Related papers: Derived categories for the working mathematician
We prove descent theorems for semiorthogonal decompositions using techniques from derived algebraic geometry. Our methods allow us to capture more general filtrations of derived categories and even marked filtrations, where one descends not…
Classical homological algebra considers chain complexes, resolutions, and derived functors in additive categories. We describe "track algebras in dimension n", which generalize additive categories, and we define higher order chain…
In this note, I define a notion of a compactly supported object in a triangulated category. I prove a number of propositions relating this to traditional notions of support and give an application to the theory of derived Morita…
We consider limits over categories of extensions and show how certain well-known functors on the category of groups turn out as such limits. We also discuss higher (or derived) limits over categories of extensions.
Cluster categories of hereditary algebras have been introduced as orbit categories of their derived categories. Keller has pointed out that for non-hereditary algebras orbit categories need not be triangulated, and he introduced the notion…
We unravel a deep connection between limits of real numbers and limits in category theory. Using a new variant of the classical characterisation of the real numbers, we characterise the category of finite-dimensional Hilbert spaces and…
The description of algebraic structure of n-fold loop spaces can be done either using the formalism of topological operads, or using variations of Segal's $\Gamma$-spaces. The formalism of topological operads generalises well to different…
This note is supposed to answer some questions on deformation theory in derived algebraic geometry. We show that derived algebraic geometry allows for a geometrical interpretation of the full cotangent complex and gives a natural setting…
We introduce and develop the notion of *displayed categories*. A displayed category over a category C is equivalent to "a category D and functor F : D --> C", but instead of having a single collection of "objects of D" with a map to the…
We define the Hall algebra associated to any triangulated category under some finiteness conditions with the $t$-periodic translation functor $T$ for odd $t>1.$ This generalizes the results in \cite{Toen2005} and \cite{XX2006}.
We give a simultaneous generalization of exact categories and triangulated categories, which is suitable for considering cotorsion pairs, and which we call extriangulated categories. Extension-closed, full subcategories of triangulated…
In this paper we consider a construction in an arbitrary triangulated category T which resembles the notion of a Moore spectrum in algebraic topology. Namely, given a compact object C of T satisfying some finite tilting assumptions, we…
We investigate the homological behaviour of compactly generated triangulated categories under separable extensions. We show that homological invariants (finiteness of global dimension, gorensteinness and regularity) are preserved under such…
In Categorial Topology, given a category (as a "geometric object") we can consider its properties preserved under continuous action (a "deformation") of a comma-propagation operation. However, the Metacategory space, valid for all…
We use the abstract setting of excisive functors in the language of $\infty$-categories to show that the best approximation to the $\ell^1$-homology functor by an excisive functor is trivial. Then we make an effort to explain the used…
This paper establishes semiorthogonal decompositions for derived Grassmannians of perfect complexes with Tor-amplitude in $[0,1]$. This result verifies the author's Quot formula conjecture [J21a] and generalizes and strengthens Toda's…
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.
Category theory provides a means through which many far-ranging fields of mathematics can be related by their similar structure. In a paper by Robinson [2], this interconnectivity afforded by categorical perspectives allowed for the…
We define triangulated factorization systems on triangulated categories, and prove that a suitable subclass thereof (the normal triangulated torsion theories) corresponds bijectively to $t$-structures on the same category. This result is…
We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double category…