We discuss the K-theory of categories of modules over a number field, but additionally equipped with a locally compact topology. Such a category has a natural exact structure, so it has a derived category, K-theory etc., all basic homological algebra is available. Studying such K-theory spectra is insipred by Dustin Clausen's work on lifting the Artin map of class field theory to a map of spectra, and is connected to André Weil's classical ideas to implement the infinite primes on an equal footing as the finite ones by using the concept of local compactness.