# The Picard crossed module of a braided tensor category

@article{Davydov2012ThePC,
title={The Picard crossed module of a braided tensor category},
author={Alexei Davydov and Dmitri Nikshych},
journal={arXiv: Quantum Algebra},
year={2012}
}
• Published 1 February 2012
• Mathematics, Psychology
• arXiv: Quantum Algebra
For a finite braided tensor category we introduce its Picard crossed module consisting of the group of invertible module categories and the group of braided tensor autoequivalences. We describe the Picard crossed module in terms of braided autoequivalences of the Drinfeld center of the braided tensor category. As an illustration, we compute the Picard crossed module of a braided pointed fusion category.
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## References

SHOWING 1-10 OF 31 REFERENCES
On braided fusion categories I
• Mathematics
• 2009
We introduce a new notion of the core of a braided fusion category. It allows to separate the part of a braided fusion category that does not come from finite groups. We also give a comprehensive and
On the Structure of Modular Categories
For a braided tensor category C and a subcategory K there is a notion of a centralizer CC K, which is a full tensor subcategory of C. A pre‐modular tensor category is known to be modular in the sense
Lectures on tensor categories and modular functors
• Mathematics
• 2000
Introduction Braided tensor categories Ribbon categories Modular tensor categories 3-dimensional topological quantum field theory Modular functor Moduli spaces and complex modular functor
Module categories, weak Hopf algebras and modular invariants
AbstractWe develop a theory of module categories over monoidal categories (this is a straightforward categorization of modules over rings). As applications we show that any semisimple monoidal
On braided tensor categories
We investigate invertible elements and gradings in braided tensor categories. This leads us to the definition of theta-, product-, subgrading and orbitcategories in order to construct new families of
Categorical Morita Equivalence for Group-Theoretical Categories
A finite tensor category is called pointed if all its simple objects are invertible. We find necessary and sufficient conditions for two pointed semisimple categories to be dual to each other with
The Brauer group of a braided monoidal category
• Mathematics
• 1998
Representation theory of finite groups prompted the introduction of the Brauer group of a field and also of its Schur subgroups. The theory of quadratic forms and spaces introduced the Zr2Z-graded
Module categories over the Drinfeld double of a finite group
Let C be a (semisimple abelian) monoidal category. A module category over C is a (semisimple abelian) category M together with a functor C×M → M and an associativity constraint (= natural isomorphism