Lax higher category theory

The Gray tensor product is an essential operation in higher category theory which systematically replaces the cartesian product. It forms a nonsymmetric biclosed monoidal structure on (infinity,infinity)-categories which corepresents functors and (op)lax natural transformations. It turns out that essentially all operations of interest in higher categorical settings are only functorial in this (op)lax sense, and that the (more) correct objects to study are categories enriched in (infinity,infinity)-categories under the Gray tensor product. We will explain this philosophy and some elementary notions in this setting, including fibrations and (op)lax (co)limits. This is joint work with Hadrian Heine.