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Register a new userPivotal tricategories and a categorification of inner-product modules
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Gregor Schaumann
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This article investigates duals for bimodule categories over finite tensor categories. We show that finite bimodule categories form a tricategory and discuss the dualities in this tricategory using inner homs. We consider inner-product bimodule categories over pivotal tensor categories with additional structure on the inner homs. Inner-product module categories are related to Frobenius algebras and lead to the notion of $*$-Morita equivalence for pivotal tensor categories. We show that inner-product bimodule categories form a tricategory with two duality operations and an additional pivotal structure. This is work is motivated by defects in topological field theories.
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We develop the rewriting theory for monoidal supercategories and 2-supercategories. This extends the theory of higher-dimensional rewriting established for (linear) 2-categories to the super setting, providing a suite of tools for constructing bases and normal forms for 2-supercategories given by generators and relations. We then employ this newly developed theory to prove the non-degeneracy conjecture for the odd categorification of quantum sl(2) from arXiv:1307.7816 and arXiv:1701.04133. As a corollary, this gives a classification of dg-structures on the odd 2-category conjectured in arXiv:1808.04924.
We show any slightly degenerate weakly group-theoretical fusion category admits a minimal extension. Let $d$ be a positive square-free integer, given a weakly group-theoretical non-degenerate fusion category $mathcal{C}$, assume that $text{FPdim}(mathcal{C})=nd$ and $(n,d)=1$. If $(text{FPdim}(X)^2,d)=1$ for all simple objects $X$ of $mathcal{C}$, then we show that $mathcal{C}$ contains a non-degenerate fusion subcategory $mathcal{C}(mathbb{Z}_d,q)$. In particular, we obtain that integral fusion categories of FP-dimensions $p^md$ such that $mathcal{C}subseteq text{sVec}$ are nilpotent and group-theoretical, where $p$ is a prime and $(p,d)=1$.
Let $mathscr{C}$ be the category of finite dimensional modules over the quantum affine algebra $U_q(widehat{mathfrak{g}})$ of a simple complex Lie algebra ${mathfrak{g}}$. Let $mathscr{C}^-$ be the subcategory introduced by Hernandez and Leclerc. We prove the geometric $q$-character formula conjectured by Hernandez and Leclerc in types $mathbb{A}$ and $mathbb{B}$ for a class of simple modules called snake modules introduced by Mukhin and Young. Moreover, we give a combinatorial formula for the $F$-polynomial of the generic kernel associated to the snake module. As an application, we show that snake modules correspond to cluster monomials with square free denominators and we show that snake modules are real modules. We also show that the cluster algebras of the category $mathscr{C}_1$ are factorial for Dynkin types $mathbb{A,D,E}$.
This is the first part of a series of two papers aiming to construct a categorification of the braiding on tensor products of Verma modules, and in particular of the Lawrence--Krammer--Bigelow representations. In this part, we categorify all tensor products of Verma modules and integrable modules for quantum $mathfrak{sl_2}$. The categorification is given by derived categories of
In our recent papers [Sh1,2], we introduced a {it twisted tensor product} of dg categories, and provided, in terms of it, {it a contractible 2-operad $mathcal{O}$}, acting on the category of small dg categories, in which the natural transformations are derived. We made use of some homotopy theory developed in [To] to prove the contractibility of the 2-operad $mathcal{O}$. The contractibility is an important issue, in vein of the theory of Batanin [Ba1,2], according to which an action of a contractible $n$-operad on $C$ makes $C$ a weak $n$-category. In this short note, we provide a new elementary proof of the contractibility of the 2-operad $mathcal{O}$. The proof is based on a direct computation, and is independent from the homotopy theory of dg categories (in particular, it is independent from [To] and from Theorem 2.4 of [Sh1]).
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