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compactly generated model category

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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

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          • Grothendieck construction for model categories

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

            • (∞,1)-category

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            • Model structures

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                for ∞-groupoids

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                  related by the Dold-Kan correspondence

                • model structure on cosimplicial simplicial sets

                • for nn-groupoids

                  • for n-groupoids/for n-types

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                    • model structure on simplicial groups

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                      general

                      • on monoids

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                      • on algebras over an operad,

                        on modules over an algebra over an operad

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                        • model structure on differential graded-commutative superalgebras

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                          • model structure on spectra

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

                            • on categories with weak equivalences

                            • Joyal model for quasi-categories

                            • on sSet-categories

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                                • 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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                                      Compact objects

                                      Contents

                                      Definition

                                      Definition

                                      A cofibrantly generated simplicial model category CC is compactly generated if

                                      • there exists a small set SObj(C)S \subset Obj(C) of objects

                                      • such that

                                        1. each KSK \in S is cofibrant;

                                        2. each KSK \in S is a homotopy compact object: for all filtered colimit diagram Y:DCY : D \to C the morphism

                                          𝕃lim iC(K,Y i)C(K,𝕃lim iY i) \mathbb{L}\lim_{\to_i} C(K, Y_i) \simeq C(K, \mathbb{L}\lim_{\to_i} Y_i)

                                          (where 𝕃lim \mathbb{L}\lim_\to denotes the Ho CHo_C is the homotopy category of CC) is a weak homotopy equivalence in sSet;

                                        3. a morphism XYX \to Y in CC is a weak equivalence precisely if for all KSK \in S the induced morphism

                                          Ho C(K,X)Ho C(K,Y) Ho_C(K, X) \to Ho_C(K,Y)

                                          is a bijection.

                                      Something needs to be added/fixed here!!

                                      See (Jardine11, page 14), (Marty, def 1.7).

                                      References

                                      Page 88 (14) of

                                      • Rick Jardine, Representability theorems for presheaves of spectra J. Pure Appl. Algebra

                                        215 (2011) (pdf)

                                      Def. 1.7 of

                                      • Florian Marty, Smoothness in relative geometry (2009) (pdf)

                                      A different meaning of “compactly generated model category” is used in Definition 5.9 of

                                      Last revised on February 10, 2014 at 06:51:02. See the history of this page for a list of all contributions to it.