Theories of Everything with Curt Jaimungal Emily Riehl Makes Infinity Categories Elementary
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Apr 6, 2026 Emily Riehl, a Johns Hopkins professor and leading category theorist, outlines making infinity category theory accessible to undergrads. She discusses rethinking foundations with homotopy type theory and simplicial type theory. The conversation covers core categorical ideas, why higher morphisms matter, and efforts to formalize these concepts in proof assistants.
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Galois Story Shows Abstraction Takes Time
- Emily Riehl tells the Galois story to illustrate how new abstractions take time but later become standard undergraduate material.
- Galois's 1832 ideas were misunderstood for decades until abstraction made Galois theory teachable in algebra courses.
Yoneda Turns Isomorphisms Into Map Correspondences
- The Yoneda lemma (innate dilemma) reduces constructing isomorphisms to giving natural bijections of maps out of objects.
- Riehl proves U⊗(V⊕W) ≅ (U⊗V)⊕(U⊗W) by showing natural bijections of linear maps into any X and applying Yoneda.
Adjunctions Encode Currying And Dualities
- Adjoint functors are captured by natural HOM-set bijections: Hom_D(F(C),D) ≅ Hom_C(C,U(D)).
- Tensoring with a fixed U is left adjoint to taking linear maps out of U, explaining currying in linear algebra.