How to construct a Hovey triple from two cotorsion pairs

By James Gillespie. 2014

51 citations recorded by the source corpus.

Let $\mathcal{A}$ be an abelian category, or more generally a weakly idempotent complete exact category, and suppose we have two complete hereditary cotorsion pairs $(\mathcal{Q}, \widetilde{\mathcal{R}})$ and $(\widetilde{\mathcal{Q}}, \mathcal{R})$ in $\mathcal{A}$ satisfying $\widetilde{\mathcal{R}} \subseteq \mathcal{R}$ and $\mathcal{Q} \cap \widetilde{\mathcal{R}} = \widetilde{\mathcal{Q}} \cap \mathcal{R}$. We show how to construct a (necessarily unique) abelian model structure on $\mathcal{A}$ with $\mathcal{Q}$ (respectively $\widetilde{\mathcal{Q}}$) as the class of cofibrant (resp. trivially cofibrant) objects and $\mathcal{R}$ (respectively $\widetilde{\mathcal{R}}$) as the class of fibrant (resp. trivially fibrant) objects.

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

View Paper: https://www.semanticscholar.org/paper/3f21eaccb168df9e855e464dc96222f921c7aabc

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Citations in the corpus

On Ding injective, Ding projective, and Ding flat modules and complexes (2015, background): “hort exact sequences 0 →A →Y →X →0 and 0 →Y ′ →X′ →A →0 with X,X′ ∈X and Y,Y′ ∈Y. Standard references include [EJ00] and [GT06] and connections to abelian model categories can be found in [Hov02] and [Gil15]. Recall that a Grothendieck category is a cocomplete abelian category with a set of generato”
Hereditary abelian model categories (2015, background): “A self-contained elementary proof is detailed in [29].”
On purity and applications to coderived and singularity categories (2014, unknown): “cally coherent, the two cotorsion pairs from Lemma 6.5 are necessarily hereditary and their mere existence implies the existence of a model structure on C(G) whose homotopy category is D(FpinjG); see [Gil14a]. However, it seems far from being obvious from there that the class of trivial objects is t”