We prove that the homotopy theory of monoidal relative categories is equivalent to that of monoidal ∞-categories, and likewise in the symmetric monoidal setting. As an application, we give a concise and complete proof of the fact that every presentably monoidal or presentably symmetric monoidal ∞-category is presented by a monoidal or symmetric monoidal model category, which, in the monoidal case, was sketched by Lurie, and in the symmetric monoidal case, was proved by Nikolaus–Sagave.
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Kensuke Arakawa
Journal of Pure and Applied Algebra
Kyoto University
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Kensuke Arakawa (Tue,) studied this question.
www.synapsesocial.com/papers/69a75b45c6e9836116a224ed — DOI: https://doi.org/10.1016/j.jpaa.2026.108183