compactly generated model category

under construction – warning – currently inconsistent


Model category theory

model category


  • category with weak equivalences

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  • Morphisms

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      • Refinements

        • monoidal model category

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        • Producing new model structures

          • on functor categories (global)

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          • Grothendieck construction for model categories

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

            • (∞,1)-category

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            • Model structures

              • Cisinski model structure
              • for \infty-groupoids

                for ∞-groupoids

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                  related by the Dold-Kan correspondence

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                      • on monoids

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                      • 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

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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 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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                                      Compact objects




                                      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).


                                      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.