Tricategorical limits and colimits (part 1 of 2)

Adrian Miranda – 5 April 2023

Bicategorical limits and colimits have a well developed theory, and are related to homotopically well-behaved limits and colimits enriched over Cat as a monoidal model category. Given a diagram F: A --> B and a weight W: A --> Cat, new data F': A' --> B' and W': A' --> Cat can be described so that the original bilimit {W, F}, if it exists, is an `up to equivalence' version of an enriched limit of F' weighted by W'. The enriched weights needed to model bicategorical notions are the projectively cofibrant ones; those which can be built out of representables using coproducts, coinserters, coequifiers and splittings of idempotents.

There is also a monoidal model structure on Gray, and one might hope to similarly develop a theory of tricategorical limits and colimits, and relate it to homotopically well-behaved notions enriched over Gray. There are some extra subtleties involved in achieving this, largely due to the fact that not all 2-categories are cofibrant. As such, the categorifications of many familiar results about bicategorical limits and colimits require extra assumptions. These assumptions include pointwise cofibrancy of weights and hom-wise cofibrancy of shapes. In these talks I will describe some progress towards developing this theory.