Skeleta in non-Archimedean and tropical geometry

I describe an algebro-geometric theory of skeleta, which provides a unified setting for the study of tropical varieties, skeleta of non-Archimedean analytic spaces, and affine manifolds with singularities. Skeleta are spaces equipped with a structure sheaf of topological semirings, and are locally modelled on the spectra of the same. The primary result of this paper is that the topological space underlying a non-Archimedean analytic space may locally be recovered from the sheaf of `pointwise valuations' of its analytic functions. In a sequel, I will apply the theory of skeleta to address a question of constructing affine manifolds from maximally degenerating Calabi-Yau manifolds, as predicted by the SYZ conjecture.