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Schemas and semantics for Higher Inductive Types

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BPRW01 - Computer-aided mathematical proof

Higher inductive types are now an established tool of homotopy type theory, but many important questions about them are still badly-understood, including:

  • can we set out a scheme defining “general HITs”, analogously to how CIC defines “general inductive types”?
  • can we find a small specific collection of HITs from which one can construct “all HITs”, analogously to how the type-formers of MLTT suffice for inductive types?
  • how can we model HITs (specific or general) in interesting homotopical settings?
I will survey these questions and present what I know of progress on them (in particular, the cell monads semantics of Lumsdaine/Shulman https://arxiv.org/abs/1705.07088); I will also open the floor for interested audience members to briefly present other current work on these topics.

This talk is part of the Isaac Newton Institute Seminar Series series.

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