Higher Dimensional Algebra: I. Braided Monoidal 2-Categories

Abstract We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2-categories and their relevance to 4d TQFTs and 2-tangles. Then we give concise definitions of semistrict monoidal 2-categories and braided monoidal 2-categories and show how these may be unpacked to give long explicit definitions similar to, but not quite the same as, those given by previously Kapranov and Voevodsky. Finally, we describe how to construct a semistrict braided monoidal 2-category Z ( C ) as the “center” of a semistrict monoidal category C , in a manner analogous to the construction of a braided monoidal category as the center of a monoidal category. As a corollary this yields a strictification theorem for braided monoidal 2-categories.

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