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

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

$$\mathrm{SegalPreOperad}\hookrightarrow [{\Omega }^{\mathrm{op}},\mathrm{sSet}]$$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 $T\in \Omega$ the powering

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

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

$$\mathrm{SegalOperad}\hookrightarrow \mathrm{SegalPreOperad}$$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 $[{\Omega }^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{gReedy}}$.

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 $\mathrm{\infty }$-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. 1 has a left and right adjoint functors

$$\mathrm{sdSet}\stackrel{\stackrel{{\gamma }_{!}}{\to }}{\stackrel{\stackrel{{\gamma }^{*}}{\leftarrow }}{\underset{{\gamma }_{*}}{\to }}}\mathrm{SegalPreOperad}.$$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 $\mathrm{SegalPreOperad}$ is a category of presheaves over the site $S(\Omega )$ which is the localization of $\Omega \times \Delta$ at morphisms of the form $(-,{\mathrm{Id}}_{\eta })$, where $\eta$ is the tree with one edge and no vertex. Write

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

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

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

$$\begin{array}{ccc}X(\eta )& \to & X(T)\\ \downarrow & & \downarrow \\ {\pi }_{0}X(\eta )& \to & {\gamma }_{!}(X)(T)\end{array}$$\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 $\mathrm{SegalPreOperad}$ 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 $\varnothing \to X$ is a normal monomorphism.

Definition

A morphism in $\mathrm{SegalPreOperad}$ is called an acyclic fibration precisely if it has the right lifting property against all normal monomorphisms, def. 2.

Definition

Say a morphism $f$ in $\mathrm{SegalPreOperad}$ is a Segal weak equivalence precisely if ${\gamma }^{*}(f)$ is a weak equivalence in the model structure for dendroidal complete Segal spaces $[{\Omega }^{\mathrm{op}}m,\mathrm{sSet}{]}_{\genfrac{}{}{0ex}{}{\mathrm{gReedy}}{\mathrm{cSegal}}}$.

Definition

Call a morphism in $\mathrm{SegalPreOperad}$

• a weak equivalence precisely if it is a Segal weak equivalence, def. 4;

• a cofibration precisely if it is a normal monomorphism, def. 2.

Theorem 1 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 $[{\Omega }^{\mathrm{op}},\mathrm{sSet}]$ is a normal monomorphism and ${\pi }_{0}X(\eta )\to {\pi }_{0}Y(\eta )$ is a monomorphism, then ${\gamma }_{!}(f)$ is normal in $\mathrm{SegalPreOperad}$.

(Cis-Moer, lemma 7.4).

Proposition

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

$$\{{\gamma }_{!}(\partial \Delta [n]\times \Omega [T]\cup \Delta [n]\times \partial \Omega [T])\to {\gamma }_{!}(\Delta [n],\Omega [T]){\}}_{n\in \Delta ,T\in \Omega ,\mid T\mid \ge 1}\cup \{\varnothing \to \eta \}$$\{ \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 $X\in [{\Omega }^{\mathrm{op}},\mathrm{sSet}{]}_{\genfrac{}{}{0ex}{}{\mathrm{gReedy}}{\mathrm{Segal}}}$ be fibrant. Then ${\gamma }_{*}X$ is a Reedy fibrant Segal operad. If $X$ is moreover fibrant in $[{\Omega }^{\mathrm{op}},\mathrm{sSet}{]}_{\genfrac{}{}{0ex}{}{\mathrm{gReedy}}{\mathrm{cSegal}}}$ then the counit ${\gamma }^{*}\gamma *X\to X$ is a weak equivalence in $[{\Omega }^{\mathrm{op}},\mathrm{sSet}{]}_{\genfrac{}{}{0ex}{}{\mathrm{gReedy}}{\mathrm{cSegal}}}$.

(Cis-Moer, prop 8.2).

Lemma

An acyclic fibration in $\mathrm{SegalPreOperad}$, def. 3, is also a weak equivalence in $[{\Omega }^{\mathrm{op}},\mathrm{sSet}{]}_{\genfrac{}{}{0ex}{}{\mathrm{gReedy}}{\mathrm{Segal}}}$.

(Cis-Moer, prop 8.12).

Of the model structure itself

Theorem

The structures in def. 5 make the category $\mathrm{SegalPreOperad}$ 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 $[{\Omega }^{\mathrm{op}},\mathrm{sSet}{]}_{\genfrac{}{}{0ex}{}{\mathrm{gReedy}}{\mathrm{cSegal}}}$)

3. the morphisms with right lifting against the normal monomorphisms are weak equivalences, by lemma 2.

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)