The periodic table of $n$-categories for low dimensions II: degenerate tricategories

Cheng, Eugenia
Gurski, Nick
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We continue the project begun in ``The periodic table of $n$-categories for low dimensions I'' by examining degenerate tricategories and comparing them with the structures predicted by the Periodic table. For triply degenerate tricategories we exhibit a triequivalence with the partially discrete tricategory of commutative monoids. For the doubly degenerate case we explain how to construct a braided monoidal category from a given doubly degenerate category, but show that this does not induce a straightforward comparison between \bfseries{BrMonCat} and \bfseries{Tricat}. We show how to alter the natural structure of \bfseries{Tricat} in two different ways to provide a comparison, but show that only the more brutal alteration yields an equivalence. Finally we study degenerate tricategories in order to give the first fully algebraic definition of monoidal bicategories and the full tricategory structure \bfseries{MonBicat}.
Comment: 51 pages
Mathematics - Category Theory, 18D05, 18D10