nLab
global model structure on simplicial presheaves

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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                                      Contenta

                                      Idea

                                      The global model structure on simplicial presheaves [C op,sSet Quillen][C^{op}, sSet_{Quillen}] on a small category CC is the global model structure on functors on CC with values in sSet QuillensSet_{Quillen}, the standard model structure on simplicial sets.

                                      It presented the (∞,1)-category of (∞,1)-functors from C opC^{op} to ∞Grpd, hence the (∞,1)-category of (∞,1)-presheaves on CC.

                                      The left Bousfield localizations of [C op,sSet] proj[C^{op}, sSet]_{proj} are, up to Quillen equivalence, precisely the combinatorial model categories.

                                      In particular, if CC carries the structure of a site, then

                                      These localizations present the topological localization and hypercompletion of the (∞,1)-topos of (,1)(\infty,1)-presheaves on CC to the corresponding (∞,1)-topos of (∞,1)-sheaves/∞-stacks on CC.

                                      Definition

                                      In every global model structure on simplicial presheaves on CC the weak equivalences are objectwise those with respect to the standard model structure on simplicial sets.

                                      A morphism f:ABf : A \to B in [C op,SSet][C^{op}, SSet] is, thus, a weak equivalence with respect to a global model structure precisely if for all UObj(C)U \in Obj(C) the morphism

                                      f(U):A(U)B(U) f(U) : A(U) \to B(U)

                                      is a weak equivalence of simplicial set (i.e. a morphism inducing isomorphisms of simplicial homotopy groups).

                                      There are several choices for how to extend this notion of weak equivalences to an entire model category structure. The two common extreme choices are

                                      The other class of morphisms (cofibrations / fibrations) is in each case fixed by the corresponding lifting property.

                                      Properties

                                      References

                                      See also model structure on simplicial presheaves.

                                      The global projective model structure is originally due to

                                      • A. K. Bousfield and D.M. Kan, Homotopy limits completions and localizations, Springer Lecture Notes in Math. 304 (2nd corrected printing), Springer-Verlag, Berlin–Heidelberg–New York (1987).

                                      The fact that the global injective model structure yields a proper simplicial cofibrantly generated model category is originally due to

                                      • Alex HellerHomotopy Theories, no. 383, Memoirs Amer.

                                        Math. Soc., Amer. Math. Soc., 1988.

                                      The fact that the global projective structure yields a proper simplicial cellular model category is due to Hirschhorn-Bousfield-Kan-Quillen

                                      • P. Hirschhorn, Localizations of Model Categories (web)

                                      A quick review of these facts is on the first few pages of

                                      • Benjamin Blander, Local projective model structure on simplicial presheaves (pdf)

                                      Details on the projective global model structure is in

                                      Last revised on September 20, 2013 at 22:57:19. See the history of this page for a list of all contributions to it.