2010/05/10 by Michael A. Shulman, Shulman, Michael A. · 1 citation
Mathematics · #18D05 (Primary) 18C15 (Secondary) #Category Theory (math.CT) #FOS: Mathematics #math.CT #msc:18C15 #msc:18D05
paper · pdf · doi:10.48550/arxiv.1005.1520
17 pages; added more explanation; final version, to appear in Adv. Math
arxiv created 2011/01/11 · arxiv updated 2011/01/12
We describe a finitary 2-monad on a locally finitely presentable 2-category for which not every pseudoalgebra is equivalent to a strict one. This shows that having rank is not a sufficient condition on a 2-monad for every pseudoalgebra to be strictifiable. Our counterexample comes from higher category theory: the strict algebras are strict 3-categories, and the pseudoalgebras are a type of semi-strict 3-category lying in between Gray-categories and tricategories. Thus, the result follows from the fact that not every Gray-category is equivalent to a strict 3-category, connecting 2-categorical and higher-categorical coherence theory. In particular, any nontrivially braided monoidal category gives an example of a pseudoalgebra that is not equivalent to a strict one.