University of Cambridge > > Logic and Semantics Seminar (Computer Laboratory) > Higher Categorical Structures, Type-Theoretically

Higher Categorical Structures, Type-Theoretically

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If you have a question about this talk, please contact Dominic Mulligan.

Category theory internally in homotopy type theory is intricate as categorical laws can only be stated ‘up to homotopy’, requiring coherence (similar to how the associator in a bicategory requires the pentagon). To avoid this, one can consider definitions with truncated types such as the univalent categories by Ahrens-Kapulkin-Shulman, which however fail to capture some important examples. A more general structure are (n,1)-categories, ideally with n = infinity, and a possible definition is given by Capriotti’s complete semi-Segal types. I will show how simplified (complete) semi-Segal types are indeed equivalent to univalent categories. Very much related is the question what an appropriate notion of a type-valued diagram is (joint work with Sattler). Using the type universe as a semi-Segal type, I will present several constructions of homotopy coherent diagrams (some of them infinite) and show, as an application, how one can construct all the degeneracies that a priori are missing in a complete semi-Segal type. With a further construction, we get a (finite) definition of simplicial types (up to a finite level).

This talk is part of the Logic and Semantics Seminar (Computer Laboratory) series.

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