Quillen model structures for relative homological algebra

By Mark Hovey. Mathematical Proceedings of the Cambridge Philosophical Society, 2000

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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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e9837dec807cf01a835bee24ceb7b273b607dd60

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Citations in the corpus

Generalized Brown representability in homotopy categories (2005, unknown): “here fn = 0 for n < −k − 1 and n > k, fn = idAn for −k ≤ n ≤ k and f−k−1 = d−k. Thus each ﬁnitely presentable complex is bounded and evidently consists of ﬁnitely presentable modules. Following [8], 2.9, Ch(R) is strongly ω-combinatorial if and only if ﬁnitely presentable complexes have ﬁnitel”
More Concise Algebraic Topology: Localization, Completion, and Model Categories (2012, methodology): “The h-model structure, which was long folklore, is due independently to Cole [30], Schwänzl and Vogt [119], and Christensen and Hovey [27, 64].”
Local and stable homological algebra in Grothendieck abelian categories (2007, methodology): “5 coincides with the model structure of Christensen and Hovey [5] in the particular case where all the elements of G are projective.”
Stable model categories are categories of modules (2003, background): “In [8], Christensen and Hovey introduce model category structures for chain complexes over an abelian category based on a projective class.”
Approximations and adjoints in homotopy categories (2010, background): “Let us exhibit the pure derived category of a module category more closely; see also [10, 38].”
Categorical Homotopy Theory (2014, background): “” There are many other model structures relevant to homological algebra, for instance on chain complexes that are bounded above or bounded below, or for unbounded chain complexes taking values in other abelian categories; see [CH02].”