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This preprint contains a part of the results of our earlier preprint arXiv:0907.3335v2 presented in a form suitable for journal publication. It covers a construction of a 2-fold monoidal structure on the category of tetramodules, with all necessary definitions, and an overview of the results of R.Taillefer [Tai1,2] on tetramodules and the Gerstenhaber-Schack cohomology [GS] (formerly served as Appendix in arXiv:0907.3335v2), as well as a computation of the Gerstenhaber-Schack cohomology for the free commutative cocommutative bialgebra S(V), for a V is a vector space.
We show that the category of positive opetopes with contraction morphisms, i.e. all face maps and some degeneracies, forms a test category. The category of positive opetopic sets pOpeSet can be defined as a full subcategory of the category of polyg
In this paper, which is subsequent to our previous paper [PS] (but can be read independently from it), we continue our study of the closed model structure on the category $mathrm{Cat}_{mathrm{dgwu}}(Bbbk)$ of small weakly unital dg categories (in the
We study properties of the category of modules of an algebra object A in a tensor category C. We show that the module category inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a by-prod
We study abelian envelopes for pseudo-tensor categories with the property that every object in the envelope is a quotient of an object in the pseudo-tensor category. We establish an intrinsic criterion on pseudo-tensor categories for the existence of
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