Cotorsion pairs and model categories

By Mark Hovey. 2007

65 citations recorded by the source corpus.

This paper is an expanded version of two talks given by the author at the Summer School on the Interactions between Homotopy Theory and Algebra at the University of Chicago, July 26 to August 6, 2004. It describes a connection between model categories, a structure invented by algebraic topologists that allows one to introduce the ideas of homotopy theory to situations far removed from topological spaces, and cotorsion pairs, an algebraic notion that simultaneously generalizes the notion of projective and injective objects. It also gives some applications of this connection, some due to the authour about model structures on Z[G]-modules for G a finite group, and some due to Jim Gillespie about flat model structures of sheaves.

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

View Paper: https://www.semanticscholar.org/paper/48a4f5056b6c7c0a954e295b7afde38c4fa948b8

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In Algebra and Number Theory

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Citations in the corpus

Deconstructibility and the Hill Lemma in Grothendieck categories (2010, methodology): “For example in: (1) Gillespie’s construction [16, 17, 18] of flat monoidal model structures on module and sheaf categories (see [23] for a nice overview).”
Hearts of twin cotorsion pairs on extriangulated categories (2017, methodology): “For example, a pair of cotorsion pairs satisfying some conditions, which we call a twin cotorsion pair, has close relationship with model structure called abelian model structure ([Ho1],[Ho2]) and exact model structure [G].”
Proper classes and Gorensteinness in extriangulated categories (2019, background): “This result is inspired from [7, 8, 9], where the case of exact categories is studied in more details.”
More Concise Algebraic Topology: Localization, Completion, and Model Categories (2012, background): “Hovey [66] has studied abelian categories with model structures in which the cofibrations are the monomorphisms with cofibrant cokernel and the fibrations are the epimorphisms with fibrant kernel.”
On exact categories and applications to triangulated adjoints and model structures (2010, methodology): “Although the main tool we use for this, the so-called Quillen’s small object argument, has been advocated by Hovey [30] and actually used for constructing approximations by Rosický [56], its real power has not been exploited.”
Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves (2013, background): “There are probably two reasons for this development: The foundation of the theory of abelian and exact model categories has been only given a decade ago by Hovey [42] (see also [43] for a nice overview), and the “implementation details” for their construction are rather recent, see [23, 27, 28, 29, ”