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model structure on dg-comodules

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

                                      A model category structure on the category of comodules in chain complexes over a dg-coalgebra.

                                      Details

                                      Let CC be a differential graded-cocommutative coalgebra? over a field.

                                      Model structure of the second kind

                                      There exists a model category structure on the category CdgCoModC dgCoMod of dg-comodules? over CC whose

                                      This is due to (Positelski 11, 8.2 Theorem (a)).

                                      Model structure of the first kind

                                      There is another model structure where the fibrations in addition satisfy the condition that their kernel KK satisfies that for all acyclic NN, then Hom̲ C(N,K)\underline{Hom}_C(N,K) is acyclic.

                                      This is due to (Positelski 11, 8.2 Remark), there called the “model category structure of the first kind”. This is also reviewed as (Pridham 13, prop. 2.2).

                                      References

                                      Last revised on June 6, 2017 at 12:22:12. See the history of this page for a list of all contributions to it.