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We construct a braid group action on a homotopy category of $p$-DG modules of a deformed Webster algebra.
We propose the notion of quasi-abelian third cohomology of crossed modules, generalizing Eilenberg and MacLane's abelian cohomology and Ospel's quasi-abelian cohomology, and classify crossed pointed categories in terms of it. We apply the…
We introduce \emph{flagged $(\infty,n)$-categories} and prove that they are equivalent to Segal sheaves on Joyal's category ${\mathbf\Theta}_n$. As such, flagged $(\infty,n)$-categories provide a model-independent formulation of Segal…
We define a family of the braid group representations via the action of the $R$-matrix (of the quasitriangular extension) of the restricted quantum $\mathfrak{sl}(2)$ on a tensor power of a simple projective module. This family is an…
We start from any small strict monoidal braided Ab-category and extend it to a monoidal nonstrict braided Ab-category which contains braided bialgebras. The objects of the original category turn out to be modules for these bialgebras
A near-group category is an additively semisimple category with a product such that all but one of the simple objects is invertible. We classify braided structures on near-group categories, and give explicit numerical formulas for their…
Tangent categories provide a categorical axiomatization of the tangent bundle. There are many interesting examples and applications of tangent categories in a variety of areas such as differential geometry, algebraic geometry, algebra, and…
Graded vector bundles over a given $\mathbb{Z}$-graded manifold can be defined in three different ways: certain sheaves of graded modules over the structure sheaf of the base graded manifold, finitely generated projective graded modules…
We give an alternate conception of string diagrams as labeled 1-dimensional oriented cobordisms, the operad of which we denote by Cob/O, where O is the set of string labels. The axioms of traced (symmetric monoidal) categories are fully…
Given certain intersection cohomology sheaves on a projective variety with a torus action, we relate the cohomology groups of their tensor product to the cohomology groups of the individual sheaves. We also prove a similar result in the…
In this paper the category of opposite brace triples is introduced in a general braided monoidal setting. Under cocommutativity, it is proved to be isomorphic to the category of Hopf braces. Furthermore, if one considers the subcategories…
We investigate the notion of associated graded coalgebra (algebra) of a bialgebra with respect to a subbialgebra (quotient bialgebra) and characterize those which are bialgebras of type one in the framework of abelian braided monoidal…
We construct a category of flat vector bundles on an elliptic curve. It arises in the representation theory of quantum affine algebras and carries meromorphic braided structure with singularities on the diagonal of the square of the curve.
Usually a name of the category is inherited from the name of objects. However more relevant for a category of objects and morphisms is an algebra of morphisms. Therefore we prefer to say a category of graphs if every morphism is a graph. In…
We define the orbit category for transitive topological groupoids and their equivariant CW-complexes. By using these constructions we define equivariant Bredon homology and cohomology for actions of transitive topological groupoids. We show…
In this work, we introduce the notion of a partial action of a group on a strict monoidal category. We propose, in the context of Monoidal categories, new constructions analogous to those existing for partial group actions over an algebra…
We classify various types of graded extensions of a finite braided tensor category $\cal B$ in terms of its $2$-categorical Picard groups. In particular, we prove that braided extensions of $\cal B$ by a finite group $A$ correspond to…
This is the second part of the paper. Results of the first part about crossed modules are applied here to study of quantum groups in braided categories. Correct cross product in the class of quantum braided groups is built. Criterion when…
We describe the category of homotopy coalgebras, concentrating on properties of relatively cofree homotopy coalgebras, morphisms and coderivations from an ordinary coalgebra to a relatively cofree homotopy coalgebra, morphisms and…
In this paper we investigate the categories of braided objects, algebras and bialgebras in a given monoidal category, some pairs of adjoint functors between them and their relations. In particular we construct a braided primitive functor…