Related papers: Accessible categories, set theory, and model theor…
We present several new model-theoretic applications of the fact that, under the assumption that there exists a proper class of almost strongly compact cardinals, the powerful image of any accessible functor is accessible. In particular, we…
The notion of a simplicial set originated in algebraic topology, and has also been utilized extensively in category theory, but until relatively recently was not used outside of those fields. However, with the increasing prominence of…
There exists a dispute in philosophy, going back at least to Leibniz, whether is it possible to view the world as a network of relations and relations between relations with the role of objects, between which these relations hold, entirely…
An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is…
A new approach is suggested to characterize algebraically automorphisms of the category of free algebras of a given variety. It gives in many cases an answer to the problem set by the first of authors, if automorphisms of such a category…
In this paper we study compact closed categories within the context of homotopical algebra. We construct two new model category structures by localizing two (Quillen equivalent) model categories of symmetric monoidal categories with the…
We introduce a new class of categories generalizing locally presentable ones. The distinction does not manifest in the abelian case and, assuming Vopenka's principle, the same happens in the regular case. The category of complete partial…
Let $\kappa$ be a regular cardinal, $\lambda<\kappa$ be a smaller infinite cardinal, and $\mathsf K$ be a $\kappa$-accessible category where colimits of $\lambda$-indexed chains exist. We show that various category-theoretic constructions…
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…
Our aim is to give a fairly complete account on the construction of compatible model structures on exact categories and symmetric monoidal exact categories, in some cases generalizing previously known results. We describe the close…
We use a category-theoretic formulation of Aczel's Fullness Axiom from Constructive Set Theory to derive the local cartesian closure of an exact completion. As an application, we prove that such a formulation is valid in the homotopy…
We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is…
We show that both the $\infty$-category of $(\infty, \infty)$-categories with inductively defined equivalences, and with coinductively defined equivalences, satisfy universal properties with respect to weak enrichment in the sense of Gepner…
A new approach to the construction of general persistent polyhierarchical classifications is proposed. It is based on implicit description of category polyhierarchy by a generating polyhierarchy of classification criteria. Similarly to…
The categoricity spectrum of a class of structures is the collection of cardinals in which the class has a single model up to isomorphism. Assuming that cardinal exponentiation is injective (a weakening of the generalized continuum…
In the context of categories equipped with a structure of nullhomotopies, we introduce the notion of homotopy torsion theory. As special cases, we recover pretorsion theories as well as torsion theories in multi-pointed categories and in…
We study multidimensional diagrams in independent amalgamation in the framework of abstract elementary classes (AECs). We use them to prove the eventual categoricity conjecture for AECs, assuming a large cardinal axiom. More precisely, we…
In this note we provide a characterization, in terms of additional algebraic structure, of those intervals (certain cocategory objects) in a symmetric monoidal closed category E that are representable in the sense of inducing on E the…
We rewrite classical topological definitions using the category-theoretic notation of arrows and are led to concise reformulations in terms of simplicial categories and orthogonality of morphisms, which we hope might be of use in the…
We study countable embedding-universal and homomorphism-universal structures and unify results related to both of these notions. We show that many universal and ultrahomogeneous structures allow a concise description (called here a finite…