Day convolution for $\infty$-categories

Given symmetric monoidal $\infty$-categories $\mathbb{C}$ and $\mathbb{D}$, subject to mild hypotheses on $\mathbb{D}$, we define an $\infty$-categorical analog of the Day convolution symmetric monoidal structure on the functor category $\mathrm{Fun}(\mathbb{C},\mathbb{D})$. An $\mathbb{E}_{\infty}$ monoid for the Day convolution product is a lax monoidal functor from $\mathbb{C}$ to $\mathbb{D}$.

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Accepted 23 March 2015

Published 12 January 2017