nLab
classical model structure on pointed topological spaces

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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                                      Contents

                                      Idea

                                      The classical model category structure on pointed topological spaces Top Quillen */Top^{\ast/}_{Quillen} is the model structure on an undercategory of the classical model structure on topological spaces Top QuillenTop_{Quillen} under the point.

                                      With the smash product this is a monoidal model category.

                                      Properties

                                      Cofibrant generation

                                      Recall that the generatic cofibrations of the classical model structure on topological spaces are

                                      I Top{S n1ι nD n} n I_{Top} \coloneqq \left\{ S^{n-1} \overset{\iota_n}{\longrightarrow} D^n \right\}_{n \in \mathbb{N}}

                                      and the generating acylic cofibrations are

                                      J Top{D n(id,δ 0)D n×I} n. J_{Top} \coloneqq \left\{ D^n \overset{(id,\delta_0)}{\longrightarrow} D^n \times I \right\}_{n \in \mathbb{N}} \,.

                                      Write

                                      () +:TopTop */ (-)_+ \;\colon\; Top \longrightarrow Top^{\ast/}

                                      for the operation of freely adjoining a basepoint.

                                      Proposition

                                      The coslice model structure (Top Quillen) */(Top_{Quillen})^{\ast/} is itself cofibrantly generated, with generating cofibrations

                                      I Top */={S + n1(ι n) +D + n} I_{Top^{\ast/}} = \left\{ S^{n-1}_+ \overset{(\iota_n)_+}{\longrightarrow} D^n_+ \right\}

                                      and generating acyclic cofibrations

                                      J Top */={D + n(id,δ 0) +(D n×I) +}. J_{Top^{\ast/}} = \left\{ D^n_+ \overset{(id, \delta_0)_+}{\longrightarrow} (D^n \times I)_+ \right\} \,.

                                      This is a special case of a general statement about cofibrant generation of coslice model structures, see this proposition.

                                      References

                                      • Mark Hovey, around corollary 2.4.20 of Model categories

                                      Last revised on April 14, 2016 at 13:33:20. See the history of this page for a list of all contributions to it.