Model structures on exact categories

By James Gillespie. 2010

134 citations recorded by the source corpus.

We define model structures on exact categories, which we call exact model structures. We look at the relationship between these model structures and cotorsion pairs on the exact category. In particular, when the underlying category is weakly idempotent complete, we get Hovey’s one-to-one correspondence between model structures and complete cotorsion pairs. We classify the right and the left homotopy relation in terms of the cotorsion pairs and look at examples of exact model structures. In particular, we see that given any hereditary abelian model category, the full subcategories of cofibrant, fibrant and cofibrant–fibrant subobjects each has natural exact model structures equivalent to the original model structure. These model structures each has interesting characteristics. For example, the cofibrant–fibrant subobjects form a Frobenius category, whose stable category is the same as the homotopy category of its model structure.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ac10e05b3bcb58a09daa5717bb2a1a76e62fd1d1

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In Geometry and Topology

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Citations in the corpus

Gorenstein complexes and recollements from cotorsion pairs (2012, methodology): “ with cokernel in Wfollowed by an epimorphism with kernel in W. Remark 1. We note that one normally assumes the category Ais bicomplete for it to qualify as a model category. However, as explained in [Gil11] an abelian category with an abelian model structure already has enough colimits and limits t”
Hereditary abelian model categories (2015, background): “This observation was first made in [28], but it is so easy that we will reprove it here to make it transparent.”
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.”
Hearts of twin cotorsion pairs on extriangulated categories (2017, unknown): “ 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]. If one considers a cotorsion pair in a triangulated category, it becomes essentially the same a”
Models for singularity categories (2012, result): “[13,6]; though they seem different at first, they are both equivalent to our definition by [13, Proposition 4.”
Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves (2013, methodology): “In this context, we denote for brevity Cac(A) by Ã (this notation has been used by Gillespie in [27, 28, 29]).”