Iterated Monoidal Categories

1998/08/18 by C. Balteanu, Z. Fiedorowicz, Balteanu, C. +5 · 2 citations
Mathematics · #18D10 #19D23 (Secondary) #55P35 (Primary) #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #math.AT #math.CT #msc:18D10 #msc:19D23 #msc:55P35

paper · pdf · doi:10.48550/arxiv.math/9808082

55 pages, 3 PostScript figures

arxiv created 1998/08/18 · arxiv updated 2009/11/30

Abstract

We develop a notion of iterated monoidal category and show that this notion corresponds in a precise way to the notion of iterated loop space. Specifically the group completion of the nerve of such a category is an iterated loop space and free iterated monoidal categories give rise to finite simplicial operads of the same homotopy type as the classical little cubes operads used to parametrize the higher H-space structure of iterated loop spaces. Iterated monoidal categories encompass, as a special case, the notion of braided tensor categories, as used in the theory of quantum groups.

Cited by

• Directional Non-Commutative Monoidal Structures with Interchange Law via Commutative Generators
• Directional Non-Commutative Monoidal Structures for Compositional Embeddings in Machine Learning

Related

• Combinatorial n-fold monoidal categories and n-fold operads
• The Combinatorics of Iterated Loop Spaces
• Group completions and the homotopical monadicity theorem
• Braided injections and double loop spaces
• Operads in iterated monoidal categories