There are various notions of colimit that play important roles in 2-category theory. One of the most important is the notion of weighted colimit arising from Cat-enriched category theory, which subsumes many of the others. A notable exception is the notion of local colimit, which appears to possess a universal property of a fundamentally different nature. This is unfortunate, as there are fundamental constructions in two-dimensional category theory that could be more elegantly treated if local and weighted colimits were both instances of a more general notion.
In this talk, I will propose a new definition of colimit for double categories that (1) when specialised to 2-categories, recovers both weighted colimits and local colimits; (2) subsumes various colimit-like notions of recent interest in double category theory.
This talk is based on ongoing joint work with Bryce Clarke.