Enriched homotopy-coherent structures

Abstract

We introduce a general notion of enrichment for homotopy-coherent algebraic structures described by Segal conditions, using the framework of "algebraic patterns" developed in our previous work. This recovers several known examples of enriched structures, including enriched $\infty$-categories, enriched $\infty$-operads, and enriched $\infty$-properads. As new examples we discuss enriched modular $\infty$-operads, enriched $n$-fold $\infty$-categories, and a non-iterative definition of enriched $(\infty,n)$-categories.