Computational Synthetic Cohomology Theory in Homotopy Type Theory

This paper discusses the development of synthetic cohomology in Homotopy Type Theory (HoTT), as well as its computer formalisation. The objectives of this paper are (1) to generalise previous work on integral cohomology in HoTT by the current authors and Brunerie (2022) to cohomology with arbitrary coefficients and (2) to provide the mathematical details of, as well as extend, results underpinning the computer formalisation of cohomology rings by the current authors and Lamiaux (2023). With respect to objective (1), we provide new direct definitions of the cohomology group operations and of the cup product, which, just as in (Brunerie et al., 2022), enable significant simplifications of many earlier proofs in synthetic cohomology theory. In particular, the new definition of the cup product allows us to give the first complete formalisation of the axioms needed to turn the cohomology groups into a graded commutative ring. We also establish that this cohomology theory satisfies the HoTT formulation of the Eilenberg-Steenrod axioms for cohomology and study the classical Mayer-Vietoris and Gysin sequences. With respect to objective (2), we characterise the cohomology groups and rings of various spaces, including the spheres, torus, Klein bottle, real/complex projective planes, and infinite real projective space. All results have been formalised in Cubical Agda and we obtain multiple new numbers, similar to the famous `Brunerie number', which can be used as benchmarks for computational implementations of HoTT. Some of these numbers are infeasible to compute in Cubical Agda and hence provide new computational challenges and open problems which are much easier to define than the original Brunerie number.