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Bourke, John and Garner, Richard 2017. Algebraic weak factorisation systems II: Categories of weak maps. Journal of Pure and Applied Algebra, Vol. 220, Issue. 1, p. 148.
ESTRADA, SERGIO GILLESPIE, JAMES and ODABAŞI, SINEM 2017. Pure exact structures and the pure derived category of a scheme. Mathematical Proceedings of the Cambridge Philosophical Society, p. 1.
Gillespie, James 2017. 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 2017. On pure derived categories. Journal of Algebra, Vol. 454, p. 252.
2017. Introduction to Abelian Model Structures and Gorenstein Homological Dimensions.
Christensen, J. Daniel and Wang, Gaohong 2017. Ghost Numbers of Group Algebras II. Algebras and Representation Theory, Vol. 18, Issue. 3, p. 849.
Rosický, J. 2017. Accessible Model Categories. Applied Categorical Structures,
Krause, Henning 2017. Approximations and adjoints in homotopy categories. Mathematische Annalen, Vol. 353, Issue. 3, p. 765.
Barnes, David and Roitzheim, Constanze 2017. Monoidality of Frankeʼs exotic model. Advances in Mathematics, Vol. 228, Issue. 6, p. 3223.
Rosický, J. 2017. On Combinatorial Model Categories. Applied Categorical Structures, Vol. 17, Issue. 3, p. 303.
Rump, Wolfgang 2017. The Absolute of a Topological Space and its Application to Abelian l-groups. Applied Categorical Structures, Vol. 17, Issue. 2, p. 153.
Chorny, Boris 2017. A generalization of Quillen's small object argument. Journal of Pure and Applied Algebra, Vol. 204, Issue. 3, p. 568.
Cole, Michael 2017. Mixing model structures. Topology and its Applications, Vol. 153, Issue. 7, p. 1016.
Chorny, Boris 2017. 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. 2017. Obstruction theory in model categories. Advances in Mathematics, Vol. 181, Issue. 2, p. 396.
Schwede, Stefan and Shipley, Brooke 2017. 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.
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