Condensed Learning Seminar

Introduce the notion of stable compactly generated $\infty$-category and discuss its properties and some examples. In particular, prove that the $\infty$-category of small idempotent complete stable $\infty$-categories is equivalent to the $\infty$-category of compactly generated stable $\infty$-categories with strongly continuous functors. After that, recall the definition of dualizable object in a symmetric monoidal ($\infty$)category. Briefly review the Lurie tensor product of (stable) presentable $\infty$-categories and then discuss the notion of dualizable stable $\infty$-categories. Prove some of its equivalent characterizations: as retracts of compactly generated stable $\infty$-categories, in terms of the double left adjoint to the Yoneda embedding, and in terms of compact maps. If time permits, prove that the $\infty$-category of sheaves of spectra on a locally compact Hausdorff topological space is dualizable.