From Analysis to Stable Homotopy Theory via lax-idempotent Monads

Recent progress in the theory of dualizable categories due to Efimov, Nikolaus, Clausen among others have made it possible to use analytic/operator-theoretic techniques in the setting of stable $\mathrm{\infty }$-categories. This theory provides a promising framework for understanding questions in geometric topology, such as assembly conjectures like the Borel, Novikov and Farrell-Jones conjectures. It was observed that dualizable categories share many formal similarities with compact Hausdorff spaces - In particular, variants of the Tychonoff Theorem and Urysohn Lemma exist for dualizable categories.

We explore these similarities using the notion of a lax-idempotent monad, also called KZ-monad, on a $2$-category. Every lax-idempotent monad has a corresponding category of continuous algebras. Examples of these are abundant: If one starts with the category of small stable $\mathrm{\infty }$-categories and the monad given by $\mathrm{Ind}$, one obtains the category of dualizable categories. If one starts with the category of distributive lattices, one obtains stably compact spaces. The category of symmetric monoidal categories with the envelope monad $\mathrm{Env}$ leads to operads. Due to the formal nature of this construction, several highly non-trivial results, such as Aoki’s Sheaves-Smashing spectrum adjunction, and Verdier Duality for locally compact Hausdorff spaces, can be understood using 2-categorical techniques, and allow straightforward generalizations.