Regular directed complexes are an order-theoretic model of (shapes of) higher-categorical diagrams. There are two natural notions of morphism between regular directed complexes: they are called "maps" and "comaps" and are dual to each other. Roughly, a map can only collapse or rigidly identify cells, while a comap can only merge cells together.
A subclass of maps---called cartesian maps---serves as a foundation for a model of (∞,n)-categories with exceptionally nice properties. In this talk, I will present a conjecture on the existence of a certain factorisation of cartesian maps against comaps, which I strongly believe to be true. This conjecture implies a (semi-)strictification theorem for (∞,n)-categories in the same explicit, combinatorial style as Mac Lane's celebrated strictification theorem for bicategories.
This talk is based on joint work with Clémence Chanavat, both past and in progress.
