Formalization of ∞-Category Theory

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Conference Proceeding

Streaming Media

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The field of category theory has served as an excellent tool to unify and translate between constructions and theorems across different subfields in mathematics, but also computer science and physics.

Informally, a category is a structure that models composition (e.g. of functions, transformations, processes, or algorithms). In many settings (e.g. algebraic geometry, quantum field theory, and homotopy theory) composition is not given by a well-defined function but rather up to higher-dimensional topological data. This gives rise to the notion of ∞-category, an infinite-dimensional structure.

I will introduce the main elements of a formal language to reason about ∞-categories in a "synthetic" way. The language is an extension of homotopy type theory à la Voevodsky and Awodey–Warren. This constitutes an alternative, arguably more slick foundational system than set theory. Moreover, the new proof assistant Rzk implements this formal language. If time permits, I'll outline the ideas and computer formalization of the Yoneda Lemma, the fundamental theorem of category theory. This result turns out to be easier to prove for ∞-categories in the synthetic setting than for 1-categories in classical set theory. The material is based on joint work with Buchholtz, Gratzer, Kudasov, and Riehl.


This presentation was part of the Orange County Inland Empire (OCIE) Seminar series in History and Philosophy of Mathematics in spring 2024.