Abstract: Usual scheme theory can be viewed as the syntactic theory of polynomial equations with coefficients in a ring, most importantly the ring of integers. But none of its most fundamental ingredients, such as faithfully flat descent, require subtraction. So we can set up a scheme theory over semirings (``rings but possibly without additive inverses’’, such as the non-negative integers or reals), thus bringing positivity in to the foundations of scheme theory. It is then reasonable to view non-negativity as integrality at the infinite place, the Boolean semiring as the residue field there, and the non-negative reals as the completion. In this talk, I'll discuss some recent developments in module theory over semirings. While the classical definitions of ``vector bundle'' are not all equivalent over semirings, the classical definitions of ``line bundle'' are all equivalent, which allows us to define Picard groups and Picard stacks. The narrow class group of a number field can be recovered as the reflexive class group of the semiring of its totally nonnegative integers, i.e. the arithmetic compactification of the spectrum of the ring of integers. This gives a scheme-theoretic definition of the narrow class group, as was done for the ordinary class group a long time ago. This is based mostly on arXiv:2405.18645, which is joint work with Jaiung Jun, and also on forthcoming paper with Johan de Jong and Ivan Zelich.