UC Berkeley

The Barr-Beck-Lurie comonadicity theorem characterizes when an adjunction $\begin{tikzcd} \mathcal{C} \arrow[r,yshift=1ex,"L"] & \mathcal{D} \arrow[l,yshift=-1ex,"R"] \end{tikzcd}$ can be used to present $\mathcal{C}$ as the category of comodules in $\mathcal{D}$ for the coalgebra object \mbox{$L\circ R \in End(\mathcal{D})$}; loosely speaking, the comodule structure supplies the instructions for how to assemble object in $\mathcal{C}$ from an object in $\mathcal{D}$. This thesis explores the proofs and applications in a number of contexts of this method for endowing categories of interest $\mathcal{C}$ with this tautological algebraically-flavored description, in the hopes of being a kind of handbook and toolkit for someone looking to demonstrate a comandicity result.

The toolkit grew out of an investigation of a fundamental comonadicity result: Zariski descent for quasicoherent sheaves. Our main effort, joint with Peng Zhou, is in (1) presenting comonadicity statements in the case where $mathcals{C}\xrightarrow{L}\mathcal{D}$ is a product of reflective localizations, and (2) applying it to deduce a descent statement for those closed covers of Lagrangian skeleta which are locally modeled on ones that arise in the coherent-constructible correspondence of Fang-Liu-Treumann-Zaslow.