Related papers: Classification of Module Categories for $SO(3)_{2m…

Let $k$ be an algebraically closed field of characteristic $p\ge 0$. Let $G$ be an affine group scheme over $k$. We classify the indecomposable exact module categories over the rigid tensor category $\text{Coh}_f(G)$ of coherent sheaves of…

Torsion pairs in the category of finitely presented modules over a noetherian ring can be parametrised by the class of cosilting modules. In this paper, we characterise such modules in terms of their indecomposable summands, providing a new…

In this paper we study the indecomposable module categories over $\mathcal{C}(\mathfrak{sl}_N, k)$, the category of integrable level-$k$ respresentations of affine Kac-Moody $\mathfrak{sl}_N$. Our first main result classifies these module…

We present a construction of a 2-Hilbert space of sections of a bundle gerbe, a suitable candidate for a prequantum 2-Hilbert space in higher geometric quantisation. We introduce a direct sum on the morphism categories in the 2-category of…

This paper proposes a new, visual method to study numerical semigroups and the Frobenius problem. The method is based on building a so-called reduction graph, whose nodes usually correspond to monogenic semigroups, and whose edges can have…

The fusion rules and braiding statistics of anyons in $(2+1)$D fermionic topological orders are characterized by the modular data of a super-modular category. On the other hand, the modular data of a super-modular category form a congruence…

We develop abstract nonsense for module categories over monoidal categories (this is a straightforward categorification of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects…

The aim of this paper is to unify classification theories of torsion classes of finite dimensional algebras and commutative Noetherian rings. For a commutative Noetherian ring $R$ and a module-finite $R$-algebra $\Lambda$, we study the set…

We establish a correspondence between modules and spans of algebras within a general monoidal 2-category $\mathfrak{C}$. Specifically, for an algebra $A$ in $\mathfrak{C}$, we construct a normalized lax 3-functor from the 2-category of…

This article develops several main results for a general theory of homological algebra in categories such as the category of sheaves of idempotent modules over a topos. In the analogy with the development of homological algebra for abelian…

We develop some techniques to the study of exact module categories over some families of pointed finite-dimensional Hopf algebras. As an application we classify exact module categories over the tensor category of representations of the…

In this paper, we classify certain subcategories of modules over a ring R. A wide subcategory of R-modules is an Abelian subcategory of R-Mod that is closed under extensions. We give a complete classification of wide subcategories of…

We discuss algebraic and representation theoretic structures in braided tensor categories C which obey certain finiteness conditions. Much interesting structure of such a category is encoded in a Hopf algebra H in C. In particular, the Hopf…

For any 0-cell $B$ in a 2-category $\Bc$ we introduce the notion of adjoint algebra $\adj_B$. This is an algebra in the center of $\Bc$. We prove that, if $\ca$ is a finite tensor category, this notion applied to the 2-category of…

Let $\CC^0_{\g}$ be the category of finite-dimensional integrable modules over the quantum affine algebra $U_{q}'(\g)$ and let $R^{A_\infty}\gmod$ denote the category of finite-dimensional graded modules over the quiver Hecke algebra of…

This is the third part in a series of papers developing a tensor product theory for modules for a vertex operator algebra. The goal of this theory is to construct a ``vertex tensor category'' structure on the category of modules for a…

We develop some foundations of commutative algebra, with a view towards algebraic geometry, in symmetric tensor categories. Most results establish analogues of classical theorems, in tensor categories which admit a tensor functor to some…

We generalize the construction of tensor categories of endomorphisms of a type III factor $M$ associated with a $G$-kernel, from the case of a discrete group $G$ to that of a compact second countable group. Our approach is based on the…

We give an algorithm to compute representatives of the conjugacy classes of semisimple square integral matrices with given minimal and characteristic polynomials. We also give an algorithm to compute the $\mathbb{F}_q$-isomorphism classes…

Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or…