On model-comparison results for $\infty$-categories

The introduction of each new definition of a weak higher
category poses an accompanying comparison problem, the challenge
being to convert between higher categories defined via the
new model and higher categories defined via pre-existing models.
A secondary question is whether proofs involving a particular model
of higher categories have any implications for other related models.
This talk will give a preliminary report on work in progress with
Dominic Verity that develops a framework in which categorical constructions
defined with one model of $(\infty, n)$-categories can be translated
into other models of (8,k)-categories. For the well-behaved
models of $(\infty,1)$-categories -- including quasi-categories, complete
Segal spaces, and Segal categories -- this work specializes to establish
the model independence of their basic category theory.