Related papers: Categorical joins
We introduce the notion of a categorical cone, which provides a categorification of the classical cone over a projective variety, and use our work on categorical joins to describe its behavior under homological projective duality. In…
Classically, the projective duality between joins of varieties and the intersections of varieties only holds in good cases. In this paper, we show that categorically, the duality between joins and intersections holds in the framework of…
We introduce a notion of Homological Projective Duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties $X$ and $Y$ in dual projective spaces are…
The notion of a duality between two derived functors as well as an extension theorem for derived functors to larger categories in which they need not be defined is introduced. These ideas are then applied to extend and study the coext…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of the desired objects, but also…
In this work, we establish a categorification of the classical Dold-Kan correspondence in the form of an equivalence between suitably defined $\infty$-categories of simplicial stable $\infty$-categories and connective chain complexes of…
An isomorphism between two hermitian unitals is proved, and used to treat isomorphisms of classical groups that are related to the isomorphism between certain simple real Lie algebras of types A and D (and rank 3).
It is known that a countable $\omega$-categorical structure interprets all finite structures primitively positively if and only if its polymorphism clone maps to the clone of projections on a two-element set via a continuous clone…
For an abelian category, a category equivalent to its derived category is constructed by means of specific projective (injective) multicomplexes, the so-called homological resolutions.
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…
We introduce a notion of join for (augmented) simplicial sets generalising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial category $\Delta$.
Following ideas of Lawvere and Linton we prove that classical varieties are precisely the exact categories with a varietal generator. This means a strong generator which is abstractly finite and regularly projective. An analogous…
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…
We construct of the main object of the Partite Lemma as the colimit over a certain diagram. This gives a purely category theoretic take on the Partite Lemma and establishes the canonicity of the object. Additionally, the categorical point…
We prove a categorical duality between a class of abstract algebras of partial functions and a class of (small) topological categories. The algebras are the isomorphs of collections of partial functions closed under the operations of…
A notion of stratification is introduced for any compactly generated triangulated category T endowed with an action of a graded commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences…
Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades,…
Many types of categorical structure obey the following principle: the natural notion of equivalence is generated, as an equivalence relation, by identifying $A$ with $B$ when there exists a strictly structure-preserving map $A \to B$ that…
Category theory has become central to certain aspects of theoretical physics. Bain [Synthese, 190:1621--1635 (2013)] has recently argued that this has significance for ontic structural realism. We argue against this claim. In so doing, we…