LOCALLY PRESENTABLE CATEGORY GENERATOR

Cambridge University Press, Cambridge (2009) Prest M.: Purity, spectra and localisation. Neeman A.: Some adjoints in homotopy categories. Neeman A.: The homotopy category of flat modules, and Grothendieck duality. Princeton University Press, Princeton (2001) Neeman A.: The derived category of an exact category. Murfet D.: The Mock Homotopy Category of Projectives and Grothendieck Duality. Makkai M., Paré R.: Accessible Categories: The Foundations of Categorical Model Theory. Ladkani S.: On derived equivalences of lines, rectangles and triangles. Krause H.: The stable derived category of a Noetherian scheme. Kiełpiński R.: On Γ-pure injective modules. Keller B.: Derived categories and their uses. Jørgensen P.: The homotopy category of complexes of projective modules. Iyengar S., Krause H.: Acyclicity versus total acyclicity for complexes over Noetherian rings. In: Representation of Algebras (Ottowa, 1974). Gabriel P., Zisman M.: Calculus of Fractions and Homotopy Theory. Gabriel P., Ulmer F.: Lokal präsentierbare Kategorien, Lecture Notes in Mathematics, vol. 194(2), 284–295 (2005)įranke J.: On the Brown representability theorem for triangulated categories. 39(3), 189–209 (1981)Įnochs E., Estrada S.: Relative homological algebra in the category of quasi-coherent sheaves. Theory 9(5), 423–430 (2006)Įnochs E.E.: Injective and flat covers, envelopes and resolvents. 133(2), 261–293 (2002)Ĭrawley-Boevey W.: Locally finitely presented additive categories. 16(1), 1–5 (2009)Ĭhristensen J.D., Hovey M.: Quillen model structures for relative homological algebra. PreprintĬasacuberta C., Neeman A.: Brown representability does not come for free. Cambridge University Press, Cambridge (1991)īican L., El Bashir R., Enochs E.: All modules have flat covers. Cambridge Studies in Advanced Mathematics, vol. 129(3), 447–475 (2000)īenson D.J.: Representations and Cohomology I: Basic Representation Theory of Finite Groups and Associative Algebras. Algebra 66(1), 61–122 (1980)īeke T.: Sheafifiable homotopy model categories. 52(2), 225–247 (2000)Īuslander M., Smalø S.O.: Preprojective modules over Artin algebras. 54(129)(3), 623–636 (2004)Īlonso Tarrío L., Jeremías López A., Souto Salorio M.J.: Localization in categories of complexes and unbounded resolutions. Cambridge University Press, Cambridge (1994)Īdámek J., Rosický J.: On pure quotients and pure subobjects. We also include a proof of the model category equivalence of modules over the Eilenberg–Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R.Adámek J., Rosický J.: Locally Presentable and Accessible Categories. This is a slight generalization of Rickard's work on derived equivalent rings.

Finally, we characterize stable model categories which represent the derived category of a ring. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories.

LOCALLY PRESENTABLE CATEGORY GENERATOR

We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring.

Abstract: A stable model category is a setting for homotopy theory where the suspension functor is invertible.

Locally presentable category