Combinatorial n-fold monoidal categories and n-fold operads

Details Combinatorial n-fold monoidal categories and n-fold operads Combinatorial n-fold monoidal categories and n-fold operads

2004-11-25

arxiv.org/abs/math/0411561v4

Mathematics

Category Theory

Corrections made: see remark 1 in comparison with old versions

Scientific paper

Operads were originally defined as V-operads, that is, enriched in a symmetric or braided monoidal category V. The symmetry or braiding in V is required in order to describe the associativity axiom the operads must obey, as well as the associativity that must be a property of the action of an operad on any of its algebras. After a review of the role of operads in loop space theory and higher categories we go over definitions of iterated monoidal categories and introduce a large family of simple examples. Then we generalize the definition of operad by defining n-fold operads and their algebras in an iterated monoidal category. We discuss examples of these that live in the previously described categories. Finally we describe the iterated monoidal category of m-fold V-operads.

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