Related papers: Notes on Triangulated Categories
The Popescu-Gabriel theorem states that each Grothendieck abelian category is a localization of a module category. In this paper, we prove an analogue where Grothendieck abelian categories are replaced by triangulated categories which are…
The aim of this paper is to develop a framework for localization theory of triangulated categories $\mathcal{C}$, that is, from a given extension-closed subcategory $\mathcal{N}$ of $\mathcal{C}$, we construct a natural extriangulated…
The aim of these notes is to provide a succinct, accessible introduction to some of the basic ideas of category theory and categorical logic. The notes are based on a lecture course given at Oxford over the past few years. They contain…
In this paper, we explore when a locally finite triangulated category has dimension zero or finite representation type. We also study generation of derived categories by orthogonal subcategories.
We review the concept of a graded bundle as a natural generalisation of a vector bundle. Such geometries are particularly nice examples of more general graded manifolds. With hindsight there are many examples of graded bundles that appear…
We use a $K$-theory recipe of Thomason to obtain classifications of triangulated subcategories via refining some standard thick subcategory theorems. We apply this recipe to the full subcategories of finite objects in the derived categories…
We explicitly present homological residue fields for tensor triangulated categories as categories of comodules in a number of examples across algebra, geometry, and topology. Our results indicate that, despite their abstract nature, they…
We show how one can do algebraic geometry with respect to the category of simplicial objects in an exact category. As a biproduct, we get a theory of derived analytic geometry.
We introduce abelian framed bicategories, which are particular framed bicategories that are locally abelian, and show that they are suitable for developing homology and cohomology theories for directed structures. This means in particular…
This paper surveys some recent results, concerning the intrinsicness of natural subcategories of weakly approximable triangulated categories. We also review the results about uniqueness of enhancements of triangulated categories, with the…
We generalise the notions of good, middling good, and Verdier good morphisms of distinguished triangles in triangulated categories, first introduced by Neeman, to the setting of $n$-angulated categories, introduced in Geiss, Keller, and…
Given an essentially small triangulated category it is possible to give a metric on it, to complete it with respect to the metric, and to look at the subcategory of objects in the completion which are compactly supported with respect to the…
We introduce Toda brackets for n-angulated categories and show that the various definitions of Toda brackets coincide. We prove juggling formulas for these Toda brackets generalizing the triangulated case. Following that, we generalize a…
We show that, with some technical conditions, an abelian category can be embedded into the category of bimodules over a ring. The case of semisimple rigid monoidal categories is studied in more detail.
Homological algebra is often understood as the translator between the world of topology and algebra. However, this branch of mathematics is worth studying by itself, given that it provides fascinating perspectives about other disciplines,…
Motivated by the problem of classifying quantum symmetries of non-semisimple, finite-dimensional associative algebras, we define a notion of connection between bounded quivers and build a bicategory of bounded quivers and quiver…
In this survey we explain the results of the recent article arXiv:1806.06471. Following a 1973 article by Lawvere one can define metrics on categories, and following Kelly's 1982 book one can complete a category with respect to its metric.…
In [1] we introduced the concept of structured space, which is a topological space that locally resembles some algebraic structures. In [2] we proceeded the study of these spaces, developing two cohomology theories. The aim of this paper is…
We prove new Brown representability theorems for triangulated categories using metric techniques as introduced in the work of Neeman. In the setting of algebraic geometry, this gives us new representability theorems for homological and…
We introduce the notion of a definable category--a category equivalent to a full subcategory of a locally finitely presentable category that is closed under products, directed colimits and pure subobjects. Definable subcategories are…