nLab model structure for Segal operads

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 $\Omega$ for the tree category, the site for dendroidal sets.

Segal operads

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

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

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 $X \in [\Omega^{op}, sSet]$ is a simplicial dendroidal set such that $X(\eta)$ is a discrete simplicial set (a plain set regarded as a simplicially constant simplicial set). Write

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

for the full subcategory on the Segal pre-operads.

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

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

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

$$SegalOperad \hookrightarrow SegalPreOperad$$

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 $[\Omega^{op}, sSet]_{gReedy}$.

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 $T$ an ∞-groupoid of operations in the operad of the shape of this tree — notably $\infty$-groupoids of $n$-ary operations if the tree is the $n$-corolla, $T = C_n$.

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

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

Proof

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

$$\gamma : \Delta \times \Omega \to S(\Omega)$$

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

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

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

Explicitly, these adjoints are given as follows.

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

$$\array{ X(\eta) &\to& X(T) \\ \downarrow && \downarrow \\ \pi_0 X(\eta) &\to& \gamma_!(X)(T) }$$

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

In words, $\gamma_!(X)$ is obtained from $X$ 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 $f$ in $SegalPreOperad$ is a normal monomorphism precisely if $\gamma^*(f)$ 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 $X$ is called normal if $\emptyset \to X$ is a normal monomorphism.

Definition

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

Definition

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

Definition

Call a morphism in $SegalPreOperad$

• 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 $f : X \to Y$ in $[\Omega^{op}, sSet]$ is a normal monomorphism and $\pi_0 X(\eta) \to \pi_0 Y(\eta)$ is a monomorphism, then $\gamma_!(f)$ is normal in $SegalPreOperad$.

(Cis-Moer, lemma 7.4).

Proposition

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

$$\{ \gamma_!(\partial \Delta[n] \times \Omega[T] \cup \Delta[n] \times \partial \Omega[T]) \to \gamma_! (\Delta[n], \Omega[T]) \}_{n \in \Delta, T \in \Omega, {\vert T\vert} \geq 1} \cup \{ \emptyset \to \eta \}$$

(Cis-Moer, prop 7.5).

Proposition

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

(Cis-Moer, prop 8.2).

Lemma

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

(Cis-Moer, prop 8.12).

Of the model structure itself

Theorem

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

• cofibrantly generated;

• left proper.

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 $[\Omega^{op}, sSet]_{gReedy \atop cSegal}$)

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

• Denis-Charles Cisinski, Ieke Moerdijk, Dendroidal Segal spaces and infinity-operads (arXiv:1010.4956)