A Tour of EM(Cat#) in Category Theory

Join a Berkeley seminar where David Spivak explores the mathematical concept of EM(Cat#), delving into the relationship between small categories and graphs with compositional path structures. Learn how small categories function as algebras for the "paths" monad on graphs, and discover how this monad relates to loose maps in the double category Cat#. Explore Lack and Street's construction of the free completion of Cat# under Eilenberg-Moore objects, examining how monads and their morphisms operate within this framework. Understand the concept of "quintets" as morphisms and their role in inducing functors between Eilenberg-Moore categories. Investigate various examples of monads in Cat#, including those derived from cofunctors, functors, Grothendieck topologies, and multicategories. Examine practical applications through examples of morphisms in EM(Cat#), such as free-forgetful adjunctions for algebras, methods for converting convex spaces to monoids, and techniques for working with copresheaf category elements.