How strict is strictification?, part 2

Alexander Campbell – 11 October 2017

Back in March, we saw that the strictification adjunction (between the category of 2-categories and 2-functors and the category of bicategories and pseudofunctors) can be enriched to an adjunction between categories (strictly!) enriched over the symmetric closed multicategory of bicategories (introduced in Verity's PhD thesis). Six months later, we revisit this topic to show that the strictification of indexed bicategories (i.e. bicategory-valued trihomomorphisms) forms a biadjunction between bicategories enriched over the Gray--Verity-style symmetric closed 2-multicategory of pseudo double categories, and hence a triadjunction between tricategories (a la Garner--Gurski on locally cubical bicategories).

This week, we will review the original strictification adjunction and show that its enrichment follows formally from its structure as an adjunction between closed multicategories. Furthermore, we will recover Gurski's "[pseudo]monoidal structure of strictification" from the observations that the strictification adjunction extends moreover to an adjunction between 2-multicategories (whose 2-cells are multivariable icons), and that the 2-multicategory structure on Bicat is birepresented (i.e. represented up to equivalence) by the cartesian product of bicategories. At various points, our categorical instincts and our desire for best-possible results will lead us to thoughts of enrichment over pseudo double categories, to which we will succumb next time.