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# Classifying Spaces for Monoidal Categories Through Geometric Nerves

Published:2004-09-01
Printed: Sep 2004
• M. Bullejos
• A. M. Cegarra
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## Abstract

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
 MSC Classifications: 18D10 - Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 18G30 - Simplicial sets, simplicial objects (in a category) [See also 55U10] 55P15 - Classification of homotopy type 55P35 - Loop spaces 55U40 - Topological categories, foundations of homotopy theory

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