In [11], Quillen defines the notion of exact category. Each such category N is an additive category together with a suitable collection of sequences called short exact sequences. Quillen constructs a space by gluing simplices to each other, using the exact sequences of N to determine the simplices and the gluing instructions. Then for each n ∈ N he defines KnN as a homotopy group of the space. ...

When the exact completion of a category with weak finite limits is a Mal’cev category, it is possible to combine the universal property of the exact completion and the universal property of the coequalizer completion. We use this fact to explain Freyd’s representation theorems in abelian and Frobenius categories. MSC 2000 : 18A35, 18B15, 18E10, 18G05.

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. Furthermore, we show the functoriality of Harder-Narasimhan filtrations (indexed by R), which can not be stated in the classical setting of Harder and Narasimhan’s...

We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3×3-lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any em...

We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3×3-lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any em...

We show that for a given exact category, there exists bijection between semibricks (pairwise Hom-orthogonal set of bricks) and length wide subcategories (exact extension-closed abelian subcategories). In particular, we category is if only simple objects form semibrick, is, the Schur's lemma holds.

We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial exact functors to or from these algebraic respectively topological triangulated categories. Introduction. Triangulated categories are ...

The purpose of this paper is the study of direct limits in category of Krasner (m, n)-hyperrings. In this regards we introduce and study direct limit of a direct system in category (m, n)-hyperrings. Also, we consider fundamental relation , as the smallest equivalence relation on an (m, n)-hyperring R such that the quotient space is an (m, n)-ring, to introdu...

For a finite group scheme G, we continue our investigation of those finite dimensional kG-modules which are of constant Jordan type. We introduce a Quillen exact category structure C(kG) on these modules and investigate K0(C(kG)). We study which Jordan types can be realized as the Jordan types of (virtual) modules of constant Jordan type. We also briefly consider thickenings of C(kG) inside the...