Strictification of indexed bicategories

Alexander Campbell – 1 November 2017

A quirk of bicategory theory is that the relation of isomorphism between objects in a bicategory is poorly behaved: it is neither reflexive nor transitive. For this reason, if one wants to work "as strictly as possible" when studying bicategories it is sometimes convenient to work with the more general pseudo double categories, which have an extra class of "vertical" morphisms which compose strictly associatively. Thus Garner and Gurski introduced the notion of "locally cubical bicategory" (that is, a bicategory enriched in the cartesian monoidal 2-category PsDblCat of pseudo double categories) to capture those three dimensional categories in which composition ought to be associative up to isomorphism (rather than merely up to equivalence, as in a tricategory), and showed that a locally cubical bicategory whose homs are "fibrant" gives rise to a tricategory. Similarly, Shulman showed that many symmetric monoidal bicategories of interest arise from symmetric monoidal fibrant pseudo double categories (in which the tensor product is associative up to vertical isomorphism).

In this talk we will show that the tricategory Hom(A,Bicat) of indexed bicategories over a Gray-category A (that is, the tricategory of trihomomorphisms A --> Bicat) arises from a locally cubical bicategory, whose vertical 2-cells are icon-component trimodifications ("tricons"), and that the pointwise strictification trihomomorphism from Hom(A,Bicat) to the Gray-category Hom(A,Gray) arises from a PsDblCat-enriched pseudofunctor, which is the left biadjoint of a PsDblCat-enriched biadjunction.