On exact categories and applications to triangulated adjoints and model structures

Abstract

We show that Quillen's small object argument works for exact categories under very mild conditions. This has immediate applications to cotorsion pairs and their relation to the existence of certain triangulated adjoint functors and model structures. In particular, the interplay of different exact structures on the category of complexes of quasi-coherent sheaves leads to a streamlined and generalized version of recent results obtained by Estrada, Gillespie, Guil Asensio, Hovey, Jørgensen, Neeman, Murfet, Prest, Trlifaj and possibly others.

38 pages; version 2: major revision, more explanation added at several places, reference list updated and extended, misprints corrected

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Cited by in corpus (41)

• Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence
• Singular compactness and definability for $\Sigma$-cotorsion and Gorenstein modules
• Deconstructibility and the Hill lemma in Grothendieck categories
• Derived equivalences induced by big cotilting modules
• Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories
• On compactly generated torsion pairs and the classification of co-t-structures for commutative noetherian rings
• Cotorsion pairs in categories of quiver representations
• Silting objects
• Definability and approximations in triangulated categories
• The tilting-cotilting correspondence
• $t$-Structures with Grothendieck hearts via functor categories
• Derived, coderived, and contraderived categories of locally presentable abelian categories
• Approximations and locally free modules
• Fp-projective periodicity
• Lifting of recollements and gluing of partial silting sets
• On the obscure axiom for one-sided exact categories
• Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity
• A homotopy theory of additive categories with suspensions
• Locally type ${\text{FP}}_n$ and $n$-coherent categories
• Brown representability often fails for homotopy categories of complexes
• Cut cotorsion pairs
• When stable short exact sequences define an exact structure on any additive category
• The derived category with respect to a generator
• A note on compactly generated co-t-structures
• Flat Mittag-Leffler modules, and their relative and restricted versions
• Homotopy in Exact Categories
• Dualizations of approximations, ${\aleph }_1$-projectivity, and Vopěnka's Principles
• Exact model structures and recollements
• Right triangulated categories: As extriangulated categories, aisles and co-aisles
• Priles of one-sided triangulated categories and exact model categories
• Generalized periodicity theorems
• The $Q$-shaped derived category of a ring
• Fishing for complements
• Tor-pairs: products and approximations
• Cotorsion pairs in Hopfological algebra
• On a fat small object argument
• Hereditary abelian model categegories
• One-sided exact categories
• Applications of Exact Structures in Abelian Categories
• Complete cotorsion pairs in exact categories
• Cellular objects and Shelah's singular compactness theorem