model structure on cellular sets


Model category theory

model category


  • 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


                      • 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

                                    • Edit this sidebar

                                      Higher category theory

                                      higher category theory

                                      Basic concepts

                                      Basic theorems





                                      Universal constructions

                                      Extra properties and structure

                                      1-categorical presentations



                                      For Θ n\Theta_n the nn-truncation of the Theta-category, an nn-cellular set, hence a presheaf on Θ n\Theta_n, may be viewed as a collection of n-morphisms of an “nn-graph” underlying an (∞,n)-category. The model structure on cellular sets (Ara) models (,n)(\infty,n)-categories this way. This model is referred to as n-quasicategories.


                                      Let Θ n\Theta_n be the Theta category restricted to nn-cells. The model structure on cellular sets is the Cisinski model structure on the category of presheaves PSh(Θ n)PSh(\Theta_n) defined by the following localizer: (…)

                                      In (Ara) the fibrant objects in this model structure are called nn-quasi-categories, see Relation to quasi-categories below.


                                      Relation to quasi-categories

                                      For n=1n = 1 we have Θ 1Δ\Theta_1 \simeq \Delta is the simplex category; and the model structure on 1-cellular sets reproduces the model structure for quasi-categories. (Ara, theorem 5.26)

                                      Relation to Θ n\Theta_n-spaces

                                      The model structure on Θ n\Theta_n-cellular sets is Quillen equivalent to that ∞-n spaces. (Ara, theorem 7.4).


                                      Last revised on October 8, 2013 at 20:35:15. See the history of this page for a list of all contributions to it.