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Explore the complex landscape of double categorical equivalences in this 58-minute colloquium talk that addresses a fundamental challenge in category theory. Delve into the flexible 2-dimensional framework of double categories, which accommodate two distinct types of morphisms between objects alongside higher-dimensional cells. Discover why, unlike most categorical structures, double categories lack a canonical notion of equivalence, requiring mathematicians to make deliberate choices when defining equivalences. Learn how homotopy theory provides a decisive solution to this problem through Campbell's gregarious equivalences, which emerge as the canonical double categorical equivalences. Examine the construction of multiple model structures on double categories, each encoding different 2-dimensional structures within their respective homotopy theories. Gain insights into ongoing research that bridges abstract category theory with practical applications in higher-dimensional mathematics, presented as part of collaborative work with Lyne Moser and Paula Verdugo.
