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Presentably symmetric monoidal \(\infty\)-categories are represented by symmetric monoidal model categories. (English) Zbl 1380.55018

It is known that every combinatorial simplicial model category \(\mathcal{M}\) has an underlying \(\infty\)-category, \(\mathcal{M}_{\infty}\), which is presentable. Moreover, every presentable \(\infty\)-category is, actually, equivalent to the \(\infty\)-category associated with a combinatorial simplicial model category. Now, if \(\mathcal{M}\) is equipped with a symmetric monoidal product that is compatible with the model structure, it is also known that the underlying \(\infty\)-category \(\mathcal{M}_{\infty}\) is a presentable symmetric monoidal \(\infty\)-category. Then, it is natural to pose the question whether every presentable symmetric monoidal \(\infty\)-category arises from a combinatorial symmetric monoidal model category and, in this paper and as its main object, the authors give an affirmative answer to this question.

MSC:

55U35 Abstract and axiomatic homotopy theory in algebraic topology
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
18G55 Nonabelian homotopical algebra (MSC2010)
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References:

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