Model structure arising from one hereditary complete cotorsion pair on extriangulated categories

Hovey’s correspondence between model structures and cotorsion pairs in the setting of abelian categories, has been generalized by Nakaoka-Palu, using two cotorsion pairs, to the setting of weakly idempotent complete extriangulated categories, and the aim of the paper is to give an analogous correspondence using one (hereditary) cotorsion pair generalizing in this setting work of Beligiannis-Reiten and Cui, Lu and Zhang. Furthermore, we provide methods to construct model structures from silting objects in weakly idempotent complete extriangulated categories and co-$t$-structures on triangulated categories.

💡 Research Summary

The paper “Model structure arising from one hereditary complete cotorsion pair on extriangulated categories” extends the celebrated Hovey correspondence between model structures and cotorsion pairs from abelian categories to the much broader setting of weakly idempotent complete extriangulated categories, using only a single hereditary complete cotorsion pair.

In the classical Hovey framework, a model structure on an abelian (or exact) category is obtained from two compatible complete cotorsion pairs. Nakaoka and Palu later generalized this to extriangulated categories, but still required two cotorsion pairs. Beligiannis and Reiten showed that, for abelian categories, a weakly projective (or ω‑model) structure can be built from a single hereditary complete cotorsion pair whose core ω is contravariantly finite. The present work lifts this “one‑pair” correspondence to extriangulated categories, thereby unifying and simplifying the construction of model structures in this setting.

The main technical result (Theorem 1.1) states that for a weakly idempotent complete extriangulated category 𝔅, given full additive subcategories 𝓧 and 𝓨 closed under direct summands and isomorphisms, with ω = 𝓧 ∩ 𝓨, the triple

• CoFib ω = {inflations f | Cone(f) ∈ 𝓧},
• Fib ω = {maps f | Hom(W,f) is surjective for every W ∈ ω}, and
• Weq ω = {maps f that appear as a deflation in a conflation whose co‑cone lies in 𝓨 and whose middle term contains a summand from ω}
  forms a model structure on 𝔅 if and only if (𝓧, 𝓨) is a hereditary complete cotorsion pair and ω is contravariantly finite. In this situation the cofibrant objects are precisely 𝓧, all objects are fibrant, the trivial objects are 𝓨, and the homotopy category Ho(𝔅) is equivalent to the additive quotient 𝓧/ω.

Theorem 1.2 (the Beligiannis‑Reiten correspondence) establishes a bijection between the class Ω of such hereditary cotorsion pairs (with contravariantly finite core) and the class Γ of weakly projective model structures on 𝔅. The forward map Φ sends (𝓧, 𝓨) to the model structure defined above, while the inverse Ψ extracts the cofibrant and trivial fibrant classes from a given model structure. This recovers the original correspondence for abelian categories and its extension to exact categories, and it further yields a one‑to‑one correspondence between weakly projective model structures on a triangulated category and co‑t‑structures whose co‑heart is contravariantly finite.

The paper also investigates the nature of weakly projective model structures. Proposition 4.4 provides several equivalent characterizations, showing that a model structure is weakly projective precisely when cofibrations are inflations with cofibrant cones, trivial fibrations are deflations with trivially fibrant cocones, and every object is fibrant. Consequently, in a weakly idempotent complete exact category the ω‑model structures coincide with the weakly projective model structures introduced by Cui, Lu, and Zhang.

A significant application is given via silting theory. Using the bijection between bounded hereditary cotorsion pairs and silting subcategories established in