Abelian categories arising from cluster tilting subcategories II: quotient functors

Abstract

In this paper, we consider a kind of ideal quotient of an extriangulated category such that the ideal is the kernel of a functor from this extriangulated category to an abelian category. We study a condition when the functor is dense and full, in another word, the ideal quotient becomes abelian. Moreover, a new equivalent characterization of cluster-tilting subcategories is given by applying homological methods according to this functor. As an application, we show that in a connected 2-Calabi-Yau triangulated category B, a functorially finite, extension closed subcategory T of B is cluster tilting if and only if B/T is an abelian category.

Associahedra for finite‐type cluster algebras and minimal relations between g‐vectors

“…Extriangulated categories, recently introduced in [72], axiomatize extension‐closed subcategories of triangulated categories in a (moderately) similar way that Quillen's exact categories axiomatize extension‐closed subcategories of abelian categories. They appear in representation theory in relation with cotorsion pairs [29, 58, 59, 104], with Auslander–Reiten theory [51], with cluster algebras, mutations, or cluster‐tilting theory [29, 63–65, 83, 106], with Cohen–Macaulay dg‐modules in the remarkable [53]. We also note the generalization, called $$n$$‐exangulated categories [47, 48], to a version suited for higher homological algebra.…”