We study certain orientation-preserving involutions on three-dimensional small covers. We prove that the quotient space of an orientable three-dimensional small cover by such an involution belonging to the 2-torus is homeomorphic to a connected sum of copies of $S^2 \times S^1$.

A classical result of Dugger and Lurie says that presentable $\infty$-categories are precisely the $\infty$-categories that underlie combinatorial model categories.

There are (at least) two generalizations of this result in the literature:

- A symmetric monoidal version of Dugger-Lurie's theorem: Presentably symmetric monoidal $\infty$-categories are exactly those that underlie combinatorial symmetric monoidal model categories (Nikolaus-Sagave, 2017).

- Lifting Dugger-Lurie's theorem to the level of entire homotopy theories: The homotopy theory of combinatorial model categories is equivalent to that of presentable $\infty$-categories (Pavlov, 2025).

As a natural meeting point of these directions, Pavlov conjectured that the homotopy theory of combinatorial symmetric monoidal model categories is equivalent to that of presentably symmetric monoidal $\infty$-categories. This would, for example, ensure the existence and uniqueness of (combinatorial) model-categorical presentations of important objects like spectra (with smash product) and operads (with the Boardman-Vogt tensor product).

In this talk, we will give an affirmative answer to this conjecture (and its monoidal version), explain the main ideas behind the proof, and sketch some applications to the theory of $\infty$-operads.

The cohomology of a moment-angle complex is given by the Koszul homology of the Stanley-Reisner ring of the underlying simplicial complex. The Hochster decomposition gives an easy way to compute this Koszul homology from the combinatorics of the simplicial complex. In this talk, we will discuss Hochster's proof of the decomposition. Then, we will look at some 'colorful' generalizations of the decomposition which correspond to cohomology of certain partial quotients of moment-angle complexes.