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Explore the mathematical landscape of arboreal categories and their coreflective properties in this Berkeley Seminar lecture. Discover how arboreal categories represent objects with intrinsic tree-like process structures that generate bisimilarity relations, which can be transported through adjunctions into extensional categories containing relational structures. Learn how arboreal adjunctions recover logical equivalence for various fragments of infinitary first-order logic and provide foundations for game comonads, enabling extensions of resource-sensitive model-theoretic results like Rossman's equirank preservation theorem. Examine the systematic correspondence between logics and arboreal adjunctions by focusing specifically on coreflective cases, where idempotent game comonads correspond to variants of basic modal logic. Understand the concept of "seeds" - full subcategories of structures that generate arboreal coreflective subcategories through colimits - and discover methods for identifying these seeds toward potential classification theorems. Investigate how density comonads can characterize coreflective subcategories without explicitly constructing the coreflector, and see concrete examples of arboreal categories obtained through these theoretical results.
