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.

arXiv comment27 pages

References in corpus (2)

• Higher enveloping algebras
• Day convolution for infinity-categories

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