Condensed Learning Seminar

Introduce the notion of nuclear object in a symmetric monoidal stable ($\infty$-)category and discuss its properties. In particular, prove that if the monoidal unit is compact, then an object is dualizable if and only if it is compact and nuclear. Introduce the notions of trace class maps and basic nuclear objects. Prove that the full subcategory of nuclear objects is generated under colimits by basic nuclear objects. Discuss the notion of inner nuclearity. Prove that for a discrete Huber pair $(A,A^+)$, the full subcategory of nuclear objects in $\mathcal{D}((A,A^+)_\blacksquare)$ is precisely the derived category $\mathcal{D}(A)$.