In this talk, I will focus on F-theory Standard Models and the computation of the exact massless vector-like spectrum. I will review that this leads to the computation of sheaf cohomologies of coherent sheaves.

On Toric spaces coherent sheaves can be represented by finitely presented graded modules. Collectively, these modules form an instance of a Freyd category, which is the universal tool to study cokernels. The computation of the sheaf cohomologies requires that the Freyd category is enriched by a monoidal structure. I will elaborate on recent work which shows that these monoidal structures can be derived from pro-monoidal structures.

This approach is completely algorithmic and has been implemented in the software-package Freyd categories of the CAP-project. Currently, this implementation is being used to study exact massless vector-like spectra in F-theory Standard Models. I will conclude this talk by elaborating on such applications.