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model structure for Segal operads

Context

Model category theory

model category

Definitions

  • category with weak equivalences

  • weak factorization system

  • homotopy

  • small object argument

  • resolution

  • Morphisms

    • Quillen adjunction

    • Universal constructions

      • homotopy Kan extension

      • homotopy limit/homotopy colimit

      • Bousfield-Kan map

      • Refinements

        • monoidal model category

        • enriched model category

        • simplicial model category

        • cofibrantly generated model category

        • algebraic model category

        • compactly generated model category

        • proper model category

        • cartesian closed model category, locally cartesian closed model category

        • stable model category

        • Producing new model structures

          • on functor categories (global)

          • on overcategories

          • Bousfield localization

          • transferred model structure

          • Grothendieck construction for model categories

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

            • (∞,1)-category

            • simplicial localization

            • (∞,1)-categorical hom-space

            • presentable (∞,1)-category

            • Model structures

              • Cisinski model structure
              • for \infty-groupoids

                for ∞-groupoids

                • on topological spaces

                  • Strom model structure?
                • Thomason model structure

                • model structure on presheaves over a test category

                • on simplicial sets, on semi-simplicial sets

                • model structure on simplicial groupoids

                • on cubical sets

                • on strict ∞-groupoids, on groupoids

                • on chain complexes/model structure on cosimplicial abelian groups

                  related by the Dold-Kan correspondence

                • model structure on cosimplicial simplicial sets

                • for nn-groupoids

                  • for n-groupoids/for n-types

                  • for 1-groupoids

                  • for \infty-groups

                    • model structure on simplicial groups

                    • model structure on reduced simplicial sets

                    • for \infty-algebras

                      general

                      • on monoids

                      • on simplicial T-algebras, on homotopy T-algebras

                      • on algebas over a monad

                      • on algebras over an operad,

                        on modules over an algebra over an operad

                      • specific

                        • model structure on differential-graded commutative algebras

                        • model structure on differential graded-commutative superalgebras

                        • on dg-algebras over an operad

                        • model structure on dg-modules

                        • for stable/spectrum objects

                          • model structure on spectra

                          • model structure on ring spectra

                          • model structure on presheaves of spectra

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

                            • on categories with weak equivalences

                            • Joyal model for quasi-categories

                            • on sSet-categories

                            • for complete Segal spaces

                            • for Cartesian fibrations

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

                              • on dg-categories
                              • for (,1)(\infty,1)-operads

                                • on operads, for Segal operads

                                  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

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

                                  • for (n,r)-categories as ∞-spaces

                                  • for weak ∞-categories as weak complicial sets

                                  • on cellular sets

                                  • on higher categories in general

                                  • on strict ∞-categories

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

                                    • on homotopical presheaves

                                    • model structure for (2,1)-sheaves/for stacks

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                                      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.