model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
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for rational equivariant $\infty$-groupoids
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for $(\infty,1)$-operads
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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. 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. .
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 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. , is also a weak equivalence in $[\Omega^{op}, sSet]_{gReedy \atop Segal}$.
The structures in def. 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 .
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
Last revised on April 2, 2012 at 15:12:11. See the history of this page for a list of all contributions to it.