The algebra of presentable infinity-categories

The category \(\mathrm{Pr}^\mathrm{L}\) of presentable \(\infty\)-categories is a place where one can do surprising amounts of "algebra", and the study of presentable categories with extra structure often helps in the study of objects within those presentable categories. This higher algebra has become more and more studied in recent times under the impetus of Efimov and his continuous K-theory, as well as Scholze and Stefanich's Gestalten, which uses this higher categorical algebra as the basic foundation of a new type of geometry.
Personally, categorification is also one of my favourite tricks in math.

• Most recently, I used the categorification of basic ideas in algebra to give a description of free rigid commutative algebras. This was one of the key technical inputs in my paper on Noshift (more about that in the "Localizing invariants" section).
• With Shachar Carmeli, Bastiaan Cnossen and Lior Yanovski, we used categorified traces to say things about Becker--Gottlieb transfers and THH of colimits (I also used this in my paper on endomorphisms of THH) in Characters and transfer maps via categorified traces.
• A symmetric monoidal fracture square: With Niko Naumann and Luca Pol, we studied a variant of the "completion" of a symmetric monoidal category, which is useful when one wants to prove results using recollements but one needs a strong symmetric monoidal functor, as opposed to a lax one.
• In the companion papers Dualizable presentable \(\infty\)-categories and Locally rigid \(\infty\)-categories, I laid some foundations of the theory of dualizable presentable \(\infty\)-categories.