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Explore the mathematical structure of cospans of finite sets in this colloquium lecture that examines their role as morphisms in a bicategory rather than a traditional category. Delve into the fundamental property that composition of cospans achieves associativity only up to natural isomorphism, creating unique categorical challenges. Learn about Steve Lack's elegant abstract characterization of the category formed by finite sets and isomorphism classes of cospans, and discover how Fong and Spivak leveraged this framework to illuminate the concept of hypergraph categories. Investigate the deeper question of characterizing the complete bicategory of finite sets, cospans, and maps between cospans, including discussion of an unproven conjecture that addresses this more complex structural problem.
