Related papers: Some remarks on phantom categories and motives

We study the Gray index of phantom maps, which is a numerical invariant of phantom maps. It is conjectured that the only phantom map with infinite Gray index between finite-type spaces is the constant map. We disprove this conjecture by…

We use chain level genus zero Gromov-Witten theory to associate to any closed monotone symplectic manifold a formal group (loosely interpreted), whose Lie algebra is the odd degree cohomology of the manifold (with vanishing bracket). When…

For the cluster category of a hereditary or a canonical algebra, equivalently for the cluster category of the hereditary category of coherent sheaves on a weighted projective line, we study the Grothendieck group with respect to an…

We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties, and that if I has these properties, then so does each of its powers. We show how…

We develop criteria for deciding the contravariant finiteness status of a subcategory $A \subseteq \Lambda\text{-mod}$, where $\Lambda$ is a finite dimensional algebra. In particular, given a finite dimensional $\Lambda$-module $X$, we…

In this article we introduce and study a motivic category in the arithmetic of function fields, namely the category of motives over an algebraic closure $L$ of a finite field with coefficients in a global function field over this finite…

Differential categories provide an axiomatization of the basics of differentiation and categorical models of differential linear logic. As differentiation is an important tool throughout quantum mechanics and quantum information, it makes…

We give a proof of the Andersen-Haboush identity that implies Kempf's vanishing theorem. Our argument is based on the structure of derived categories of coherent sheaves on flag varieties over $\mathbb Z$.

We introduce the category of finite strings and study its basic properties. The category is closely related to the augmented simplex category, and it models categories of linear representations. Each lattice of non-crossing partitions…

We show that very general noncommutative projective planes do not admit phantom categories.

We establish and advocate for a novel branch of category theory, centered around strong dinatural transformations (herein known as "paranatural transformations"). Paranatural transformations generalize natural transformations to…

Voevodsky's derived category of motives is the main arena today for the study of algebraic cycles and motivic cohomology. In this paper we study whether the inclusions of three important subcategories of motives have a left or right…

We introduce in this work the notion of the category of pure $\mathbf{E}$-Motives, where $\mathbf{E}$ is a motivic strict ring spectrum and construct twisted $\mathbf{E}$-cohomology by using six functors formalism of J. Ayoub. In…

We explain the sense in which a warping on a monoidal category is the same as a pseudomonad on the corresponding one-object bicategory, and we describe extensions of this to the setting of skew monoidal categories: these are a…

We define duality triples and duality pairs in compactly generated triangulated categories and investigate their properties. This enables us to give an elementary way to determine whether a class is closed under pure subobjects, pure…

The concept of category from mathematics happens to be useful to computer programmers in many ways. Unfortunately, all "good" explanations of categories so far have been designed by mathematicians, or at least theoreticians with a strong…

We define a symmetric monoidal category modelling fuzzy concepts and fuzzy conceptual reasoning within G\"ardenfors' framework of conceptual (convex) spaces. We propose log-concave functions as models of fuzzy concepts, showing that these…

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…

We prove a few results concerning the notions of finite dimensionality of mixed Tate motives in the sense of Kimura and O'Sullivan. It is shown that being oddly or evenly finite dimensional is equivalent to vanishing of certain cohomology…

In the first part, we determine conditions on spectra X and Y under which either every map from X to Y is phantom, or no nonzero maps are. We also address the question of whether such all or nothing behaviour is preserved when X is replaced…