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Search: MSC category 18D10 ( Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] )

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1. CMB Online first

Hassanzadeh, Mohammad; Khalkhali, Masoud; Shapiro, Ilya
Monoidal Categories, 2-Traces, and Cyclic Cohomology
In this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category $(\mathcal{C}, \otimes)$ endowed with a symmetric $2$-trace, i.e. an $F\in Fun(\mathcal{C}, \operatorname{Vec})$ satisfying some natural trace-like conditions, one can attach a cyclic (resp.cocyclic) module, and therefore speak of the (co)cyclic homology of the (co)algebra ``with coefficients in $F$". Furthermore, we observe that if $\mathcal{M}$ is a $\mathcal{C}$-bimodule category and $(F, M)$ is a stable central pair, i.e., $F\in Fun(\mathcal{M}, \operatorname{Vec})$ and $M\in \mathcal{M}$ satisfy certain conditions, then $\mathcal{C}$ acquires a symmetric 2-trace. The dual notions of symmetric $2$-contratraces and stable central contrapairs are derived as well. As an application we can recover all Hopf cyclic type (co)homology theories.

Keywords:monoidal category, abelian and additive category, cyclic homology, Hopf algebra
Categories:16T05, 18D10, 19D55

2. CMB 2014 (vol 57 pp. 721)

Bruillard, Paul; Galindo, César; Hong, Seung-Moon; Kashina, Yevgenia; Naidu, Deepak; Natale, Sonia; Plavnik, Julia Yael; Rowell, Eric C.
Classification of Integral Modular Categories of Frobenius--Perron Dimension $pq^4$ and $p^2q^2$
We classify integral modular categories of dimension $pq^4$ and $p^2q^2$, where $p$ and $q$ are distinct primes. We show that such categories are always group-theoretical except for categories of dimension $4q^2$. In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara-Yamagami categories and quantum groups. We show that a non-group-theoretical integral modular category of dimension $4q^2$ is equivalent to either one of these well-known examples or is of dimension $36$ and is twist-equivalent to fusion categories arising from a certain quantum group.

Keywords:modular categories, fusion categories

3. CMB 2013 (vol 57 pp. 506)

Galindo, César
On Braided and Ribbon Unitary Fusion Categories
We prove that every braiding over a unitary fusion category is unitary and every unitary braided fusion category admits a unique unitary ribbon structure.

Keywords:fusion categories, braided categories, modular categories
Categories:20F36, 16W30, 18D10

4. CMB 2004 (vol 47 pp. 321)

Bullejos, M.; Cegarra, A. M.
Classifying Spaces for Monoidal Categories Through Geometric Nerves
The usual constructions of classifying spaces for monoidal categories produce CW-complexes with many cells that, moreover, do not have any proper geometric meaning. However, geometric nerves of monoidal categories are very handy simplicial sets whose simplices have a pleasing geometric description: they are diagrams with the shape of the 2-skeleton of oriented standard simplices. The purpose of this paper is to prove that geometric realizations of geometric nerves are classifying spaces for monoidal categories.

Keywords:monoidal category, pseudo-simplicial category,, simplicial set, classifying space, homotopy type
Categories:18D10, 18G30, 55P15, 55P35, 55U40

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