Mathematics Ph.D. candidate Kevin Pomorski defends his dissertation on "On Braided Monoidal 2-Categories" on March 3 at 5 p.m. via Teams.

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The dissertation committee is Alexei Davydov , Marcel Bischoff , Sergio Lopez-Permouth , and Nancy Sandler .

Abstract : In Topological Field Theories (TFTs), there is a well documented correlation between 3-dimensional TFTs and braided monoidal categories. While braided monoidal 2-categories have been expected to have applications to 4-dimensional TFTs, there are very few known examples of braided monoidal 2-categories. The major goal of this dissertation is to present a categorification of a result of Pareigis. Namely, that modules over a commutative algebra in a braided monoidal category form a braided monoidal category. The categorified statement is that pseudomodules over a braided pseudomonoid in a braided monoidal 2-category form a braided monoidal 2-category. This result would be an example of constructing braided monoidal 2-categories from existing braided monoidal 2-categories. We approach this by constructing a new language which simplifies some of the complexities coming from relative tensor products of pseudomodules. The relationships between monoidal categories and multicategories has been well documented e.g. by Leinster. We define the notion of a multi-2-category, as well as the notion of a braided multi-2-category. We then construct braided pseudomonoids in braided multi-2-categories and examine pseudomodules over them. The main theorem of the dissertation is that local pseudomodules over a braided pseudomonoid in a braided multi-2-category forms a braided multi-2-category. A second result is a revision to the Joyal-Street definition of a balanced 2-category. We give a coherence for balanced 2-categories which was omitted in the original definition.