nLab
model structure for Segal operads

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Higher algebra

Contents

Idea

The model structure for Segal operads is a presentation of the (∞,1)-category of (∞,1)-operads regarding these as ∞Grpd-enriched operads.

It is the operadic analog of the model structure for Segal categories: its fibrant objects are operadic analogs of Segal categories.

Definition

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

Segal operads

Write η\eta for the tree with a single edge and no vertices. Write

sdSet:=[Ω op,sSet] sdSet := [\Omega^{op}, sSet]

for the category of simplicial presheaves on the tree category – simplicial dendroidal sets or dendroidal simplicial sets (see model structure for complete dendroidal Segal spaces for more on this).

Definition

A Segal pre-operad X[Ω op,sSet]X \in [\Omega^{op}, sSet] is a simplicial dendroidal set such that X(η)X(\eta) is a discrete simplicial set (a plain set regarded as a simplicially constant simplicial set). Write

SegalPreOperad[Ω op,sSet] SegalPreOperad \hookrightarrow [\Omega^{op}, sSet]

for the full subcategory on the Segal pre-operads.

A Segal operad is a Segal pre-operad such that for every tree TΩT \in \Omega the powering

X Ω[T]X Sp(T)sSet X^{\Omega[T]} \to X^{Sp(T)} \in sSet

of the spine inclusion (Sp(T)T)(Sp(T) \hookrightarrow T) \in dSet into XX is an acyclic Kan fibration. Write

SegalOperadSegalPreOperad SegalOperad \hookrightarrow SegalPreOperad

for the full subcategory on the Segal operads.

A Reedy-fibrant Segal operad is a Segal operad which is moreover fibrant in the generalized Reedy model structure [Ω op,sSet] gReedy[\Omega^{op}, sSet]_{gReedy}.

This is (Cisinski-Moerdijk, def. 7.1, def. 8.1).

Remark

The definition of Segal pre-operads encodes a set of colors of an operad, together with for each tree TT an ∞-groupoid of operations in the operad of the shape of this tree — notably \infty-groupoids of nn-ary operations if the tree is the nn-corolla, T=C nT = C_n.

The condition on Segal operads encodes the existence of composition of these operad operations by ∞-anafunctors. See the discussion at Segal category for more on this.

The Reedy fibrancy condition is mostly a technical convenience.

Obervation

The inclusion def. has a left and right adjoint functors

sdSetγ *γ *γ !SegalPreOperad. sdSet \stackrel{\overset{\gamma_!}{\to}}{\stackrel{\overset{\gamma^*}{\leftarrow}}{\underset{\gamma_*}{\to}}} SegalPreOperad \,.
Proof

One way to see the existence of the adjoints is to note that SegalPreOperadSegalPreOperad is a category of presheaves over the site S(Ω)S(\Omega) which is the localization of Ω×Δ\Omega \times \Delta at morphisms of the form (,Id η)(-,Id_\eta), where η\eta is the tree with one edge and no vertex. Write

γ:Δ×ΩS(Ω) \gamma : \Delta \times \Omega \to S(\Omega)

for the localization functor, then the inclusion of Segal pre-operads is the precomposition with this functor

γ *:SegalPreOperad[S(Ω) op,sSet][Ω op,sSet]. \gamma^* : SegalPreOperad \simeq [S(\Omega)^{op}, sSet] \hookrightarrow [\Omega^{op}, sSet] \,.

Therefore the left and right adjoint to γ *\gamma^* are given by left and right Kan extension along γ\gamma.

Explicitly, these adjoints are given as follows.

For X[Ω op,sSet]X \in [\Omega^{op}, sSet], the Segal pre-operad γ !(X)\gamma_!(X) sends a tree TT either to X(T)X(T), if TT is non-linear, hence if it admits no morphism to η\eta, or else to the pushout

X(η) X(T) π 0X(η) γ !(X)(T) \array{ X(\eta) &\to& X(T) \\ \downarrow && \downarrow \\ \pi_0 X(\eta) &\to& \gamma_!(X)(T) }

in sSet, where the top morphism is X(Tη)X(T \to \eta) for the unique morphism to η\eta.

In words, γ !(X)\gamma_!(X) is obtained from XX precisely by contracting the simplicial set of colors to its set of connected components.

Special morphisms

We discuss morphisms between Segal pre-operads with special properties, which will appear in the model structure.

Definition

Say a morphism ff in SegalPreOperadSegalPreOperad is a normal monomorphism precisely if γ *(f)\gamma^*(f) is a normal monomorphism (see generalized Reedy model structure), which in turn is the case if it is simplicial-degreewise a normal morphisms of dendroidal sets (see there for details).

Correspondingly, a Segal pro-operad XX is called normal if X\emptyset \to X is a normal monomorphism.

Definition

A morphism in SegalPreOperadSegalPreOperad is called an acyclic fibration precisely if it has the right lifting property against all normal monomorphisms, def. .

Definition

Say a morphism ff in SegalPreOperadSegalPreOperad is a Segal weak equivalence precisely if γ *(f)\gamma^*(f) is a weak equivalence in the model structure for dendroidal complete Segal spaces [Ω opm,sSet] gReedycSegal[\Omega^{op}m, sSet]_{gReedy \atop cSegal}.

Definition

Call a morphism in SegalPreOperadSegalPreOperad

  • a weak equivalence precisely if it is a Segal weak equivalence, def. ;

  • a cofibration precisely if it is a normal monomorphism, def. .

Theorem below asserts that this is indeed a model category struture whose fibrant objects are the Segal operads.

Properties

Of the various classes of morphisms

Lemma

If f:XYf : X \to Y in [Ω op,sSet][\Omega^{op}, sSet] is a normal monomorphism and π 0X(η)π 0Y(η)\pi_0 X(\eta) \to \pi_0 Y(\eta) is a monomorphism, then γ !(f)\gamma_!(f) is normal in SegalPreOperadSegalPreOperad.

(Cis-Moer, lemma 7.4).

Proposition

The class of normal monomorphisms in SegalPreOperadSegalPreOperad is generated (under pushout, transfinite composition and retracts) by the set

{γ !(Δ[n]×Ω[T]Δ[n]×Ω[T])γ !(Δ[n],Ω[T])} nΔ,TΩ,|T|1{η} \{ \gamma_!(\partial \Delta[n] \times \Omega[T] \cup \Delta[n] \times \partial \Omega[T]) \to \gamma_! (\Delta[n], \Omega[T]) \}_{n \in \Delta, T \in \Omega, {\vert T\vert} \geq 1} \cup \{ \emptyset \to \eta \}

(Cis-Moer, prop 7.5).

Proposition

Let X[Ω op,sSet] gReedySegalX \in [\Omega^{op}, sSet]_{gReedy \atop Segal} be fibrant. Then γ *X\gamma_* X is a Reedy fibrant Segal operad. If XX is moreover fibrant in [Ω op,sSet] gReedycSegal[\Omega^{op}, sSet]_{gReedy \atop cSegal} then the counit γ *γ*XX\gamma^* \gamma* X \to X is a weak equivalence in [Ω op,sSet] gReedycSegal[\Omega^{op}, sSet]_{gReedy \atop cSegal}.

(Cis-Moer, prop 8.2).

Lemma

An acyclic fibration in SegalPreOperadSegalPreOperad, def. , is also a weak equivalence in [Ω op,sSet] gReedySegal[\Omega^{op}, sSet]_{gReedy \atop Segal}.

(Cis-Moer, prop 8.12).

Of the model structure itself

Theorem

The structures in def. make the category SegalPreOperadSegalPreOperad a model category which is

This is (Cis-Moer, theorem 8.13).

Proof

The existence of the cofibrantly generated model structure follows with Smith’s theorem: by the discussion there it is sufficient to notice that

  1. the Segal equivalences are an accessibly embedded accessible full subcategory of the arrow category;

  2. the acyclic cofibrations are closed under pushout and retract;

    (both of these because these morphisms come from the combinatorial model category [Ω op,sSet] gReedycSegal[\Omega^{op}, sSet]_{gReedy \atop cSegal})

  3. the morphisms with right lifting against the normal monomorphisms are weak equivalences, by lemma .

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 dendroidal complete Segal spaces

(…) model structure for dendroidal complete Segal spaces

References

Last revised on April 2, 2012 at 15:12:11. See the history of this page for a list of all contributions to it.