Related papers: Classification of grouplike categories
The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A…
We classify the matrices M which correspond to finite categories
We present here definitions and constructions basic for the theory of monoidal and tensor categories. We provide references to the original sources, whenever possible. Group-theoretical categories are used as examples
We introduce a monoidal category whose morphisms are finite partial orders, with chosen minimal and maximal elements as source and target respectively. After recalling the notion of presentation of a monoidal category by the means of…
This is the first part of a series of three strongly related papers in which three equivalent structures are studied: - internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans -…
The main purpose of this paper is to introduce the structure of soft group category. In this category, we determine some special objects and morphisms having a universal structure such as the final object and product. Therefore, the…
We introduce and investigate the category $\mathsf{AtoMon}$ of atomic monoids and atom-preserving monoid homomorphisms, which is a (non-full) subcategory of the usual category of monoids. In particular, we compute all limits and colimits,…
In this paper, we state and prove precise theorems on the classification of the category of (braided) categorical groups and their (braided) monoidal functors, and some applications obtained from the basic studies on monoidal functors…
We explore the category of internal categories in the usual category of (right) group-sets, whose objects are referred to as categorified group-sets. More precisely, we develop a new Burnside theory, where the equivalence relation between…
We define a notion of grading of a monoid T in a monoidal category C, relative to a class of morphisms M (which provide a notion of M-subobject). We show that, under reasonable conditions (including that M forms a factorization system),…
A variety of algebras is called limit if it is non-finitely based but all its proper subvarieties are finitely based. A monoid is aperiodic if all its subgroups are trivial. We classify all limit varieties of aperiodic monoids with…
The article presents a complete classification of $n$-valued monoids and groups of order 3. Important corollaries of this result are discussed.
In this article we introduce and study a class of finite groups for which the orders of normal subgroups satisfy a certain inequality. It is closely connected to some well-known arithmetic classes of natural numbers.
We define and study the Picard group of a monoid scheme and the class group of a normal monoid scheme. To do so, we develop some ideal theory for (pointed abelian) noetherian monoids, including primary decomposition and discrete valuations.…
In this paper, we define indexed type theories which are related to indexed ($\infty$-)categories in the same way as (homotopy) type theories are related to ($\infty$-)categories. We define several standard constructions for such theories…
We study a class of finite groups, called almost monomial groups, which generalize the class of monomial groups and it is connected with the theory of Artin L-functions. Our method of research is based on finding similarities with the…
The main objective of the paper is to define the category of monoids as a weighted limit. We also define the category of actions of monoids along the action of a monoidal category as a weighted limit.
Bimonoidal categories are categorical analogues of rings without additive inverses. They have been actively studied in category theory, homotopy theory, and algebraic $K$-theory since around 1970. There is an abundance of new applications…
It is known that monoidal categories have a finite definition, whereas multicategories have an infinite (albeit finitary) definition. Since monoidal categories correspond to representable multicategories, it goes without saying that…
Pursuing a generalization of group symmetries of modular categories to category symmetries in topological phases of matter, we study linear Hopf monads. The main goal is a generalization of extension and gauging group symmetries to category…