Enriched homotopy-coherent structures

TL;DR

This paper develops a general framework for enriching homotopy-coherent algebraic structures using algebraic patterns, unifying known examples and introducing new enriched structures like modular $ ext{infty}$-operads and $( ext{infty},n)$-categories.

Contribution

It introduces a universal notion of enrichment for homotopy-coherent structures via algebraic patterns, encompassing and extending existing examples.

Findings

Unified framework for enriched $ ext{infty}$-categories, operads, and properads.

New examples include enriched modular $ ext{infty}$-operads and $ ext{infty}$-categories.

Provides a non-iterative approach to defining enriched $( ext{infty},n)$-categories.

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.