Stable $\infty$-Operads and the multiplicative Yoneda lemma

Abstract

We construct for every $\infty$-operad $\mathcal{O}^\otimes$ with certain finite limits new $\infty$-operads of spectrum objects and of commutative group objects in $\mathcal{O}$. We show that these are the universal stable resp. additive $\infty$-operads obtained from $\mathcal{O}^\otimes$. We deduce that for a stably (resp. additively) symmetric monoidal $\infty$-category $\mathcal{C}$ the Yoneda embedding factors through the $\infty$-category of exact, contravariant functors from $\mathcal{C}$ to the $\infty$-category of spectra (resp. connective spectra) and admits a certain multiplicative refinement. As an application we prove that the identity functor Sp $\to$ Sp is initial among exact, lax symmetric monoidal endofunctors of the symmetric monoidal $\infty$-category Sp of spectra with smash product.