sSet]_{cSegal}

Proof

Since cofibrations and simplicial enrichment are preserved by left Bousfield localization, this follows from the analogous statements for [Ω op,sSet] gReedy[\Omega^{op}, sSet]_{gReedy}.

Fibrant objects

Remark

An object X[Ω op,sSet] gReedyX \in [\Omega^{op}, sSet]_{gReedy} is fibrant, precisely if for every tree TΩT \in \Omega, the morphism

X (Ω[T]Ω[T]):X(T)X Ω[T] X^{(\partial \Omega[T] \hookrightarrow \Omega[T])} : X(T) \to X^{\partial \Omega[T]}

is a Kan fibration.

Proof

By prop. 4, using Y=*Y = *.

Proposition

Let X[Ω op,sSet] gReedyX \in [\Omega^{op}, sSet]_{gReedy} be fibrant. Then the following conditions are equivalent

  • XdsSetX \in dsSet is a dendroidal Segal space, hence fibrant in [Ω op,sSet] Segal[\Omega^{op}, sSet]_{Segal};

  • for every spine inclusion Sp[T]Ω[T]Sp[T] \hookrightarrow \Omega[T], the induced morphism X Ω[T]X Sp[T]X^{\Omega[T]} \to X^{Sp[T]} is an acyclic Kan fibration;

  • for every inner horn inclusion Λ e[T]Ω[T]\Lambda^e[T] \hookrightarrow \Omega[T], the induced morphism X Ω[T]X Λ e[T]X^{\Omega[T]} \to X^{\Lambda^e[T]} is an acyclic Kan fibration.

This appears as (Cisinski-Moerdijk, cor. 5.6).

Proof

By prop. 1 Segal objects are equivalently spine-local and horn-local. By prop. 6 both the spine and the horn inclusion are morphisms between cofibrant objects in [Ω op,sSet] gReedy[\Omega^{op}, sSet]_{gReedy}. By the general properties of left Bousfield localization and using that [Ω op,sSet] gReedy[\Omega^{op}, sSet]_{gReedy} is a simplicial model category by prop. 7, it follows that a fibrant object X[Ω op,sSet] gReedyX \in [\Omega^{op}, sSet]_{gReedy} is local with respect to the spine / horn inclusions precisely if powering these into this object, remark 2, is a weak equivalence of simplicial sets. Since moreover the horn and spine inclusions are cofibrations, by prop. 6, this will necessarily be an acyclic Kan fibration (by the dual of the pushout-product axiom in a simplicial model category).

Let S={AB}S = \{A \to B\} be a set of generating acyclic cofibrations for the model structure on dendroidal sets, dSet CMdSet_{CM}, chosen such that all domains and codomains are normal, hence cofibrant.

Proposition

An object X[Ω op,sSet] cSegalX \in [\Omega^{op}, sSet]_{cSegal} is fibrant precisely if

  1. it is fibrant in [Ω op,sSet] Segal[\Omega^{op}, sSet]_{Segal};

  2. it has the right lifting property against the set

    {(AB)¯(Δ[n]Δ[n])} (AB)S,n. \{ (A \to B) \bar \cdot (\partial \Delta[n] \to \Delta[n]) \}_{(A \to B) \in S, n \in \mathbb{N}} \,.

(Cis-Moer, cor. 6.5)

Proof

By prop. 3 and the basic nature of left Bousfield localization.

Weak equivalences

Proposition

A morphism f:XYf : X \to Y between dendroidal Segal spaces is a weak equivalence in [Ω op,sSet] Segal[\Omega^{op}, sSet]_{Segal}, and hence in [Ω op,sSet] cSegal[\Omega^{op}, sSet]_{cSegal} precisely if its components on the trees η\eta and C nC_n for all nn, def. 2, are weak homotopy equivalences of simplicial sets.

This appears as (Cisinski-Moerdijk, prop. 5.7).

Proof

By general properties of left Bousfield localization, a morphism between local objects is a weak equivalence precisely if it is so already in the unlocalized model structure [Ω op,sSet] genReedy[\Omega^{op}, sSet]_{genReedy}. There the weak equivalences are the morphisms that are so over every tree. But by prop. 8 these are already implied by weak equivalences over the spines. These are, finally, colimits which happen to be homotopy colimits of η\eta and of corollas, and hence it suffices to have weak equivalences over these components in order to have them over all components.

Proposition

A morphism f:XYf : X \to Y of dendroidal Segal spaces is a weak equivalence in [Ω op,sSet] Segal[\Omega^{op}, sSet]_{Segal} precisely if it is

  1. fully faithful, def. 4;

  2. essentially surjective in that f(η):X(η)Y(η)f(\eta) : X(\eta) \to Y(\eta) is a weak equivalence of simplicial sets.

(See also equivalence of categories.)

This appears as (Cisinski-Moerdijk, cor. 5.10).

Proof

Being essentially surjective is equivalent to f(η)f(\eta) being an equivalence. By prop. 10 it only remains to check that in this situation ff being fully faithful is equivalent to f(C n)f(C_n) being an equivalence, for all nn.

By remark 1, of f(C n):X(C n)Y(C n)f(C_n) : X(C_n) \to Y(C_n) is a weak equivalence for all nn then ff is fully faithful, since weak equivalence are preserved by homotopy pullback.

For the converse, consider for each nn the inclusion of all input and output colors

(x 1,,x n;x)*X(η) n+1 \coprod_{(x_1, \cdots, x_n; x)} * \to X(\eta)^{n+1}

and similarly for YY. Since this evidently hits all connected components of X(η) n+1X(\eta)^{n+1}, it is an effective epimorphism in an (∞,1)-category in ∞Grpd. These are stable under homotopy pullback, and so also

(x 1,,x n;x)X(x 1,,x n;x)X(C n) \coprod_{(x_1, \cdots, x_n; x)} X(x_1, \cdots, x_n; x) \to X(C_n)

is an effective epimorphism, and similarly for YY. If now ff is fully faithful, then by the definition of effective epimorphism in an (∞,1)-category, this exhibits f(C n)f(C_n) as the homotopy colimit of a diagram of equivalences. Hence f(C n)f(C_n) is itself a weak equivalence.

Relation to other model structures

We discuss the relation to various other model structures for operads. For an overview see table - models for (infinity,1)-operads.

To complete Segal spaces

Write ηΩdSetdsSet\eta \in \Omega \hookrightarrow dSet \hookrightarrow dsSet for the tree with a single edge and no non-trivial vertex.

Then slice category of dsSetdsSet over η\eta is evidently equivalent to that of bisimplicial sets

ssSetdsSet /ηdsSet. ssSet \simeq dsSet_{/\eta} \hookrightarrow dsSet \,.

By restriction along this inclusion, the above model structure reproduces the model structure for complete Segal spaces.

To dendroidal sets / quasi-operads

The model structure for dendroidal complete Segal spaces is Quillen equivalent to the model structure on dendroidal sets, whose fibrant objects are the “quasi-operads” (the operadic generalization of quasi-categories).

We discuss in fact two Quillen equivalences, with right adjoints going in both directions:

  1. From quasi-operads to dendroidal complete Segal spaces

    (|| JSing J):dSet CMSing j|| J[Ω op,sSet] cSegal. ({|-|_J} \dashv Sing_J) : dSet_{CM} \stackrel{\overset{{|-|_J}}{\leftarrow}}{\underset{{Sing_j}}{\to}} [\Omega^{op}, sSet]_{cSegal} \,.
  2. From dendroidal complete Segal spaces to quasi-operads

    (i)[Ω op,sSet] cSegalidSet (i \dashv ) [\Omega^{op}, sSet]_{cSegal} \stackrel{\overset{i}{\leftarrow}}{\underset{}{\to}} dSet
Quasi-operads to dendroidal complete Segal spaces

Recall from complete Segal space the basic example Categories as complete Segal spaces which shows how an ordinary small category CC is regarded as a complete Segal space Sing J(C)Sing_J(C) by setting

Sing J(C):nN(Core(C Δ[n])). Sing_J(C) : n \mapsto N(Core(C^{\Delta[n]})) \,.

Recall also that this and its generalization to Complete Segal spaces of quasi-categories, amounts to simply forming a double-nerve with respect to the invertible interval object. We consider here the operadic generalization of this construction.

Definition

Write

Δ J:ΔsSet \Delta_J : \Delta \to sSet

for the cosimplicial simplicial set that in degree nn is the nerve of the free groupoid on Δ[n]\Delta[n]

Δ J(n):=N({0n}). \Delta_J(n) := N ( \{0 \stackrel{\simeq}{\to} \cdots \stackrel{\simeq}{\to} n \} ) \,.

We use the same symbol for the further prolongation to a cosimplicial dendroidal set

Δ J:ΔsSeti !dSet. \Delta_J : \Delta \to sSet \stackrel{i_!}{\hookrightarrow} dSet \,.

Moreover, we use the same symbol also for

Δ J:Δ×ΩdSet \Delta_J : \Delta \times \Omega \to dSet
Δ J:([n],T)Δ J[n] BVΩ[T] \Delta_J : ([n], T) \mapsto \Delta_J[n] \otimes_{BV} \Omega[T]

(where BV\otimes_{BV} is the Boardman-Vogt tensor product on dendroidal sets).

The induced nerve and realization adjunction we denote

(|| JSing J):dSet Sing j|| J[Ω op,sSet] . ({|-|_J} \dashv Sing_J) : dSet_{} \stackrel{\overset{{|-|_J}}{\leftarrow}}{\underset{{Sing_j}}{\to}} [\Omega^{op}, sSet]_{} \,.

So for XdSetX \in dSet

Sing J(X):(T,[n])Hom dSet(Δ J[n]Ω[T],X). Sing_J(X) : (T, [n]) \mapsto Hom_{dSet}(\Delta_J[n]\otimes \Omega[T], X) \,.

This appears as (Cis-Moer, 6.10).

Example

For

CCatOperadN ddSet C \in Cat \stackrel{}{\hookrightarrow} Operad \stackrel{N_d}{\hookrightarrow} dSet

a small category, we have

Sing J(C):i !(nN(Core(C Δ[n]))). Sing_J(C) : i_!( n \mapsto N(Core(C^{\Delta[n]})) ) \,.
Proposition

The nerve and realization adjunction, def. 5 constitutes a Quillen equivalence to the model structure on dendroidal sets.

(|| JSing J):dSet CMSing j|| J[Ω op,sSet] cSegal. ({|-|_J} \dashv Sing_J) : dSet_{CM} \stackrel{\overset{{|-|_J}}{\leftarrow}}{\underset{{Sing_j}}{\to}} [\Omega^{op}, sSet]_{cSegal} \,.

This appears as (Cis-Moer, prop. 6.11).

Proof

First we show that || J{\vert -\vert_J} is a left Quillen functor. Since dSet CMdSet_{CM} is a monoidal model category, it follows from the pushout-product axiom in (dSet CM, BV)(dSet_{CM}, \otimes_{BV}) that || J{\vert -\vert_J} sends the generating (acyclic) cofibrations of [Ω op,sSet] Reedy[\Omega^{op}, sSet]_{Reedy} from prop. 5 to (acyclic) cofibrations in dSet CMdSet_{CM}. Since the cofibrations of [Ω op,sSet] cSegal[\Omega^{op}, sSet]_{cSegal} are the same as those of [Ω op,sSet] Reedy[\Omega^{op}, sSet]_{Reedy}, it is sufficient to see that || J{\vert -\vert_J} sends the morphisms that define the localization, def. 1, to weak equivalences in dSet CMdSet_{CM}. But since these moprhisms are in the image of the inclusion dSet[Ω op,sSet]dSet \hookrightarrow [\Omega^{op}, sSet], the functor indeed sends them to themselves, and they are indeed weak equibalences in dSet CMdSet_{CM} (since all inner anodyne morphisms are – this gives that Λ e[T]Ω[T]\Lambda^e[T] \to \Omega[T] is a weak equivalence – and all equivalences in the canonical model structure on operads are – this gives that Ω[T] BVJΩ[T]\Omega[T] \otimes_{BV} J \to \Omega[T] is).

So far this shows that || J{\vert - \vert_J} is left Quillen. To see that it is a Quillen equivalence, use that its composition with the left Quillen functor i:dSet CM[Ω op,sSet] gReedyi : dSet_{CM} \to [\Omega^{op}, sSet]_{gReedy} discussed in the companion section is evidently a Quillen equivalence.

(…)

Observation

If we write (as here), for AdSetA \in dSet normal and XdSetX \in dSet fibrant

k(A,X):=Core(i *[A,X] ) k(A,X) := Core(i^* [A,X]_{\otimes})

for the maximax Kan complex inside the internal hom of (dSet, BV)(dSet, \otimes_{BV}), then, still for XX fibrant, we have

Sing J(X):(T,n)k(Ω[T],X) n. Sing_J(X) : (T, n) \mapsto k(\Omega[T], X)_n \,.
Complete dendroidal Segal spaces to quasi-operads

Write

i:dSet=[Ω op,Set][Ω op,sSet] i : dSet = [\Omega^{op}, Set] \hookrightarrow [\Omega^{op}, sSet]

for the evident full subcategory inclusion of dendroidal sets into dendroidal simplicial sets induced by regarding a set as a discrete object in simplicial sets.

Theorem

The inclusion

i:dSet CM[Ω op,sSet] cSegal i : dSet_{CM} \hookrightarrow [\Omega^{op}, sSet]_{cSegal}

is the left adjoint of a Quillen equivalence from the model structure on dendroidal sets to the model structure for dendroidal complete Segal spaces, def. 1.

This is (Cisinski-Moerdijk, prop. 4.8, theorem 6.6).

The following proof proceeds by passing through another Bousfield localization of a global model structure on dendroidal simplicial sets.

Definition

Let [Δ op,dSet CM] Reedy[\Delta^{op}, dSet_{CM}]_{Reedy} be the Reedy model structure on simplicial objects in the model structure on dendroidal sets.

Write

[Δ op,dSet CM] LocConstidid[Δ op,dSet CM] Reedy [\Delta^{op}, dSet_{CM}]_{LocConst} \stackrel{\overset{id}{\leftarrow}}{\underset{id}{\to}} [\Delta^{op}, dSet_{CM}]_{Reedy}

for its left Bousfield localization at the set

S={Δ[n]Ω[T]Ω[T]} nΔ,TΩ. S = \{\Delta[n] \cdot \Omega[T] \to \Omega[T]\}_{n \in \Delta, T \in \Omega} \,.

We call this the locally constant model structure on simplicial dendroidal sets.

(Cis-Moer, def. 4.6)

Proposition

The functors

(constev 0):[Δ op,dSet CM] LocConstev 0constdSet CM (const \dashv ev_0) : [\Delta^{op}, dSet_{CM}]_{LocConst} \stackrel{\overset{const}{\leftarrow}}{\underset{ev_0}{\to}} dSet_{CM}

constitute a Quillen equivalence.

(Cis-Moer, prop. 4.8)

Proof

The set {Ω[T]} TΩ\{\Omega[T]\}_{T \in \Omega} is a set of generators, in that a morphism f:XYf : X \to Y in dSet CMdSet_{CM} is a weak equivalence precisely if under the derived hom space functor Hom(Ω[T],f)\mathbb{R}Hom(\Omega[T], f) is a weak equivalence, for all TT. Therefore the localization in def. 6 is of the general kind discussed at simplicial model category in the section Simplicial Quillen equivalent models. The above statement is thus a special case of the general theorem discussed there.

Proposition

The fibrant objects in [Δ op,dSet CM] LocConst[\Delta^{op}, dSet_{CM}]_{LocConst} are precisely

  • the Reedy fibrant simplicial dendroidal sets XX,

  • such that for every nn \in \mathbb{N} the morphism X nX 0X_n \to X_0 is a weak equivalence in the model structure on dendroidal sets;

(Cis-Moer, 4.7 ii)+iii)).

Proof

The proof is again a special case of the general discussion at Simplicial Quillen equivalent models. Here is a self-contained proof, for completeness.

By standard facts of left Bousfield localization a simplicial dendroidal set is fibrant in the locally constant model structure, def. 6, precisely if it is fibrant in [Δ op,[Ω op,Set] CM] Reedy[\Delta^{op}, [\Omega^{op}, Set]_{CM}]_{Reedy} and moreover the derived hom-space functor Hom [Δ op,dSet CM] Reedy((Δ[n]Ω[T]Ω[T]),X)\mathbb{R}Hom_{[\Delta^{op},dSet_{CM}]_{Reedy}}((\Delta[n]\cdot \Omega[T] \to \Omega[T]), X) is a weak equivalence for all nn \in \mathbb{N}.

We compute this derived hom space now in a maybe slightly non-obvious way, in order to get the result in a form that we can compare to the derived hom in dSet CMdSet_{CM}. First of all, since the derived hom space only depends on the weak equivalences, we may compute it working with the projective model structure on functors [Δ op,[Ω op,Set] CM] proj[\Delta^{op}, [\Omega^{op}, Set]_{CM}]_{proj}. Here in turn we use as framing X^\hat X of that:

(constXX^)[Δ op,[Δ op,[Ω op,Set] CM] proj] Reedy. (const X \stackrel{\simeq}{\to}\hat X ) \in [\Delta^{op}, [\Delta^{op}, [\Omega^{op}, Set]_{CM}]_{proj}]_{Reedy} \,.

Since Δ[n]Ω[T]\Delta[n]\cdot \Omega[T] is cofibrant in [Δ op,[Ω op,Set] CM] proj[\Delta^{op}, [\Omega^{op}, Set]_{CM}]_{proj} (because Δ[n]\Delta[n] is representable and Ω[T]dSet\Omega[T] \in dSet is normal), also constΔ[n]Ω[T]const \Delta[n]\cdot \Omega[T] is cofibrant in [Δ op,[Δ op,dSet CM] proj] Reedy[\Delta^{op}, [\Delta^{op}, dSet_{CM}]_{proj}]_{Reedy}, and so we have that

Hom(Δ[n]Ω[T],X)([n]Hom [Δ op,dSet](const(Δ[n]Ω[T]),X^)). \mathbb{R}Hom(\Delta[n] \cdot\Omega[T], X) \simeq \left( [n] \mapsto Hom_{[\Delta^{op}, dSet]}(const(\Delta[n]\cdot \Omega[T]), \hat X) \right) \,.

We claim now that such a resolution X^\hat X is given by (using the notation for core simplicial enrichement of dSetdSet here)

X^:[n]X (Δ[n]). \hat X : [n] \mapsto X^{(\Delta[n])} \,.

To see that this is indeed Reedy fibrant, notice that this is so precisely if for all kk \in \mathbb{N} the morphism

X (Δ[k])X (Δ[k]) X^{(\Delta[k])} \to X^{(\partial \Delta[k])}

is fibrant in [Δ op,[Ω op,Set] CM] proj[\Delta^{op}, [\Omega^{op}, Set]_{CM}]_{proj}, which is the case precisely if for all nΔn \in \Delta the morphism

X k (Δ[n])X k (Δ[n]) X^{(\Delta[n])}_k \to X^{(\partial \Delta[n])}_k

is a fibration in dSet CMdSet_{CM}. But using that the Reedy fibrant XX is in particular projectively fibrant (see Reedy model structure), hence that X kdSet CMX_k \in dSet_{CM} is fibrant (is a quasi-operad) for all kk, this is indeed the case, by discussion here at model structure on dendroidal sets.

So finally we find that we may compute the derived hom as

Hom [Δ op,dSet CM] proj(Δ[n]×Ω[T],X) =([k]Hom [Δ op,dSet](Δ[n]×Ω[T],X (Δ[k]))) =([k]Hom dSet CM(Ω[T],X n (Δ[k]))). \begin{aligned} \mathbb{R}Hom_{[\Delta^{op}, dSet_{CM}]_{proj}}(\Delta[n]\times \Omega[T], X) & = \left( [k] \mapsto Hom_{[\Delta^{op}, dSet]}( \Delta[n] \times \Omega[T], X^{(\Delta[k])} ) \right) \\ & = \left( [k] \mapsto Hom_{dSet_{CM}}(\Omega[T], X^{(\Delta[k])}_n) \right) \end{aligned} \,.

The right hand here is now manifestly the derived hom in dSet CMdSet_{CM}, from Ω[T]\Omega[T] to X nX_n, computed itself by a framing resolution.

Therefore we have found that XX is fibrant in the locally constant model structure, def. 6, precisely if for all nn and TT the morphisms

Hom dSet CM(Ω[T],X n)Hom dSet CM(Ω[T],X 0) \mathbb{R}Hom_{dSet_{CM}}(\Omega[T], X_n) \to \mathbb{R}Hom_{dSet_{CM}}(\Omega[T], X_0)

are weak equivalences. Since the {Ω[T]} TΩ\{\Omega[T]\}_{T \in \Omega} form a set of generators, this is the case precisely if X nX 0X_n \to X_0 is already a weak equivalence in dSet CMdSet_{CM}.

Now for the equivalence to the second item.

By Joyal-Tierney calculus the morphisms in question are of the form

(Λ k[n]Δ[n])ׯ(Ω[T]Ω[T]). (\Lambda^k[n] \to \Delta[n]) \bar \times (\partial \Omega[T] \to \Omega[T]) \,.

Since the horn inclusions generate the acyclic monomorphisms, a morphism X*X \to * that has right lifting against this set also has right lifting against

(Δ[0]Δ[n])ׯ(Ω[T]Ω[T]). (\Delta[0] \to \Delta[n]) \bar \times (\partial \Omega[T] \to \Omega[T]) \,.

This in turn means that X nX 0X_n \to X_0 has the right lifting property against the tree boundary inclusions. Since these are the generating cofibrations in the model structure on dendroidal sets, this implies that X nX 0X_n \to X_0 is an equivalence.

For the converse, it is sufficient to see that all the morphisms in the localizing set are acyclic cofibrations in the locally constant model structure. This follows with the discussion here at model structure on dendroidal sets.

Proposition

The fibrant objects in [Δ op,dSet CM] LocConst[\Delta^{op}, dSet_{CM}]_{LocConst} are also precisely

  • the Reedy fibrant simplicial dendroidal sets XX,

  • such that the morphism X*X \to * has the right lifting property against the set of pushout product morphisms

    {(Λ k[n]Δ[T])¯(Ω[T]Ω[T])} TΩ,n1,0kn={Λ k[n]Ω[T]Δ[n]Ω[T]Δ[n]Ω[T]} TΩ,n1,0kn. \{ (\Lambda^k[n] \to \Delta[T]) \bar \cdot (\partial \Omega[T] \to \Omega[T]) \}_{T \in \Omega, n \geq 1 , 0 \leq k \leq n} = \{ \Lambda^k[n] \cdot \Omega[T] \cup \Delta[n] \cdot \partial \Omega[T] \to \Delta[n]\cdot\Omega[T] \}_{T \in \Omega, n \geq 1 , 0 \leq k \leq n} \,.

(Cis-Moer, 4.7 i)+iii)).

Proof

Since the simplicial horn inclusions generate all acyclic cofibrations in sSet QillensSet_{Qillen}, it follows that a (Reedy fibrant) object XX which has right lifting against {(Λ k[n]Δ[n])¯(Ω[T]Ω[T])}\{(\Lambda^k[n] \to \Delta[n]) \bar \cdot (\partial \Omega[T] \to \Omega[T])\} also has right lifting against {(Δ[0]Δ[n])¯(Ω[T]Ω[T])}\{(\Delta[0] \to \Delta[n]) \bar \cdot (\partial \Omega[T] \to \Omega[T]) \}. This means that X 0X nX_0 \to X_n is an acyclic fibration for all nn, in particular a weak equivalence, hence XX is fibrant in the locally constant structure by 15.

Conversely, one finds with … and … that the morphisms in the above set are acyclic cofibrations in the locally constant model structure, hence if an object is locally constant fibrant, it lifts against these.

Proposition

Under the canonical identification of categories

[Δ op,dSet][Ω op,sSet] [\Delta^{op}, dSet] \simeq [\Omega^{op}, sSet]

the two model structures [Δ op,dSet CM] LocConst[\Delta^{op}, dSet_{CM}]_{LocConst}, def. 6 and [Ω op,sSet] cSegal[\Omega^{op}, sSet]_{cSegal}, def. 1, coincide.

Proof

By a standard fact (see at model category the section Redundancy of the axioms) it is sufficient to show that the cofibrations and the fibrant objects coincide.

By prop. 5 we know the generating cofibrations of [Ω op,sSet] cSegal[\Omega^{op}, sSet]_{cSegal}. By the same kind of argument we find the cofibrations of [Δ op,dSet] Reedy[\Delta^{op}, dSet]_{Reedy}, and hence of [Δ op,dSet] LocConst[\Delta^{op}, dSet]_{LocConst}:

by definition of Reedy model structure, a morphism f:XYf : X \to Y here is an acyclic fibration if for all nΔn \in \Delta the morphism

X Δ[n]X Δ[n]× Y Δ[n]Y Δ[n] X^{\Delta[n]} \to X^{\partial \Delta[n]} \times_{Y^{\partial \Delta[n]}} Y^{\Delta[n]}

is an acyclic fibration in dSet CMdSet_{CM}. Since dSet CMdSet_{CM} has generating cofibrations given by the set of tree boundary inclusions {Ω[T]Ω[T]} TΩ\{\partial \Omega[T] \hookrightarrow \Omega[T]\}_{T \in \Omega}, one finds as in the proof of prop. 5 that f:XYf : X \to Y is an acyclic fibration precisely if it has the right lifting property against the morphisms in the set

{Δ[n]Ω[T]Δ[n]Ω[T]Δ[n]Ω[T]} nΔ,TΩ. \{ \partial \Delta[n] \cdot \Omega[T] \cup \Delta[n] \cdot \partial \Omega[T] \to \Delta[n] \cdot \Omega[T] \}_{n \in \Delta, T \in \Omega} \,.

Therefore the cofibrations in the two model structures do coincide.

(Notice that a similar statement holds for the acyclic cofibrations, only that the generating set of acyclic cofibrations in dSet CMdSet_{CM} is, while known to exist, not known explicitly.)

Next, to see that the fibrant objects also coincide, write again S={AB}S = \{A \to B\} for a choice of set of generating acyclic cofibrations for dSet CMdSet_{CM} between normal dendroidal sets.

By prop. 16 the fibrant objects of [Δ op,dSet CM] LocConst[\Delta^{op}, dSet_{CM}]_{LocConst} are those that

  1. are Reedy fibrant over Δ op\Delta^{op}, meaning that they have the right lifting property against

    {(AB)¯(Δ[n]Δ[n])} (AB)S,n; \{ (A \to B) \bar \cdot (\partial \Delta[n] \to \Delta[n]) \}_{(A \to B) \in S, n \in \mathbb{N}} \,;
  2. are local, meaning, by prop. 16, that they have the right lifting property against

    {(Λ k[n]Δ[n])¯(Ω[T]Ω[T])}. \{ (\Lambda^k[n] \to \Delta[n]) \bar \cdot (\partial \Omega[T] \to \Omega[T]) \} \,.

On the other hand, the fibrant objects in [Ω op,sSet] cSegal[\Omega^{op}, sSet]_{cSegal} are those

  1. that are Reedy fibrant over Ω op\Omega^{op}, meaning that they have the right lifting property against

    {(Λ k[n]Δ[n])¯(Ω[T]Ω[T])} n,0kn,TΩ, \{ (\Lambda^k[n] \to \Delta[n]) \bar \cdot (\partial \Omega[T] \to \Omega[T]) \}_{n \in \mathbb{N}, 0 \leq k \leq n, T \in \Omega} \,,
  2. are Segal local, meaning by prop. 1 that they have right lifting against

    {(Δ[n]Δ[n])¯(Λ e[T]Ω[T])} \{ (\partial \Delta[n] \to \Delta[n]) \bar \cdot ( \Lambda^e [T] \to \Omega[T] ) \}
  3. are complete Segal local, meaning by prop. 9 that they have right lifting property against

    {(AB)¯(Δ[n]Δ[n])} (AB)S,n. \{ (A \to B) \bar \cdot (\partial \Delta[n] \to \Delta[n]) \}_{(A \to B) \in S, n \in \mathbb{N}} \,.

The union of the three respective sets coincides in both cases.

Proof

of theorem 1

Combining prop. 14 and prop. 17 we have a total Quillen equivalence

const:dSet CM[Δ op,dSet CM] LocConst[Ω op,sSet] cSegal. const : dSet_{CM} \to [\Delta^{op}, dSet_{CM}]_{LocConst} \simeq [\Omega^{op}, sSet]_{cSegal} \,.

To Segal operads

(…) model structure for Segal operads

References

The model structure for dendroidal complete Segal spaces was introduced in

Revised on April 18, 2012 17:02:47 by Urs Schreiber (89.204.137.12)