Foundations 3: Universal Constructions
In this series we develop an understanding of the modern foundations of pure mathematics, starting from first principles. We start with intuitive ideas about set theory, and introduce notions from category theory, logic and type theory, until we are in a position to understand dependent type theory, and in particular, homotopy type theory, which promises to replace set theory as the foundation of modern mathematics. We also take an interest in computer science, and how to write computer programming languages to formalize mathematics.
In this video, we introduce initial objects, products, coproducts, and xponential objects. We describe the universal constructions for these structures in category theory, and describe their appearance in set theory. We also sketch how these ideas correspond with logic in the Curry-Howard-Lambek isomorphism (computational trinitarianism).
The following videos from my Category For Beginners Playlist provide extra information about the topics covered:
Category Theory For Beginners: Products • Category Theory For Beginners: Products
Category Theory For Beginners: Duality And Functors • Category Theory For Beginners: Dualit...
Category Theory For Beginners: Exponential Objects • Category Theory For Beginners: Expone...
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