On the equivalence of two approaches to multiplicative homotopy theories

TL;DR

This paper proves an equivalence between two frameworks for multiplicative homotopy theories, resolving conjectures and problems in the field.

Contribution

It establishes a main theorem showing the equivalence of presentably symmetric monoidal $$-categories and combinatorial symmetric monoidal model categories.

Findings

Solved Pavlov's conjecture.

Provided a solution to a special case of Hovey's 10th problem.

Proved variations including non-symmetric monoidal semi-model categories.

Abstract

We study the relation of two frameworks for multiplicative homotopy theories: Presentably symmetric monoidal $\infty$-categories and combinatorial symmetric monoidal model categories. Our main theorem establishes an equivalence of their homotopy theories. As consequences, we solve Pavlov's conjecture and obtain a solution to a special case of Hovey's 10th problem. We also prove several variations of the main theorem, such as an analog for non-symmetric monoidal semi-model categories.