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Explore advanced concepts in categorical algebra through this Berkeley seminar lecture focusing on Segal conditions and algebraic patterns as presented by David Jaz Myers. Dive into alternative methods of presenting algebraic structures categorically, moving beyond traditional approaches like monads, Lawvere theories, and limit sketches. Learn how pasting diagrams serve as fundamental building blocks for composition, constructed from elementary features such as boxes and wires, and understand their ability to incorporate other diagrams through "swallowing" operations. Examine the organization of diagram inclusions and swallowings into factorization systems, and discover how presheaves meeting Segal conditions function as algebras for composition. Study practical applications through detailed examples of categories and double categories, while investigating the derivation of virtual double categories and contemplating the nature of virtual triple categories within this theoretical framework.
