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Explore the mathematical framework of virtual double categories through this Berkeley seminar that characterizes pro-representable virtual double categories. Discover how virtual double categories serve as an effective foundation for formal category theory, including their applications to adjoints, liftings, pointwise Kan extensions, and Yoneda structures. Learn about the structural properties that make certain virtual double categories exponentiable, with concrete examples including pseudo double categories and cospan virtual double categories. Examine a counterexample demonstrating that not all virtual double categories possess this exponentiable property. Gain insights into the enrichment structure underlying virtual double categories and explore potential alternative internal-hom constructions through the lens of loose bimodule theory developed by Libkind, Myers, Carlson, and Brown.
