We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Bourke, John and Garner, Richard 2016. Algebraic weak factorisation systems II: Categories of weak maps. Journal of Pure and Applied Algebra, Vol. 220, Issue. 1, p. 148.
Gillespie, James 2016. The derived category with respect to a generator. Annali di Matematica Pura ed Applicata (1923 -), Vol. 195, Issue. 2, p. 371.
Zheng, Yuefei and Huang, Zhaoyong 2016. On pure derived categories. Journal of Algebra, Vol. 454, p. 252.
Christensen, J. Daniel and Wang, Gaohong 2015. Ghost Numbers of Group Algebras II. Algebras and Representation Theory, Vol. 18, Issue. 3, p. 849.
Rosický, J. 2015. Accessible Model Categories. Applied Categorical Structures,
Krause, Henning 2012. Approximations and adjoints in homotopy categories. Mathematische Annalen, Vol. 353, Issue. 3, p. 765.
Barnes, David and Roitzheim, Constanze 2011. Monoidality of Frankeʼs exotic model. Advances in Mathematics, Vol. 228, Issue. 6, p. 3223.
Rosický, J. 2009. On Combinatorial Model Categories. Applied Categorical Structures, Vol. 17, Issue. 3, p. 303.
Rump, Wolfgang 2009. The Absolute of a Topological Space and its Application to Abelian l-groups. Applied Categorical Structures, Vol. 17, Issue. 2, p. 153.
Chorny, Boris 2006. A generalization of Quillen's small object argument. Journal of Pure and Applied Algebra, Vol. 204, Issue. 3, p. 568.
Cole, Michael 2006. Mixing model structures. Topology and its Applications, Vol. 153, Issue. 7, p. 1016.
Chorny, Boris 2005. Localization with respect to a class of maps I — Equivariant localization of diagrams of spaces. Israel Journal of Mathematics, Vol. 147, Issue. 1, p. 93.
Christensen, J.Daniel Dwyer, William G. and Isaksen, Daniel C. 2004. Obstruction theory in model categories. Advances in Mathematics, Vol. 181, Issue. 2, p. 396.
Schwede, Stefan and Shipley, Brooke 2003. Stable model categories are categories of modules. Topology, Vol. 42, Issue. 1, p. 103.
An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category [Ascr ] is exactly the information needed to do homological algebra in [Ascr ]. The main result is that, under weak hypotheses, the category of chain complexes of objects of [Ascr ] has a model category structure that reflects the homological algebra of the projective class in the sense that it encodes the Ext groups and more general derived functors. Examples include the ‘pure derived category’ of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and co-homology. We characterize the model structures that are cofibrantly generated, and show that this fails for many interesting examples. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories.
Email your librarian or administrator to recommend adding this journal to your organisation's collection.
