Categorified convergence: dropping the 'ultra' from 'virtual ultracategories'

Virtual ultracategories are a recent categorification of topological spaces seen as sets of points together with a convergence relation between points and ultrafilters. However, a topology can also be described in terms of convergence of filters: in this paper, we thus introduce an analogue of virtual ultracategories based on filters rather than ultrafilters. At the topological level, convergence of filters is completely determined by its restriction to ultrafilters. Our main result shows that, under the assumption of tautness, the same phenomenon holds at the categorified level, thereby offering a non-ultra-biased perspective on virtual ultracategories.