$\infty$-operads via symmetric sequences

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.

arXiv comment50 pages. v2: Substantially rewritten with an improved version of the Day convolution construction, which produces a double $\infty$-category, v3: Various corrections