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We introduce a new concept of s-recollements of extriangulated categories, which generalizes recollements of abelian categories, recollements of triangulated categories, as well as recollements of extriangulated categories. Moreover, some basic properties of s-recollements are presented. We also discuss the behavior of the localization theory on the adjoint pair of exact functors. Finally, we provide a method to obtain a recollement of triangulated categories from s-recollements of extriangulated categories via the localization theory.
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In this article, we prove that if $(mathcal A ,mathcal B,mathcal C)$ is a recollement of extriangulated categories, then torsion pairs in $mathcal A$ and $mathcal C$ can induce torsion pairs in $mathcal B$, and the converse holds under natural assumptions. Besides, we give mild conditions on a cluster tilting subcategory on the middle category of a recollement of extriangulated categories, for the corresponding abelian quotients to form a recollement of abelian categories.
A notion of balanced pairs in an extriangulated category with a negative first extension is defined in this article. We prove that there exists a bijective correspondence between balanced pairs and proper classes $xi$ with enough $xi$-projectives and enough $xi$-injectives. It can be regarded as a simultaneous generalization of Fu-Hu-Zhang-Zhu and Wang-Li-Huang. Besides, we show that if $(mathcal A ,mathcal B,mathcal C)$ is a recollement of extriangulated categories, then balanced pairs in $mathcal B$ can induce balanced pairs in $mathcal A$ and $mathcal C$ under natural assumptions. As a application, this result gengralizes a result by Fu-Hu-Yao in abelian categories. Moreover, it highlights a new phenomena when it applied to triangulated categories.
In this paper, let $(mathcal{A},mathcal{B},mathcal{C})$ be a recollement of extriangulated categories. We introduce the global dimension and extension dimension of extriangulated categories, and give some upper bounds of global dimensions (resp. extension dimensions) of the categories involved in $(mathcal{A},mathcal{B},mathcal{C})$, which give a simultaneous generalization of some results in the recollement of abelian categories and triangulated categories.
It is known that the Grothendieck group of the category of Schur functors is the ring of symmetric functions. This ring has a rich structure, much of which is encapsulated in the fact that it is a plethory: a monoid in the category of birings with its substitution monoidal structure. We show that similarly the category of Schur functors is a 2-plethory, which descends to give the plethory structure on symmetric functions. Thus, much of the structure of symmetric functions exists at a higher level in the category of Schur functors.
For a model category, we prove that taking the category of coalgebras over a comonad commutes with left Bousfield localization in a suitable sense. Then we prove a general existence result for left-induced model structure on the category of coalgebras over a comonad in a left Bousfield localization. Next we provide several equivalent characterizations of when a left Bousfield localization preserves coalgebras over a comonad. These results are illustrated with many applications in chain complexes, (localized) spectra, and the stable module category.
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