Oriented Category Theory

Abstract

As categorical dimension increases, classical categorical concepts often become too rigid and must be replaced by appropriately lax analogues. A basic manifestation of this principle is that higher-categorical versions of familiar geometric constructions -- such as cylinders, cones, and suspensions -- are not functorial in the classical sense. To remedy this situation, we extend the theory of higher categories to a framework of so-called oriented and antioriented categories, which provides a unified formalism for lax and oplax phenomena and reveals the geometric nature inherent in higher category theory. We characterize (anti)oriented categories as deformations of higher categories in which the various compositions only commute up to coherent (anti)oriented interchange law, and provide a geometric presentation of (anti)oriented categories as sheaves on a suitable thickening of the simplex category. We also construct an embedding of the category of $(\infty,\infty)$-categories into the category of (anti)oriented categories and characterize the image as those $(\infty,\infty)$-categories satisfying a strict interchange law.