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This is essentially an illustration for the general technology of homotopical enhancements developed recently in arxiv:2409.17489. We take the derived category of an abelian category, and we look at the full subcategory spanned by complexes…

In this paper we introduce $n\mathbb{Z}$-abelian and $n\mathbb{Z}$-exact categories by axiomatising properties of $n\mathbb{Z}$-cluster tilting subcategories. We study this categories and show that every $n\mathbb{Z}$-cluster tilting…

To do homological algebra with unbounded chain complexes one needs to first find a way of constructing resolutions. Spaltenstein solved this problem for chain complexes of R-modules by truncating further and further to the left, resolving…

The work is devoted to the extension groups in the category of functors from a small category to an additive category with an Abelian structure in the sense of Heller. It is constructed a spectral sequence which converges to the extension…

To characterize categorical constraints - associativity, commutativity and monoidality - in the context of quasimonoidal categories, from a cohomological point of view, we define the notion of a parity (quasi)complex. Applied to groups…

For a Serre subcategory $\mathscr L$ and a resolving subcategory $\mathscr A$ of an abelian category, we show that the derived equivalence $D^b(\overline{\mathscr A} \cap \mathscr L) \simeq D^b_{\mathscr L}(\mathscr A)$ holds under certain…

Let $m$ be a positive integer and $D_m(\mathcal {A})$ be the $m$-periodic derived category of a finitary hereditary abelian category $\mathcal {A}$. Applying the derived Hall numbers of the bounded derived category $D^b(\mathcal {A})$, we…

Let A be an exact category, that is, an extension-closed full subcategory of an abelian category. Firstly, we give some necessary and sufficient conditions for A to have almost split sequences. Then, we study when an almost split sequence…

We develop a general theory of cosimplicial resolutions, homotopy spectral sequences, and completions for objects in model categories, extending work of Bousfield-Kan and Bendersky-Thompson for ordinary spaces. This is based on a…

We extend the homotopy theories based on point reduction for finite spaces and simplicial complexes to finite acyclic categories and $\Delta$-complexes, respectively. The functors of classifying spaces and face posets are compatible with…

Suppose that $\mathcal{A}$ is an abelian category whose derived category $\mathcal{D}(\mathcal{A})$ has $Hom$ sets and arbitrary (small) coproducts, let $T$ be a (not necessarily classical) ($n$-)tilting object of $\mathcal{A}$ and let…

This paper is devoted to the comparison of different localized categories of differential complexes. The first result is that the canonical functor from the category of complexes of differential operators of order one (defined by Herrera…

We introduce signed exceptional sequences as factorizations of morphisms in the cluster morphism category. The objects of this category are wide subcategories of the module category of a hereditary algebra. A morphism $[T]:\mathcal A\to…

We develop some new aspects of cohomology in the context of semi-abelian categories: we establish a Hochschild-Serre 5-term exact sequence extending the classical one for groups and Lie algebras; we prove that an object is perfect if and…

In this work we propose a realization of Lurie's prediction that inner fibrations $p: X \rightarrow A$ are classified by $A$-indexed diagrams in a ``higher category" whose objects are $\infty$-categories, morphisms are correspondences…

We construct a surjective homomorphism from Somekawa's K-group associated to a finite collection of semi-abelian varieties over a perfect field to a corresponding Hom group in Voevodsky's triangulated category of effective motivic…

$N$-complexes have been argued recently to be algebraic structures relevant to the description of higher spin gauge fields. $N$-complexes involve a linear operator $d$ that fulfills $d^N = 0$ and that defines a generalized cohomology. Some…

In this work, we provide a simple way to construct $d$-abelian categories via bounded derived categories for certain values of $d$. Namely, let ${\mathcal C}$ be an abelian category, and let ${\mathcal C}[0,m]$ denote the full subcategory…

Derived categories were invented by Grothendieck and Verdier around 1960, not very long after the "old" homological algebra (of derived functors between abelian categories) was established. This "new" homological algebra, of derived…

Given a tensor functor between tensor categories $\mathcal{C}$ and $\mathcal{D}$, we give criteria that, under certain assumptions, the Frobeniusness of $\mathcal{C}$ or $\mathcal{D}$ implies the Frobeniusness of the other one. We also give…