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