Relative stable categories and birationality

Relative Compactness and Product Stable Quotient Maps

Triangulated Categories and Stable Model Categories

X id → X → 0→ · For any morphism u : X → Y , there is an object Z (called a mapping cone of the morphism u) fitting into a distinguished triangle X u − → Y → Z → · Any triangle isomorphic to a distinguished triangle is distinguished. This means that if X u − → Y v − → Z w −→ X[1] is a distinguished triangle, and f : X → X, g : Y → Y , and h : Z → Z are isomorphisms, then X′ gu f −1 −−−−→ Y ′ hv...

Relative Mal’tsev Categories

We define relative regular Mal’tsev categories and give an overview of conditions which are equivalent to the relative Mal’tsev axiom. These include conditions on relations as well as conditions on simplicial objects. We also give various examples and counterexamples.

Relative Homological Categories

We introduce relative homological and weakly homological categories (C,E), where “relative” refers to a distinguished class E of normal epimorphisms in C. It is a generalization of homological categories, but also protomodular categories can be regarded as examples. We indicate that the relative versions of various homological lemmas can be proved in a relative homological category.

Relative Semi-abelian Categories

We introduce a relative semi-abelian category as a pair (C,E), where C is a pointed category with finite limits, and E is a class of normal epimorphisms in C satisfying certain conditions, stronger than those defining a relative homological category [2]. In the absolute case, i.e. when E is the class of all regular epimorphisms in C, the pair (C,E) is relative semi-abelian if and only if C is s...