Infinitesimal 2-braidings and differential crossed modules |
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Authors: | Lucio Simone Cirio João Faria Martins |
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Institution: | 1. Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, Germany;2. Departamento de Matemática and Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia (Universidade Nova de Lisboa), Quinta da Torre, 2829-516 Caparica, Portugal |
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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. |
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Keywords: | primary 16T25 18D05 secondary 20F36 17B37 |
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