Related papers: Enhanced finite triangulated categories
A finite pre-tensor category is a finite abelian category equipped with a right exact tensor product for which every projective object has duals. Finite tensor categories, for which every object has duals, are notable examples. More…
We develop Morita theory for finitary additive 2-representations of finitary 2-categories. As an application we describe Morita equivalence classes for 2-categories of projective functors associated to finite dimensional algebras and for…
Graduated locally finitely presentable categories are introduced, examples include categories of sets, vector spaces, posets, presheaves and Boolean algebras. A finitary functor between graduated locally finitely presentable categories is…
These notes are meant to provide a rapid introduction to triangulated categories. We start with the definition of an additive category and end with a glimps of tilting theory. Some exercises are included.
In this paper, we study metric completions of triangulated categories in a representation-theoretic context. We provide a concrete description of completions of bounded derived categories of hereditary finite dimensional algebras of finite…
In this paper we relate triangulated category structures to the cohomology of small categories and define initial obstructions to the existence of an algebraic or topological enhancement. We show that these obstructions do not vanish in an…
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…
This is the first of a series of papers on enriched infinity categories, seeking to reduce enriched higher category theory to the higher algebra of presentable infinity categories, which is better understood and can be approached via…
The basic notions of category theory, such as limit, adjunction, and orthogonality, all involve assertions of the existence and uniqueness of certain arrows. Weak notions arise when one drops the uniqueness requirement and asks only for…
In a triangulated symmetric monoidal closed category, there are natural dualities induced by the internal Hom. Given a monoidal functor f^* between two such catgories and adjoint couples (f^*,f_*) and (f_*,f^!), we prove the necessary…
In this paper we present background results in enriched category theory and enriched model category theory necessary for developing model categories of enriched functors suitable for doing functor calculus.
For a perfect field $k$, we construct a triangulated category of mixed motives over $k[t]/{(t^{m+1})}$. The ext groups in this category are given by higher Chow groups, and additive higher Chow groups.
In this article, we investigate the category $\mathcal{A}^G$ of equivariant objects of an additive category $\mathcal{A}$ with respect to an action of a finite group $G$. We show that if $G$ is solvable then we can reconstruct $\mathcal{A}$…
We characterize virtual double categories of enriched categories, functors, and profunctors by introducing a new notion of double-categorical colimits. Our characterization is strict in the sense that it is up to equivalence between virtual…
The additivity of traces in certain tensor triangulated categories for endomorphisms of finite order of distinguished triangles is investigated. For the identity endomorphism this has been fully established by J. P. May ("The additivity of…
This paper is devoted to establishing an enhanced Fritz John type first-order necessary condition for a general constrained nonlinear infinite-dimensional optimization problem. Unlike traditional constraint qualifications in optimization…
We propose a new look on triangulated categories, which is based on the second Hochschild cohomology.
We define mutation pair in a pseudo-triangulated category. We prove that under certain conditions, for a mutation pair in a pseudo-triangulated category, the corresponding quotient category carries a natural triangulated structure. This…
The extension dimensions of an Artin algebra give a reasonable way of measuring how far an algebra is from being representation-finite. In this paper we mainly study extension dimensions linked by recollements of derived module categories…
We initiate the study of derived functors in the setting of extriangulated categories. By using coends, we adapt Yoneda's theory of higher extensions to this framework. We show that, when there are enough projectives or enough injectives,…