Related papers: Combinatorial categorical equivalences of Dold-Kan…

We show that torsion pairs in Krull--Schmidt abelian categories induce an equivalence between the subcategory of torsion-free objects admitting universal extensions to the torsion subcategory, and a quotient of the ext-orthogonal complement…

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…

We thoroughly treat several familiar and less familiar definitions and results concerning categories, functors and distributors enriched in a base quantaloid Q. In analogy with V-category theory we discuss such things as adjoint functors,…

The additivity theorem for derivateurs associated to complicial biWaldhausen categories is proved. Also, to any exact category in the sense of Quillen a K-theory space is associated. This K-theory is shown to satisfy the additivity,…

If a Quillen model category can be specified using a certain logical syntax (intuitively, ``is algebraic/combinatorial enough''), so that it can be defined in any category of sheaves, then the satisfaction of Quillen's axioms over any site…

We show that for the path algebra $A$ of an acyclic quiver, the singularity category of the derived category $\mathsf{D}^{\rm b}(\mathsf{mod}\,A)$ is triangle equivalent to the derived category of the functor category of…

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…

We exhibit a canonical equivalence between the hermitian $K$-theory (alias Grothendieck-Witt) spectrum of an exact form category and that of its derived Poincar\'e $\infty$-category, with no assumptions on the invertibility of $2$. Along…

In this paper, we study operations on functors in the category of abelian groups simplar to the derivation in the sense of Dold-Puppe. They are defined as derived limits of a functor applied to the relation subgroup over a category of free…

It is shown that every two-variable adjunction in categories enriched in a commutative quantale serves as a base for constructing Isbell adjunctions between functor categories, and Kan adjunctions are precisely Isbell adjunctions…

This expository article sets forth a self-contained and purely algebraic proof of a deep result of Quillen stating that the category of simplicial commutative algebras over a commutative ring is a model category. This is accomplished by…

Homology is characterized by the Eilenberg-Steenrod axioms. We define homology of higher categories via a categorical analogue of the Eilenberg-Steenrod axioms. We prove a categorical Dold-Kan correspondence, providing a combinatorial…

We propose a new description of Endofunctors of Module Categories, based upon a combinatorial category comprising finite sets and so-called mazes. Polynomial and numerical functors both find a natural interpretation in this frame-work.…

Auslander's formula shows that any abelian category C is equivalent to the category of coherent functors on C modulo the Serre subcategory of all effaceable functors. We establish a derived version of this equivalence. This amounts to…

Let $\mathcal{X}$ be a resolving and contravariantly finite subcategory of $\rm{mod}\mbox{-}\Lambda$, the category of finitely generated right $\Lambda$-modules. We associate to $\mathcal{X}$ the subcategory…

We give a description of simple functors taking finitely generated values, from a small additive category to the category of vector spaces over a field. This result is analogous to Steinberg's tensor product theorems in group representation…

We develop a theory of adjunctions in semigroup categories, i.e. monoidal categories without a unit object. We show that a rigid semigroup category is promonoidal, and thus one can naturally adjoin a unit object to it. This extends the…

We introduce the notion of algebraic fibrant objects in a general model category and establish a (combinatorial) model category structure on algebraic fibrant objects. Based on this construction we propose algebraic Kan complexes as an…

We introduce higher analogs for cleavages in the context of (Kan) simplicial fibrations. We apply them to obtain geometric models for representations up to homotopy of (higher) Lie groupoids. Concretely, we set an equivalence between…

Building on work of Derksen-Fei and Plamondon, we formulate a conjectural correspondence between additive and monoidal categorifications of cluster algebras, which reveals a new connection between the additive reachability conjecture and…