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The structure of a k-fold monoidal category as introduced by Balteanu, Fiedorowicz, SchwÃ¤nzl and Vogt in [2] can be seen as a weaker structure than a symmetric or even braided monoidal category. In this paper we show that it is still sufficient to permit a good definition of (n-fold) operads in a k-fold monoidal category which generalizes the definition ofâ€¦ (More)

- Jacob A. Siehler, Edward R. Green, +5 authors Julia Siehler
- 2002

We consider the possibility of semisimple tensor categories whose fusion rule includes exactly one noninvertible simple object. Conditions are given for the existence or nonexistence of coherent associative structures for such fusion rules, and an explicit construction of matrix solutions to the pentagon equations in the cases where we establish existence.â€¦ (More)

A near-group category is an additively semisimple category with a product such that all but one of the simple objects is invertible. We classify braided structures on near-group categories, and give explicit numerical formulas for their associativity and commutativity morphisms. 1. Results We consider near-group categories; that is, semisimple monoidalâ€¦ (More)

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. A sequence of categorical typesâ€¦ (More)

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. A sequence of categorical typesâ€¦ (More)

- Jacob A. Siehler
- The American Mathematical Monthly
- 2012

We present a construction for tiling the 24-cell with congruent copies of a single Hamiltonian cycle, using the algebra of quaternions. http://dx.doi.org/10.4169/amer.math.monthly.119.10.872 MSC: Primary 00A08, Secondary 05C45; 51M20 872 c Â© THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 119 This content downloaded from 195.187.72.155 on Wed, 19 Feb 2014â€¦ (More)

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