A general framework for cluster tilting is set up by showing that any quotient of a triangulated category modulo a tilting subcategory (that is, a maximal oneorthogonal subcategory) carries an induced abelian structure. These abelian quotients turn out to be module categories of Gorenstein algebras of dimension at most one.

We put cluster tilting in a general framework by showing that any quotient of a triangulated category modulo a tilting subcategory (that is, a maximal one-orthogonal subcategory) carries an induced abelian structure. These abelian quotients turn out to be module categories of Gorenstein algebras of dimension at most one.

The ideas behind splitting classes were introduced by Freyd and Scedrov in [1] and expanded by Lippincott in [2]. In the latter work, Lippincott proves that there are exactly six splitting class pairs in the category of sets, but uncountably many in the category of groups. In this paper, we prove much more generally that any category containing the category of finite abelian p-groups as a full ...

We give here some examples of non pointed protomodular categories C satisfying a property similar to the property of representation of actions which holds for the pointed protomodular category Gp of groups: any slice category of Gp, any category of groupoids with a fixed set of objects, any essentially affine category. This property gives rise to an internal construction of the center of any ob...

Here we take the view that small abelian categories are certain categorical versions of rings and the collection of Serre subcategories of a small abelian category is then analogous to the set of ideals of a commutative ring. We consider two topologies on the collection of Serre subcategories; one is a very direct lift of the Zariski topology, the other, dual, topology appeared in the context o...

In [6] Higson showed that the formal properties of the Kasparov KK -theory groups are best understood if one regards KK (A, B) for separable C∗-algebras A, B as the morphism set of a category KK . In category language the composition and exterior KK product give KK the structure of a symmetric monoidal category which is enriched over abelian groups. We show that the enrichment of KK can be lift...

The theory of protomodular categories provides a simple and general context in which the basic theorems needed in homological algebra of groups, rings, Lie algebras and other non-abelian structures can be proved [2] [3] [4] [5] [6] [7] [9] [20]. An interesting aspect of the theory comes from the fact that there is a natural intrinsic notion of normal monomorphism [4]. Since any internal reflexi...