  Introduced independently by Grothendieck and Heller in the 1980s, derivators
provide a formal way to study homotopy theories by working in some quotient
category such as the homotopy category of a model category. In 2015 Riehl and
Verity introduced $\infty$-cosmoi, which are particular $(\infty,2)$-categories
where one can develop $(\infty,1)$-category theory in a synthetic way. They
noticed that much of the theory of $\infty$-cosmoi can be developed inside a
quotient, the homotopy $2$-category. In the following, we begin a program that
aims to formalise the $\infty$-cosmological approach to $\infty$-category
theory in a derivator-like framework. In this paper we introduce some axioms
and demonstrate they hold in a variety of models, including common models
related to $\infty$-category theory. We also prove that these axioms are stable
under a particular shift operation.
