• Communications of the Korean Mathematical Society
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• v.27 no.1
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• pp.23-36
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• 2012

Dual monoidal category $\mathcal{C}^*$ of a monoidal functor F : $\mathcal{C}\;{\rightarrow}\;\mathcal{V}$ has been constructed by S. Majid. In this paper, we extend the construction of dual structures for an Ann-functor F : $\mathcal{B}\;{\rightarrow}\;\mathcal{A}$. In particular, when F = $id_{\mathcal{A}}$, then the dual category $\mathcal{A}^*$ is indeed the center of $\mathcal{A}$ an this is a braided Ann-category.

• Journal of the Korean Mathematical Society
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• v.50 no.4
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• pp.865-877
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• 2013

The disjoint union of mapping class groups of surfaces forms a braided monoidal category $\mathcal{M}$, as the disjoint union of the braid groups $\mathcal{B}$ does. We give a concrete and geometric meaning of the braidings ${\beta}_{r,s}$ in $\mathcal{M}$. Moreover, we find a set of elements in the mapping class groups which correspond to the standard generators of the braid groups. Using this, we can define an obvious map ${\phi}\;:\;B_g{\rightarrow}{\Gamma}_{g,1}$. We show that this map ${\phi}$ is injective and nongeometric in the sense of Wajnryb. Since this map extends to a braided monoidal functor ${\Phi}\;:\;\mathcal{B}{\rightarrow}\mathcal{M}$, the integral homology homomorphism induced by ${\phi}$ is trivial in the stable range.

• Journal of the Korean Mathematical Society
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• v.37 no.3
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• pp.491-502
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• 2000

The disjoint union of mapping class groups g,1 gives us a braided monoidal category so that it gives rise to a double loop space structure. We show that there exists a natural twist in this category, so it gives us a ribbon category. We show that there exists a natural twist in this category, so it gives us a ribbon category. We explicitly express this structure by showing how the twist acts on the fundamental group of the surface Sg,l. We also make an explicit description of this structure in terms of the standard Dehn twists, as well as in terms of Wajnryb's Dehn twists. We show that the inverse of the twist g for the genus g equals the Dehn twist along the fixed boundary of the surface Sg,l.