$\infty$-operads via symmetric sequences

TL;DR

This paper develops a generalized Day convolution for double ∞-categories, enabling a new ∞-categorical description of operads as associative algebras in symmetric sequences, broadening the operadic framework.

Contribution

It introduces a generalized convolution construction for double ∞-categories and characterizes ∞-operads as associative algebras within this new setting.

Findings

Generalized Day convolution for double ∞-categories

∞-operads described as associative algebras in symmetric collections

Extension of operad theory to enriched ∞-categories

Abstract

We construct a generalization of the Day convolution tensor product of presheaves that works for certain double $\infty$-categories. Using this construction, we obtain an $\infty$-categorical version of the well-known description of (one-object) operads as associative algebras in symmetric sequences; more generally, we show that (enriched) $\infty$-operads with varying spaces of objects can be described as associative algebras in a double $\infty$-category of symmetric collections.