Classification problems in K-categories
In the classical Witt theory over a field F, the study of quadratic forms begins with two simple invariants: the dimension of a form modulo 2, called the dimension index and denoted e⁰: W(F) → ℤ/2, and the discriminant e¹ with values in k₁(F) = F*/F*², which behaves well on the fundamental ideal I(F)= ker(e⁰). Here a more sophisticated situation is considered, of quadratic forms over a scheme and, more generally, over an exact category with duality. Our purposes are: ...
Extriangulated categories were introduced by Nakaoka and Palu by extracting the similarities between exact categories and triangulated categories. A notion of homotopy cartesian square in an extriangulated category is defined in this article. We prove that in an extriangulated category with enough projective objects, the extension subcategory of two covariantly finite subcategories is covariantly finite. As an application, we give a simultaneous generalization of a result of X. W. Chen (2009) and...
Let be a triangulated category and be a cluster tilting subcategory of . Koenig and Zhu showed that the quotient category is Gorenstein of Gorenstein dimension at most one. But this is not always true when becomes an exact category. The notion of an extriangulated category was introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. Now let be an extriangulated category with enough projectives and enough injectives, and a cluster...
For any positive integer , let be a linearly oriented quiver of type with vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories and , where and are the two extriangulated categories corresponding to the representation category of and the morphism category of projective representations of , respectively. As a by-product,...
Let be an abelian category, or more generally a weakly idempotent complete exact category, and suppose we have two complete hereditary cotorsion pairs and in satisfying and . We show how to construct a (necessarily unique) abelian model structure on with (resp. ) as the class of cofibrant (resp. trivially cofibrant) objects, and (resp. ) as the class of fibrant (resp. trivially fibrant) objects.