As Koenig and Zhu showed, quotient of a triangulated category by a maximal 1-orthogonal subcategory becomes an abelian category. In this paper, we generalize this result to a maximal n-orthogonal subcategory for an arbitrary positive integer n.

The bounded derived category of coherent sheaves D(X) is a natural triangulated category which can be associated with an algebraic variety X. It happens sometimes that two different varieties have equivalent derived categories of coherent sheaves D(X) ≃ D(Y ). There arises a natural question: can one say anything about motives of X and Y in that case? The first such example (see [4]) – abelian ...

We study internal structures in the category of algebras for an operad, and show that these themselves admit an operadic description. The main case of interest is where the operad is on an abelian category, and the internal structures in question are those of internal category, internal n-category, or internal (cubical) n-tuple category. This allows an operadic treatment of crossed modules and ...

For any finite group G, we define a bifunctor from the Dress category of finite G-sets to the conjugation biset category, whose objects are subgroups of G, and whose morphisms are generated by certain bifree bisets. Any additive functor from the conjugation biset category to abelian groups yields a Mackey functor by composition. We characterize the Mackey functors which arise in this way.

In this paper we show that to a unital associative algebra object (resp. co-unital coassociative co-algebra object) of any abelian monoidal category (C,⊗) endowed with a symmetric 2-trace, i.e. an F ∈ Fun(C,Vec) satisfying some natural trace-like conditions, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the (co)cyclic homology of the (co)algebra “with coefficients in F...

We describe the author’s research with Neil Strickland on the global algebra and global homological algebra of the category of BP∗BP comodules. We show, following [HS02a], that the category of E(n)∗E(n)comodules is a localization, in the abelian sense, of the category of BP∗BP comodules. This gives analogues of the usual structure theorems, such as the Landweber filtration theorem, for E(n)∗E(n...

It is well known that Barr and Beck’s definition of comonadic homology makes sense also with a functor of coefficients taking values in a semi-abelian category instead of an abelian one. The question arises whether such a homology theory has the same convenient properties as in the abelian case. Here we focus on independence of the chosen comonad: conditions for homology to depend on the induce...

This article treats the problem of deriving the reflector of a semi-abelian category A onto a Birkhoff subcategory B of A. Basing ourselves on Carrasco, Cegarra and Grandjeán’s homology theory for crossed modules, we establish a connection between our theory of Baer invariants with a generalization—to semi-abelian categories— of Barr and Beck’s cotriple homology theory. This results in a semi-a...

We examine the homotopy theory of simplicial graded abelian Hopf algebras over a prime field Fp, p > 0, proving that two very different notions of weak equivalence yield the same homotopy category. We then prove a splitting result for the Postnikov tower of such simplicial Hopf algebras. As an application, we show how to recover the homotopy groups of a simplicial Hopf algebra from its André-Qu...