Related papers: A guide to tensor-triangular classification

I wrote this book in a "do-it-yourself" style so that I give only a draft of tensor theory, which includes formulating definitions and theorems and giving basic ideas and formulas. All other work such as proving consistence of definitions,…

We investigate to what extent we can descend the classification of localizing, smashing and thick ideals in a presentably symmetric monoidal stable $\infty$-category $\mathscr{C}$ along a descendable commutative algebra $A$. We establish…

The paper aims to propose a suitable method in finding the solution of tensor complementarity problem. The tensor complementarity problem is a subclass of nonlinear complementarity problems for which the involved function is defined by a…

We describe indecomposable objects in Deligne's category $\underline{\operatorname{Re}}\!\operatorname{p}(O_\delta)$ and explain how to decompose their tensor products. We then classify thick ideals in…

We give a homotopy classification of the global defects in ordered media, and explain it via the example of biaxial nematic liquid crystals, i.e., systems where the order parameter space is the quotient of the $3$-sphere $S^3$ by the…

We give a classification of all quasitriangular structures and ribbon elements of $\mathcal{D}(G)$ explicitly in terms of group homomorphisms and central subgroups. This can equivalently be interpreted as an explicit description of all…

We consider embeddings of a finite complex in a sphere. We give a homotopy theoretic classification of such embeddings in a wide range.

We systematically develop a theory of stratification in the context of tensor triangular geometry and apply it to classify the localizing tensor-ideals of certain categories of spectral $G$-Mackey functors for all finite groups $G$. Our…

We develop a theory of descent and forms of tensor categories over arbitrary fields. We describe the general scheme of classification of such forms using algebraic and homotopical language, and give examples of explicit classification of…

We give an elementary introduction to the theory of triangulated categories covering their axioms, homological algebra in triangulated categories, triangulated subcategories, and Verdier localization. We try to use a minimal set of axioms…

Nozaki et.~al.\ gave a homotopy classification of the knotted defects of ordered media in three-dimensional space by considering continuous maps from complements of spatial graphs to the order parameter space modulo a certain equivalence…

The Hopf theorem states that homotopy classes of continuous maps from a closed connected oriented smooth $n$-manifold $M$ to the $n$-sphere are classified by their degree. Such a map is equivalent to a section of the trivial $n$-sphere…

We introduce a new type of categorical object called a \emph{hom-tensor category} and show that it provides the appropriate setting for modules over an arbitrary hom-bialgebra. Next we introduce the notion of \emph{hom-braided category} and…

For any essentially small triangulated category the centre of its lattice of thick subcategories is introduced; it is a spatial frame and yields a notion of central support. A relative version of this centre recovers the support theory for…

We define and study homotopy groups of cubical sets. To this end, we give four definitions of homotopy groups of a cubical set, prove that they are equivalent, and further that they agree with their topological analogues via the geometric…

The homotopy theory of topological defects is a powerful tool for organizing and unifying many ideas across a broad range of physical systems. Recently, experimental progress has been made in controlling and measuring colloidal inclusions…

This article investigates the homotopy theory of simplicial commutative algebras with a view to homological applications.

Exact indecomposable module categories over the tensor category of representations of Hopf algebras that are liftings of quantum linear spaces are classified.

We define the spectrum of a tensor triangulated category $K$ as the set of so-called prime ideals, endowed with a suitable topology. In this very generality, the spectrum is the universal space in which one can define supports for objects…

This paper is devoted to the classification of embeddings of higher dimensional manifolds. We study the case of embeddings $S^p\times S^q\to S^m$, which we call knotted tori. The set of knotted tori in the the space of sufficiently high…