Cirio, LS and Faria Martins, J orcid.org/0000-0001-8113-3646 (2015) Infinitesimal 2-braidings and differential crossed modules. Advances in Mathematics, 277. pp. 426-491. ISSN 0001-8708
Abstract
We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relations, leading to six categorified relations. We prove that any infinitesimal 2-braiding gives rise to a flat and fake flat 2-connection in the configuration space of n particles in the complex plane, hence to a categorification of the Knizhnik–Zamolodchikov connection. We discuss infinitesimal 2-braidings in a certain monoidal 2-category naturally assigned to every differential crossed module, leading to the notion of a symmetric quasi-invariant tensor in a differential crossed module. Finally, we prove that symmetric quasi-invariant tensors exist in the differential crossed module associated to Wagemann's version of the String Lie-2-algebra. As a corollary, we obtain a more conceptual proof of the flatness of a previously constructed categorified Knizhnik–Zamolodchikov connection with values in the String Lie-2-algebra
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2015 Elsevier Inc. This is an author produced version of a paper published in Advances in Mathematics. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Higher gauge theory; Categorification; Knizhnik–Zamolodchikov equations; Zamolodchikov tetrahedron equation; Infinitesimal braiding; Braided monoidal 2-category; Crossed module; Lie-2-algebra; 4-term relations |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 06 Apr 2017 12:20 |
Last Modified: | 11 May 2019 22:50 |
Published Version: | https://doi.org/10.1016/j.aim.2015.03.006 |
Status: | Published |
Publisher: | Elsevier |
Identification Number: | 10.1016/j.aim.2015.03.006 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:112770 |