Related papers: A cocomplete but not complete abelian category

We give some examples of abelian categories A for which the derived category D(A) is not left-complete. Perhaps the most natural of these is where A is the category of representations of the additive group G_a over a field k of…

We give an example of a finitely presented simple group containing a finitely generated subgroup which is not finitely presented.

This is mostly an overview. Given finitely presentable abelian categories $A$ and $B$, we sketch the construction of an abelian category of continuous functors from $A$ to $B$ that has nice $2$-categorical behaviour and gives an explicit…

The purpose of this article is to study the existence of Deligne's tensor product of abelian categories by comparing it with the well-known ten- sor product of finitely cocomplete categories. The main result states that the former exists…

We prove that an abelian category equipped with an ample sequence of objects is equivalent to the quotient of the category of coherent modules over the corresponding algebra by the subcategory of finite-dimensional modules. In the…

We construct a bounded plane domain which is Bergman complete but for which the Bergman kernel does not tend to infinity as the point approaches the boundary.

A completeness conjecture is advanced concerning the free small-colimit completion P(A) of a (possibly large) category A. The conjecture is based on the existence of a small generating-cogenerating set of objects in A. We sketch how the…

For any finite totally ordered set, the multisets of intervals form an abelian category. Various classes of subcategories admit natural combinatorial descriptions, and counting them yields familiar integer sequences. Surprisingly, in some…

For an abelian category, a category equivalent to its derived category is constructed by means of specific projective (injective) multicomplexes, the so-called homological resolutions.

Several important types of categories have been shown to be both exact and coexact (in the sense of Barr). The first type consists of abelian categories, which due to their self-dual definition, can be seen to be both exact and coexact by…

We demonstrate equivalence between two definitions of lower finite highest weight categories. We also show that, in the presence of a duality, a lower finite highest weight structure on a category is unique. Finally, we give a new proof for…

Let $G$ be a finite abelian group and $A$ be a subset of $G$. We say that $A$ is complete if every element of $G$ can be represented as a sum of different elements of $A$. In this paper, we study the following question: {\it What is the…

We prove that a category which is symmetric (relaxed) monoidal closed, (small) complete, well-powered and has a small cogenerating family, is cocomplete.

We prove that the derived categories of abelian categories have unique enhancements -- all of them, the unbounded, bounded, bounded above and bounded below derived categories. The unseparated and left completed derived categories of a…

The complete affine structures on abelian Lie algebras in small dimensions are well known. In this paper we are interested by the non complete case. In particular we classify all these structures in dimensions 2 and 3.

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…

We construct 3-manifolds which have at least two inequivalent embeddings such that both complementary regions have abelian fundamental group.

The goal of this note is to spell out the (apparently well-known and intuitively clear) notion of abelian category over an algebraic stack. In the future we will discuss the (much less evident) notion, when instead of an abelian category…

We consider three (2-)categories and their (anti-)equivalence. They are the category of small abelian categories and exact functors, the category of definable additive categories and interpretation functors, the category of locally coherent…

This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented…