We prove that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is still covariantly finite. This extends a result by Sikko and Smalø. We also prove a triangulated version of the result. As applications, we obtain short proofs to a classical result by Ringel and a recent result by Krause and Solberg. 1. Main Theorems Let C b...

We show that the quotient of a Hom-finite triangulated category C by the kernel of the functor HomC(T, −), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admit a calculus of left and right fractions. It follows that the Gabriel-Zisman localisation of the quotient at the class of regular morphisms is abelian. We show that it is equiv...

We define the concept of a regular object with respect to another object in an arbitrary category. We present basic properties of regular objects and we study this concept in the special cases of abelian categories and locally finitely generated Grothendieck categories. Applications are given for categories of comodules over a coalgebra and for categories of graded modules, and a link to the th...

1 Basic Properties Definition 1. Let X be a scheme. We denote the category of sheaves of OX -modules by OXMod or Mod(X). The full subcategories of qausi-coherent and coherent modules are denoted by Qco(X) and Coh(X) respectively. Mod(X) is a grothendieck abelian category, and it follows from (AC,Lemma 39) and (H, II 5.7) that Qco(X) is an abelian subcategory of Mod(X). If X is noetherian, then ...

Higher homological algebra was introduced by Iyama. It is also known as n-homological algebra where n > 2 is a fixed integer, and it deals with n-cluster tilting subcategories of abelian categories. All short exact sequences in such a subcategory are split, but it has nice exact sequences with n + 2 objects. This was recently formalised by Jasso in his theory of n-abelian categories. There is a...

The purpose of this paper is to describe a connection between model categories, a structure invented by algebraic topologists that allows one to introduce the ideas of homotopy theory to situations far removed from topological spaces, and cotorsion pairs, an algebraic notion that simultaneously generalizes the notion of projective and injective objects. In brief, a model category structure on a...

The object of this paper is to prove that the standard categories in which homotopy theory is done, such as topological spaces, simplicial sets, chain complexes of abelian groups, and any of the various good models for spectra, are all homotopically self-contained. The left half of this statement essentially means that any functor that looks like it could be a tensor product (or product, or sma...

In conventional gauge theory, a charged point particle is described by a representation of the gauge group. If we propagate the particle along some path, the parallel transport of the gauge connection acts on this representation. The Lagrangian density of the gauge field depends on the curvature of the connection which can be calculated from the holonomy around (infinitesimal) loops. For Abelia...

This paper studies the algebraic aspect of a general abelian coset theory with [DL2] as our main motivation. It is proved that the vacuum space ΩV (or the space of highest weight vectors) of a Heisenberg algebra in a general vertex operator algebra V has a natural generalized vertex algebra structure in the sense of [DL2] and that the vacuum space ΩW of a V -module W is a natural ΩV -module. Th...

We define a new topology, weaker than the gap topology, on the space of selfadjoint unbounded operators on a separable Hilbert space. We show that the subspace of selfadjoint Fredholm operators represents the functor K from the category of compact spaces to the category of abelian groups and prove a similar result for K. We define the spectral flow of a continuous path of selfadjoint Fredholm o...