Additive invariants of dg-categories. (Invariants additifs de dg-catégories.)(French)Zbl 1094.18006
In his book [“Model categories”, Math. Surv. Monogr. 63 (Providence, RI): American Mathematical Society (AMS) (1999;Zbl 0909.55001)]M. Hovey gave a necessary and sufficient criterion when a category admitting small inductive and projective limits carries a cofibrantly generated closed model category structure with prescribed cofibrations and weak equivalences. In the paper under review the author verifies this criterion in several situations for the category of the small dg-categories in the sense ofB. Keller [Ann. Sci. Éc. Norm. Supér., IV. Sér. 27, 63–102 (1994;Zbl 0799.18007)].
In the first case the weak equivalences are the dg-functors \(F:{\mathcal C}\rightarrow {\mathcal D}\) inducing a quasi-isomorphism between the complexes \(\operatorname{Hom}_{\mathcal C}(c_1,c_2)\) and \(\operatorname{Hom}_{\mathcal D}(Fc_1,Fc_2)\), and so that it induces a an essentially surjective functor between the categories \(H^0(\text{pre-tr}({\mathcal C}))\) and \(H^0(\text{pre-tr}({\mathcal D}))\). For a certain choice of cofibrant objects one gets informations on the pre-triangulated hull of a small dg-category.
In the second case the weak equivalences are those functors \(F\) which induce again quasi-isomorphisms on the \(\operatorname{Hom}\)-spaces and where the second condition on \(F\) holds for the idempotent closure of the corresponding categories. Such functors are exactly those which induce equivalences of the derived categories. For the same cofibrant objects in this setting the author verifies Hovey’s criterion. As a corollary, Hochschild homology, cyclic homology, algebraic \(K\)-theory and Chern characters can be interpreted in the resulting Quillen structure.
In the first case the weak equivalences are the dg-functors \(F:{\mathcal C}\rightarrow {\mathcal D}\) inducing a quasi-isomorphism between the complexes \(\operatorname{Hom}_{\mathcal C}(c_1,c_2)\) and \(\operatorname{Hom}_{\mathcal D}(Fc_1,Fc_2)\), and so that it induces a an essentially surjective functor between the categories \(H^0(\text{pre-tr}({\mathcal C}))\) and \(H^0(\text{pre-tr}({\mathcal D}))\). For a certain choice of cofibrant objects one gets informations on the pre-triangulated hull of a small dg-category.
In the second case the weak equivalences are those functors \(F\) which induce again quasi-isomorphisms on the \(\operatorname{Hom}\)-spaces and where the second condition on \(F\) holds for the idempotent closure of the corresponding categories. Such functors are exactly those which induce equivalences of the derived categories. For the same cofibrant objects in this setting the author verifies Hovey’s criterion. As a corollary, Hochschild homology, cyclic homology, algebraic \(K\)-theory and Chern characters can be interpreted in the resulting Quillen structure.
Reviewer: Alexander Zimmermann (Amiens)
MSC:
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 18D15 | Closed categories (closed monoidal and Cartesian closed categories, etc.) |
