on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
symmetric monoidal (∞,1)-category of spectra
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.
Write $\Omega$ for the tree category, the site for dendroidal sets.
Write $\eta$ for the tree with a single edge and no vertices. Write
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).
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
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
of the spine inclusion $(Sp(T) \hookrightarrow T) \in$ dSet into $X$ is an acyclic Kan fibration. Write
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).
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.
The inclusion def. 1 has a left and right adjoint functors
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
for the localization functor, then the inclusion of Segal pre-operads is the precomposition with this functor
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
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.
We discuss morphisms between Segal pre-operads with special properties, which will appear in the model structure.
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.
A morphism in $SegalPreOperad$ is called an acyclic fibration precisely if it has the right lifting property against all normal monomorphisms, def. 2.
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 $[\Omega^{op}m, sSet]_{gReedy \atop cSegal}$.
Call a morphism in $SegalPreOperad$
Theorem 1 below asserts that this is indeed a model category struture whose fibrant objects are the Segal operads.
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 $SegalPreOperad$.
The class of normal monomorphisms in $SegalPreOperad$ is generated (under pushout, transfinite composition and retracts) by the set
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 $[\Omega^{op}, sSet]_{gReedy \atop cSegal}$.
An acyclic fibration in $SegalPreOperad$, def. 3, is also a weak equivalence in $[\Omega^{op}, sSet]_{gReedy \atop Segal}$.
The structures in def. 5 make the category $SegalPreOperad$ a model category which is
This is (Cis-Moer, theorem 8.13).
The existence of the cofibrantly generated model structure follows with Smith’s theorem: by the discussion there it is sufficient to notice that
the Segal equivalences are an accessibly embedded accessible full subcategory of the arrow category;
the acyclic cofibrations are closed under pushout and retract;
(both of these because these morphisms come from the combinatorial model category $[\Omega^{op}, sSet]_{gReedy \atop cSegal}$)
the morphisms with right lifting against the normal monomorphisms are weak equivalences, by lemma 2.
We discuss the relation to various other model structures for operads. For an overview see table - models for (infinity,1)-operads.
(…) model structure for dendroidal complete Segal spaces