Related papers: Remarks on partially abelian exact categories

We define fully exact module categories, a subclass of exact module categories over a finite braided tensor category that is stable under the relative Deligne product. In contrast, we demonstrate with examples in both zero and non-zero…

This paper studies abelian categories that can be decomposed into smaller abelian categories via iterated recollements - such a decomposition we call a stratification. Examples include the categories of (equivariant) perverse sheaves and…

The theorem of the title is deduced from the equivalence between crossed complexes and cubical $\omega$-groupoids with connections proved by the authors in 1981. In fact we prove the equivalence of five categories defined internally to an…

We continue our study of relatively divisible and relatively flat objects in exact categories in the sense of Quillen with several applications to exact structures on finitely accessible additive categories and module categories. We derive…

We develop some new aspects of cohomology in the context of semi-abelian categories: we establish a Hochschild-Serre 5-term exact sequence extending the classical one for groups and Lie algebras; we prove that an object is perfect if and…

Many kinds of categorical structure require the existence of finite limits, of colimits of some specified type, and of "exactness" conditions between the finite limits and the specified colimits. Some examples are the notions of regular, or…

Let A be an exact category, that is, an extension-closed full subcategory of an abelian category. Firstly, we give some necessary and sufficient conditions for A to have almost split sequences. Then, we study when an almost split sequence…

We introduce abelian framed bicategories, which are particular framed bicategories that are locally abelian, and show that they are suitable for developing homology and cohomology theories for directed structures. This means in particular…

We study properties of relative modular categories and derive sufficient conditions for their existence. In particular, we derive sufficient conditions for relative pre-modular categories to be non-degenerate and relative modular, and for…

We define model structures on exact categories which we call exact model structures. We look at the relationship between these model structures and cotorsion pairs on the exact category. In particular, when the underlying category is weakly…

Using the Morita-type embedding, we show that any exact category with enough projectives has a realization as a (pre)resolving subcategory of a module category. When the exact category has enough injectives, the image of the embedding can…

We study the properties of the relative derived category $D_{\mathscr{C}}^{b}$($\mathscr{A}$) of an abelian category $\mathscr{A}$ relative to a full and additive subcategory $\mathscr{C}$. In particular, when $\mathscr{A}=A{\text -}\mod$…

We prove that for any finite-dimensional differential graded algebra with separable semisimple part the category of perfect modules is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme with…

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…

We propose a generalization of Quillen's exact category -- arithmetic exact category and we discuss conditions on such categories under which one can establish the notion of Harder-Narasimhan filtrations and Harder-Narsimhan polygons.…

We introduce the notion of a "category with path objects", as a slight strengthening of Kenneth Brown's classic notion of a "category of fibrant objects". We develop the basic properties of such a category and its associated homotopy…

This paper consists of three results on Frobenius categories: (1) we give sufficient conditions on when a factor category of a Frobenius category is still a Frobenius category; (2) we show that any Frobenius category is equivalent to an…

Given any additive category $\mathcal{C}$ with split idempotents, pseudokernels and pseudocokernels, we show that a subcategory $\mathcal{B}$ is coreflective if, and only if, it is precovering, closed under direct summands and each morphism…

We introduce a new class of categories generalizing locally presentable ones. The distinction does not manifest in the abelian case and, assuming Vopenka's principle, the same happens in the regular case. The category of complete partial…

This dissertation has two main parts. The first part deals with questions relating to Haghverdi and Scott's notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every…