Related papers: The index in $d$-exact categories

Cluster algebras are categorified by cluster categories, and $g$-vectors are categorified by the classic index with respect to cluster tilting subcategories. However, the recently introduced completed discrete cluster categories of Dynkin…

Palu defined the index with respect to a cluster tilting object in a suitable triangulated category, in order to better understand the Caldero-Chapoton map that exhibits the connection between cluster algebras and representation theory. We…

We prove that any weakly idempotent complete $d$-exact category is equivalent to a $d$-cluster tilting subcategory of a weakly idempotent complete exact category, and that any weakly idempotent complete algebraic $(d+2)$-angulated category…

We introduce the notion of an $n$-exact dg-category. This notion provides a higher analogue of Chen's exact dg-category, in the sense that the case where $n$ equals 1 recovers exact dg-categories. We prove that, under a suitable vanishing…

Index theory has had profound impact on many branches of mathematics. In this note we discuss the context for a new kind of index theorem. We begin, however, with some operator theoretic results. In [11] Berger and Shaw established that…

We introduce the notion of an exact dg category, which is a simultaneous generalization of the notions of exact category in the sense of Quillen and of pretriangulated dg category in the sense of Bondal--Kapranov. It is also a differential…

We introduce the notion of balanced pair of additive subcategories in an abelian category. We give sufficient conditions under which the balanced pair of subcategories gives rise to equivalent homotopy categories of complexes. As an…

Let $\mathscr{C}$ be a 2-Calabi-Yau triangulated category, and let $\mathscr{T}$ be a cluster tilting subcategory of $\mathscr{C}$. An important result from Dehy and Keller tells us that a rigid object $c \in \mathscr{C}$ is uniquely…

Connections between heaps of modules and (affine) modules over rings are explored. This leads to explicit, often constructive, descriptions of some categorical constructions and properties that are implicit in universal algebra and…

We construct an "almost involution" assigning a new DG-category to a given one, and use this construction to recover, say, the abelian category of graded modules over the graded ring $R^*$ from the DG-category of DG-modules over a DG-ring…

Quillen's Resolution Theorem in algebraic $K$-theory provides a powerful computational tool for calculating $K$-groups of exact categories. At the level of $K_0$, this result goes back to Grothendieck. In this article, we first establish an…

We study the category of $\mathbb{Z}$-indexed sequences over an abelian category and certain generalized homology functors for this category of sequences which are indexed by positive integers $a$ and $b$. By looking at the corresponding…

We consider three categories arising from the higher Auslander algebras of type $A$ in relation to $d$-dimensional cluster combinatorics: $d$-exact subcategory of the module category of $A^d_{n+1}$ generated by the $d$-cluster-tilting…

We study index theory on homogeneous spaces associated to an almost connected Lie group in terms of the topological aspect and the analytic aspect. On the topological aspect, we obtain a topological formula as a result of the Riemann-Roch…

Let $(\mathcal{A},\mathcal{E})$ be an exact category. We establish basic results that allow one to identify sub(bi)functors of $\operatorname{Ext}_{\mathcal{E}}(-,-)$ using additivity of numerical functions and restriction to subcategories.…

Given an additive category $\mathcal{C}$ and an integer $n\geqslant 2$. We form a new additive category $\mathcal{C}[\epsilon]^n$ consisting of objects $X$ in $\mathcal{C}$ equipped with an endomorphism $\epsilon_X$ satisfying…

We consider a notion of exact sequences in any -not necessarily exact- pointed category relative to a given (E;M)-factorization structure. We apply this notion to introduce and investigate a new notion of exact sequences of semimodules over…

In this note we prove that the set of all uniformly continuous units on a product system over a C* algebra B can be endowed with the structure of left right B - B Hilbert module after identifying similar units by the suitable equivalence…

Category theory is the language of homological algebra, allowing us to state broadly applicable theorems and results without needing to specify the details for every instance of analogous objects. However, authors often stray from the realm…

Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed…