Given a presentably symmetric monoidal \(\infty\)-category \(\mathcal C\) and a small \(\infty\)-operad \(\mathcal O\), the \(\infty\)-category \(\mathrm{Alg}_{\mathcal O}(\mathcal C)\) is presentable and carries a symmetric monoidal structure. However, it is not necessarily presentably symmetric monoidal: The symmetric monoidal structure on \(\mathrm{Alg}_{\mathcal O}(\mathcal C)\) is not necessarily compatible with finite coproducts (though it is always compatible with sifted colimits). An easy counterexample is \(\mathcal C= \mathrm{Set}\) and \(\mathcal O=\mathbb E_1\).
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2