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Explore category theory concepts in this advanced mathematics lecture that examines the relationship between cartesian and symmetric monoidal categories through natural transformations. Delve into the taxonomy created by relaxing naturality requirements, analyzing the progression from gs-monoidal and cd-categories to restriction and Markov categories. Learn how order-enriched categories connect through commutative monads and understand the structure of arrows in free categories generated by algebraic signatures. Gain insights into the mathematical folklore surrounding cartesian categories and their representation using diagonals and projections within symmetric monoidal frameworks.
