symmetric monoidal (∞,1)-category of spectra
A (complete) archimedean valued field is a field equipped with an archimedean absolute value (and complete with respect to it).
A non-archimedean field is one that is not, hence one whose norm satisfies the ultrametric triangle inequality.
One of Ostrowski's theorems says that for $k$ a field complete with respect to an absolute value ${\vert - \vert}$ either the absolute value is archimedean, in which case $k$ is either the field of real numbers or of complex numbers, or the absolute value is non-archimedean.