Oriented Category Theory

We develop foundations for oriented category theory, an extension of $(\infty,\infty)$-category theory obtained by systematic usage of the Gray tensor product, in order to study lax phenomena in higher category theory. As categorical dimension increases, classical category-theoretic concepts generally prove too rigid or fully break down and must be replaced by oriented versions, which allow more flexible notions of naturality and coherence. Oriented category theory provides a framework to address these issues. The main objects of study are oriented, and their conjugate antioriented, categories, which are deformations of $(\infty,\infty)$-categories where the various compositions only commute up to a coherent (anti)oriented interchange law. We give a geometric description of (anti)oriented categories as sheaves on a deformation of the simplex category $Δ$ in which the linear graphs are weighted by (anti)oriented cubes. To demonstrate the utility of our theory, we show that the categorical analogues of fundamental constructions in homotopy theory, such as cylinder and path objects, join and slice, and suspension and loops, are not functors of $(\infty, \infty)$-categories, but only of (anti)oriented categories, generalizing work of Ara, Guetta, and Maltsiniotis in the strict setting. As a main result we construct an embedding of the theory of $(\infty,\infty)$-categories into the theory of (anti)oriented categories and characterize the image, which we call (anti)oriented spaces. We provide an algebraic description of (anti)oriented spaces as (anti)oriented categories satisfying a strict (anti)oriented interchange law and a geometric description as sheaves on suitable categories of (anti)oriented polytopes, generalizing Grothendieck's philosophy of test categories to higher categorical dimension and refining Campion's work on lax cubes and suitable sites.