Related papers: Locally coherent exact categories

The concept of an abelian DG-category, introduced by the first-named author in arXiv:2110.08237, unites the notions of abelian categories and (curved) DG-modules in a common framework. In this paper we consider coderived and contraderived…

We show that Quillen's small object argument works for exact categories under very mild conditions. This has immediate applications to cotorsion pairs and their relation to the existence of certain triangulated adjoint functors and model…

We continue our study of relatively divisible and relatively flat objects in exact categories in the sense of Quillen with several applications to exact structures on finitely accessible additive categories and module categories. We derive…

Using generalized enriched categories, in this paper we show that Rosick\'{y}'s proof of cartesian closedness of the exact completion of the category of topological spaces can be extended to a wide range of topological categories over…

One-sided exact categories appear naturally as instances of Grothendieck pretopologies. In an additive setting they are given by considering the one-sided part of Keller's axioms defining Quillen exact categories. We study one-sided exact…

We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair $(\mathcal{A},\mathcal{B})$ in an exact category $\mathcal{C}$, $\mathcal{A}$ coincides…

We introduce the notion of a "category with path objects", as a slight strengthening of Kenneth Brown's classic notion of a "category of fibrant objects". We develop the basic properties of such a category and its associated homotopy…

In the context of ideally exact categories, we introduce the notions of internal coherent action and internal ideal action that generalise different aspects of unital actions of rings and algebras. We prove that every ideal action is…

Many kinds of categorical structure require the existence of finite limits, of colimits of some specified type, and of "exactness" conditions between the finite limits and the specified colimits. Some examples are the notions of regular, or…

We prove an analogue of the Gabriel--Quillen embedding theorem for exact $\infty$-categories, giving rise to a presentable version of Klemenc's stable envelope of an exact $\infty$-category. Moreover, we construct a symmetric monoidal…

We generalize the concepts of locally presentable and accessible categories. Our framework includes such categories as small presheaves over large categories and ind-categories. This generalization is intended for applications in the…

For a category $\mathcal E$ with finite limits and well-behaved countable coproducts, we construct a model structure, called the effective model structure, on the category of simplicial objects in $\mathcal E$, generalising the Kan--Quillen…

In categorical realizability, it is common to construct categories of assemblies and categories of modest sets from applicative structures. These categories have structures corresponding to the structures of applicative structures. In the…

We introduce and describe the $2$-category $\mathsf{Grt}_{\flat}$ of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories $\boxtimes$ restricts nicely to…

An additive category in which each object has a Krull-Remak-Schmidt decomposition -- that is, a finite direct sum decomposition consisting of objects with local endomorphism rings -- is known as a Krull-Schmidt category. A Hom-finite…

We explain why the naive definition of a natural exact category structure on complete, separated topological vector spaces with linear topology fails. In particular, contrary to arXiv:0711.2527, the category of such topological vector…

We construct an exact completion for regular categories enriched in the cartesian closed category $\mathsf{Pos}$ of partially ordered sets and monotone functions by employing a suitable calculus of relations. We then characterize the…

Among cocomplete categories, the locally presentable ones can be defined as those with a strong generator consisting of presentable objects. Assuming Vop{\v{e}}nka's Principle, we prove that a cocomplete category is locally presentable iff…

We show that for a given exact category, there exists a bijection between semibricks (pairwise Hom-orthogonal set of bricks) and length wide subcategories (exact extension-closed length abelian subcategories). In particular, we show that a…

We study the question when a category of ind-objects is abelian. Our answer allows a further generalization of the notion of weakly Tannakian categories introduced by the author. As an application we show that, under suitable conditions,…