<p>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. One of the advantages of derivator theory is that they enable a
calculus of homotopy Kan extensions that relies almost entirely on ordinary category
theory (with a bit of 2-category theory). They can also be seen as an
approximation of (1; 1)-categories, a concept which
has been realized using a range of combinatorial and homotopy theoretic models.
Quasi-categories are presumably the best developed between such models, and
their theory has been established in the 2000s by Joyal and Lurie. In 2015 Riehl
and Verity introduced 1-cosmoi, which are particular (1; 2)-categories
where one can develop (1; 1)-category theory in a synthetic
way. They noticed that much of the theory of 1-cosmoi
can be developed inside a quotient, the homotopy 2-category. Inspired by this
philosophy, we introduce a set of axioms that mirror key properties of the 1-cosmological
approach to 1-category theory and demonstrate they hold in a variety of models,
including common models related to 1-category theory. We also
prove that these axioms are stable under a particular shift operation.</p>
