Homotopy Type Theory

For people interested in doing homotopy theory in homotopy type theory, Chapter 8 of the HoTT Book is a pretty good record of a lot of what was accomplished during the IAS year. However, there are a few things it’s missing, so I thought it would be a good idea to record some of those for the benefit of those wanting to push the theory further. (Hopefully at some point, papers will be written about all of these things…)

Today I want to talk about cohomology. Chapter 8 of the book focuses mostly on calculating homotopy groups, which are an important aspect of homotopy theory, but most working algebraic topologists spend more time on homology and cohomology, which (classically) are more easily computable. It’s an open question whether they will be similarly easier in homotopy type theory, but we should still be interested in defining and studying them.

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