Coproduct idempotent algebras over internal operads in enriched $\infty$-categories

Abstract

In arXiv:1712.00555, H. Heine shows that given a symmetric monoidal $\infty$-category $\mathcal{V}$ and a weakly $\mathcal{V}$-enriched monad $T$ over an $\infty$-category $\mathcal{C}$, then there is an induced action of $\mathcal{V}$ on $LMod_T(\mathcal{C})$. Moreover, properties like tensoring or enrichment can be transferred from the action on $\mathcal{C}$ to that on $LMod_T(\mathcal{C})$. We see that the action of an internal operad $O \in Alg(sSeq(\mathcal{C}))$ can be interpreted as the action of a monad $T_O$, such that $Alg_O(\mathcal{C})\cong LMod_{T_O}(\mathcal{C})$. We can then prove that, under a presentability assumption, if the category $\mathcal{C}$ admits cotensors with respect to the action of $\mathcal{V}$, then so does $Alg_O(\mathcal{C})\cong LMod_{T_O}(\mathcal{C})$. This is used to show that the coproduct-idempotent algebras are fixed by the induced tensoring action. We apply this to the stable motivic homotopy category and prove that the tensor of any motivic sphere with rational motivic cohomology is equivalent to the latter.